From 8b97f75bf1b4c5bdcebc0483a5bcb69fe98d7ba0 Mon Sep 17 00:00:00 2001
From: Jerry James <loganjerry@gmail.com>
Date: Sun, 14 Jan 2024 11:54:36 -0700
Subject: [PATCH] Remove same-alphabet requirement from ccatlen

---
 discouraged |    5 +-
 set.mm      | 2902 +++++++++++++++++++++++++--------------------------
 2 files changed, 1435 insertions(+), 1472 deletions(-)

diff --git a/discouraged b/discouraged
index f2c7f41e59..fb90b12f0f 100755
--- a/discouraged
+++ b/discouraged
@@ -398,7 +398,6 @@
 "4syl" is used by "qtopcmap".
 "4syl" is used by "qtopf1".
 "4syl" is used by "restmetu".
-"4syl" is used by "revccat".
 "4syl" is used by "revrev".
 "4syl" is used by "rpvmasum2".
 "4syl" is used by "rpvmasumlem".
@@ -406,7 +405,6 @@
 "4syl" is used by "sdclem2".
 "4syl" is used by "sdomsdomcard".
 "4syl" is used by "serf0".
-"4syl" is used by "signstfvp".
 "4syl" is used by "signstres".
 "4syl" is used by "smoiso".
 "4syl" is used by "srng0".
@@ -415,7 +413,6 @@
 "4syl" is used by "stoweidlem11".
 "4syl" is used by "stoweidlem14".
 "4syl" is used by "subfacp1lem5".
-"4syl" is used by "swrdccat2".
 "4syl" is used by "symgtrinv".
 "4syl" is used by "tmsxms".
 "4syl" is used by "tocyc01".
@@ -13191,7 +13188,7 @@ New usage of "4atex2-0aOLDN" is discouraged (1 uses).
 New usage of "4atex2-0bOLDN" is discouraged (0 uses).
 New usage of "4atex2-0cOLDN" is discouraged (0 uses).
 New usage of "4ipval2" is discouraged (2 uses).
-New usage of "4syl" is discouraged (193 uses).
+New usage of "4syl" is discouraged (190 uses).
 New usage of "5oai" is discouraged (0 uses).
 New usage of "5oalem1" is discouraged (1 uses).
 New usage of "5oalem2" is discouraged (2 uses).
diff --git a/set.mm b/set.mm
index 810ed64a54..bcfddad7b5 100644
--- a/set.mm
+++ b/set.mm
@@ -149665,15 +149665,15 @@ computer programs (as last() or lastChar()), the terminology used for
       RVAWAVBWBVAVDABQTVBVEACQTUKPVHVDVEULUMUNUOUPAVFVOUQURUS $.
 
     $( The length of a concatenated word.  (Contributed by Stefan O'Rear,
-       15-Aug-2015.) $)
-    ccatlen $p |- ( ( S e. Word B /\ T e. Word B ) ->
+       15-Aug-2015.)  (Revised by JJ, 1-Jan-2024.) $)
+    ccatlen $p |- ( ( S e. Word A /\ T e. Word B ) ->
         ( # ` ( S ++ T ) ) = ( ( # ` S ) + ( # ` T ) ) ) $=
-      ( vx cword wcel wa cconcat co chash cfv cc0 caddc cfzo cmin wceq fvex cn0
-      cv lencl cif cmpt ccatfval fveq2d wfn ifex eqid fnmpti hashfn mp1i syl2an
-      nn0addcl hashfzo0 syl 3eqtrd ) BAEZFZCUPFZGZBCHIZJKDLBJKZCJKZMIZNIZDSZLVA
-      NIFZVEBKZVEVAOIZCKZUAZUBZJKZVDJKZVCUSUTVKJDBCUPUPUCUDVKVDUEVLVMPUSDVDVJVK
-      VFVGVIVEBQVHCQUFVKUGUHVDVKUIUJUSVCRFZVMVCPUQVARFVBRFVNURABTACTVAVBULUKVCU
-      MUNUO $.
+      ( vx cword wcel wa cconcat co chash cfv cc0 caddc cfzo cv wceq fvex lencl
+      cn0 cmin cif cmpt ccatfval fveq2d wfn ifex eqid fnmpti hashfn mp1i syl2an
+      nn0addcl hashfzo0 syl 3eqtrd ) CAFZGZDBFZGZHZCDIJZKLEMCKLZDKLZNJZOJZEPZMV
+      COJGZVGCLZVGVCUAJZDLZUBZUCZKLZVFKLZVEVAVBVMKECDUQUSUDUEVMVFUFVNVOQVAEVFVL
+      VMVHVIVKVGCRVJDRUGVMUHUIVFVMUJUKVAVETGZVOVEQURVCTGVDTGVPUTACSBDSVCVDUMULV
+      EUNUOUP $.
 
     $( The concatenation of two words is empty iff the two words are empty.
        (Contributed by AV, 4-Mar-2022.) $)
@@ -149682,9 +149682,9 @@ computer programs (as last() or lastChar()), the terminology used for
       ( wcel wa co chash cfv cc0 wceq c0 wb hasheq0 syl cle wbr cn0 lencl nn0re
       cr cword cconcat ccatcl caddc ccatlen eqeq1d nn0ge0 add20 syl2an bi2anan9
       jca 3bitrd bitr3d ) BAUAZDZCUNDZEZBCUBFZGHZIJZURKJZBKJZCKJZEZUQURUNDUTVAL
-      ABCUCURUNMNUQUTBGHZCGHZUDFZIJZVEIJZVFIJZEZVDUQUSVGIABCUEUFUOVETDZIVEOPZEZ
-      VFTDZIVFOPZEZVHVKLUPUOVEQDZVNABRVRVLVMVESVEUGUKNUPVFQDZVQACRVSVOVPVFSVFUG
-      UKNVEVFUHUIUOVIVBUPVJVCBUNMCUNMUJULUM $.
+      ABCUCURUNMNUQUTBGHZCGHZUDFZIJZVEIJZVFIJZEZVDUQUSVGIAABCUEUFUOVETDZIVEOPZE
+      ZVFTDZIVFOPZEZVHVKLUPUOVEQDZVNABRVRVLVMVESVEUGUKNUPVFQDZVQACRVSVOVPVFSVFU
+      GUKNVEVFUHUIUOVIVBUPVJVCBUNMCUNMUJULUM $.
 
     $d x I $.
     $( Value of a symbol in the left half of a concatenated word.  (Contributed
@@ -149738,7 +149738,7 @@ computer programs (as last() or lastChar()), the terminology used for
                             -> N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) ) $=
     ( cword wcel wa cc0 chash cfv cfz co caddc cconcat wi lencl elfz0add syl2an
     cn0 ccatlen oveq2d eleq2d sylibrd ) ADEZFZBUDFZGZCHAIJZKLFZCHUHBIJZMLZKLZFZ
-    CHABNLIJZKLZFUEUHSFUJSFUIUMOUFDAPDBPUHUJCQRUGUOULCUGUNUKHKDABTUAUBUC $.
+    CHABNLIJZKLZFUEUHSFUJSFUIUMOUFDAPDBPUHUJCQRUGUOULCUGUNUKHKDDABTUAUBUC $.
 
   ${
     $d A x $.  $d B x $.  $d V x $.
@@ -149758,29 +149758,28 @@ computer programs (as last() or lastChar()), the terminology used for
   ccatsymb $p |- ( ( A e. Word V /\ B e. Word V /\ I e. ZZ )
                    -> ( ( A ++ B ) ` I ) = if ( I < ( # ` A ) ,
                                    ( A ` I ) , ( B ` ( I - ( # ` A ) ) ) ) ) $=
-    ( wcel cz w3a cfv wbr co cc0 cle wa wceq 3adant3 wb syl cr c0 adantr clt wi
-    cword chash cmin cif cconcat id ad2antrl simpr anim2i simp3 0zd lencl nn0zd
-    cfzo 3ad2ant1 3jca elfzo mpbird df-3an sylanbrc ccatval1 eqcomd ex zre 0red
-    wn jca 3ad2ant3 ltnle wo 3adant2 animorrl wrdsymb0 sylc ccatcl eqtr4d com12
-    sylbird pm2.61i nn0red syl2an biimpar ancomd zaddcl ccatval2 readdcl lenltd
-    caddc lenlt ccatlen olcd simp2 zsubcl sylan2 ancoms leaddsub2 syl3an biimpa
-    eqbrtrd ifeqda ) ADUCZEZBXCEZCFEZGZCAUDHZUAIZCAHZCXHUEJZBHZUFCABUGJZHZXGXIX
-    JXLXNKCLIZXGXIMZXJXNNZUBXOXPXQXOXPMZXDXECKXHUPJEZGZXQXRXDXEMZXSXTXGYAXOXIXD
-    XEYAXFYAUHOZUIXRXSXOXIMZXPXIXOXGXIUJUKXRXFKFEZXHFEZGZXSYCPXGYFXOXIXGXFYDYEX
-    DXEXFULZXGUMXDXEYEXFXDXHDAUNZUOZUQZURUICKXHUSQUTXDXEXSVAVBXTXNXJDABCVCVDQVE
-    XPXOVHZXQXGYKXQUBXIXGYKCKUAIZXQXGCREZKREZMZYLYKPXFXDYOXEXFYMYNCVFZXFVGVIVJC
-    KVKQXGYLXQXGYLMZXJSXNYQXDXFMZYLXHCLIZVLXJSNXGYRYLXDXFYRXEYRUHVMTXGYLYSVNCDA
-    VOVPYQXMXCEZXFMZYLXMUDHZCLIZVLZXNSNZXGUUAYLXGYTXFXDXEYTXFDABVQOYGVIZTXGYLUU
-    CVNCDXMVOZVPVRVEVTTVSWAXGXIVHZMZXNXLCXHBUDHZWJJZUAIZUUIXNXLNZUBUULUUIUUMUUL
-    UUIMZXDXECXHUUKUPJEZGZUUMUUNYAUUOUUPXGYAUULUUHYBUIUUNUUOYSUULMZUUNUULYSUUIY
-    SUULXGYSUUHXDXFYSUUHPZXEXDXHREZYMUURXFXDXHYHWBZYPXHCWKWCVMWDUKWEUUNXFYEUUKF
-    EZGZUUOUUQPXGUVBUULUUHXGXFYEUVAYGYJXDXEUVAXFXDYEUUJFEUVAXEYIXEUUJDBUNZUOXHU
-    UJWFWCOURUICXHUUKUSQUTXDXEUUOVAVBDABCWGQVEUUIUULVHZUUMXGUVDUUMUBUUHXGUVDUUK
-    CLIZUUMXGUUKCXDXEUUKREZXFXDUUSUUJREZUVFXEUUTXEUUJUVCWBZXHUUJWHWCOXFXDYMXEYP
-    VJWIXGUVEUUMXGUVEMZXNSXLUVIUUAUUDUUEXGUUAUVEUUFTUVIUUCYLUVIUUBUUKCLXGUUBUUK
-    NZUVEXDXEUVJXFDABWLOTXGUVEUJXAWMUUGVPUVIXEXKFEZMZXKKUAIZUUJXKLIZVLXLSNXGUVL
-    UVEXGXEUVKXDXEXFWNXDXFUVKXEXFXDUVKXDXFYEUVKYICXHWOWPWQVMVITUVIUVNUVMXGUVEUV
-    NXDUUSXEUVGXFYMUVEUVNPUUTUVHYPXHUUJCWRWSWTWMXKDBVOVPVRVEVTTVSWAVDXBVD $.
+    ( wcel cz co cfv clt wbr wceq wa cc0 cle simpr wb ex c0 adantr sylc cconcat
+    cword chash cmin cif cfzo w3a simprll anim2i 0zd lencl nn0zd ad2antrr elfzo
+    syl3anc ad2antrl mpbird df-3an sylanbrc ccatval1 eqcomd syl zre 0red ltnled
+    wi wn adantl wo simpl anim1i animorrl wrdsymb0 ccatcl sylbird com12 adantrd
+    eqtr4d pm2.61i caddc id nn0red lenlt syl2an adantlr biimpar anim12ci zaddcl
+    ccatval2 readdcl simplr zsubcld jca ad2antlr leaddsub2d biimpa olcd ccatlen
+    cr eqbrtrd ifeqda 3impa ) ADUBZEZBXCEZCFEZCABUAGZHZCAUCHZIJZCAHZCXIUDGZBHZU
+    EZKXDXELZXFLZXNXHXPXJXKXMXHMCNJZXPXJLZXKXHKZVFXQXRXSXQXRLZXDXECMXIUFGEZUGZX
+    SXTXOYAYBXQXOXFXJUHXTYAXQXJLZXRXJXQXPXJOUIXPYAYCPZXQXJXPXFMFEXIFEZYDXOXFOZX
+    PUJXDYEXEXFXDXIDAUKZULZUMZCMXIUNUOUPUQXDXEYAURUSYBXHXKDABCUTVAVBQXQVGZXPXSX
+    JXPYJXSXPYJCMIJZXSXFYKYJPXOXFCMCVCZXFVDVEVHXPYKXSXPYKLZXKRXHYMXDXFLZYKXICNJ
+    ZVIXKRKXPYNYKXOXDXFXDXEVJVKSXPYKYOVLCDAVMTYMXGXCEZXFLZYKXGUCHZCNJZVIZXHRKZX
+    PYQYKXOYPXFDABVNVKZSXPYKYSVLCDXGVMZTVRQVOVPVQVSCXIBUCHZVTGZIJZXPXJVGZLZXMXH
+    KZVFUUFUUHUUIUUFUUHLZXDXECXIUUEUFGEZUGZUUIUUJXOUUKUULUUFXOXFUUGUHUUJUUKYOUU
+    FLZUUFUUFUUHYOUUFWAXPYOUUGXDXFYOUUGPZXEXDXIWSEZCWSEZUUNXFXDXIYGWBZYLXICWCWD
+    WEWFWGXPUUKUUMPZUUFUUGXPXFYEUUEFEZUURYFYIXOUUSXFXDYEUUDFEUUSXEYHXEUUDDBUKZU
+    LXIUUDWHWDSCXIUUEUNUOUPUQXDXEUUKURUSUULXHXMDABCWIVAVBQUUFVGZXPUUIUUGXPUVAUU
+    IXPUVAUUECNJZUUIXOUUEWSEZUUPUVBUVAPXFXDUUOUUDWSEZUVCXEUUQXEUUDUUTWBZXIUUDWJ
+    WDYLUUECWCWDXPUVBUUIXPUVBLZXMRXHUVFXEXLFEZLZXLMIJZUUDXLNJZVIXMRKXPUVHUVBXPX
+    EUVGXDXEXFWKXDXFUVGXEYNCXIXDXFOXDYEXFYHSWLWEWMSUVFUVJUVIXPUVBUVJXPXIUUDCXDU
+    UOXEXFUUQUMXEUVDXDXFUVEWNXFUUPXOYLVHWOWPWQXLDBVMTUVFYQYTUUAXPYQUVBUUBSUVFYS
+    YKUVFYRUUECNXOYRUUEKXFUVBDDABWRUMXPUVBOWTWQUUCTVRQVOVPVQVSXAVAXB $.
 
   $( The first symbol of a concatenation of two words is the first symbol of
      the first word if the first word is not empty.  (Contributed by Alexander
@@ -149844,42 +149843,41 @@ computer programs (as last() or lastChar()), the terminology used for
     ccatass $p |- ( ( S e. Word B /\ T e. Word B /\
           U e. Word B ) -> ( ( S ++ T ) ++ U ) =
           ( S ++ ( T ++ U ) ) ) $=
-      ( wcel cc0 chash cfv caddc co cfzo cconcat wfn ccatcl wceq oveq2d syl3anc
-      adantr syl cmin vx cword w3a wf stoic3 wrdf ffn 3syl ccatlen oveq1d eqtrd
-      3adant3 fneq2d mpbid simp1 3adant1 syl2anc lencl 3ad2ant1 nn0cnd 3ad2ant2
-      cn0 3ad2ant3 addassd 3eqtr4d cv wo cz nn0zd fzospliti ancoms sylan simpl1
-      simpl2 simpr ccatval1 simpl3 cuz wss uzid uzaddcl fzoss2 sseqtr4d zaddcld
-      sselda ccatval2 fzosubel3 eqtr4d fzoss1 nn0uz eleq2s elfzoelz zcnd adantl
-      wa subsub4d fveq2d eleq2d biimpa fzosubel2 syl13anc oveq12d jaodan syldan
-      cc biimpar eqfnfvd ) BAUBZEZCXHEZDXHEZUCZUAFBGHZCGHZIJZDGHZIJZKJZBCLJZDLJ
-      ZBCDLJZLJZXLXTFXTGHZKJZMZXTXRMXLXTXHEZYDAXTUDYEXIXJXSXHEZXKYFABCNZAXSDNUE
-      AXTUFYDAXTUGUHXLYDXRXTXLYCXQFKXLYCXSGHZXPIJZXQXIXJYGXKYCYJOYHAXSDUIUEXLYI
-      XOXPIXIXJYIXOOXKABCUIULZUJZUKPUMUNXLYBFYBGHZKJZMZYBXRMXLYBXHEZYNAYBUDYOXL
-      XIYAXHEZYPXIXJXKUOZXJXKYQXIACDNUPZABYANUQAYBUFYNAYBUGUHXLYNXRYBXLYMXQFKXL
-      XMYAGHZIJZXMXNXPIJZIJZYMXQXLYTUUBXMIXJXKYTUUBOXIACDUIUPPZXLXIYQYMUUAOYRYS
-      ABYAUIUQXLXMXNXPXLXMXIXJXMVBEZXKABURUSZUTZXLXNXJXIXNVBEZXKACURVAZUTZXLXPX
-      KXIXPVBEZXJADURVCZUTVDZVEPUMUNXLUAVFZXREZUUNFXMKJZEZUUNXMXQKJZEZVGZUUNXTH
-      ZUUNYBHZOZXLXMVHEZUUOUUTXLXMUUFVIZUUOUVDUUTUUNFXQXMVJVKVLXLUUQUVCUUSXLUUQ
-      WOZUUNXSHZUUNBHZUVAUVBUVFXIXJUUQUVGUVHOXIXJXKUUQVMZXIXJXKUUQVNXLUUQVOZABC
-      UUNVPQUVFYGXKUUNFYIKJZEZUVAUVGOZXLYGUUQXIXJYGXKYHULZRXIXJXKUUQVQXLUUPUVKU
-      UNXLUUPFXOKJZUVKXLXOXMVRHZEZUUPUVOVSXLXMUVPEZUUHUVQXLUVDUVRUVEXMVTSUUIXNX
-      MXMWAUQZXMFXOWBSXLYIXOFKYKPZWCWEAXSDUUNVPZQUVFXIYQUUQUVBUVHOUVIXLYQUUQYSR
-      UVJABYAUUNVPQVEXLUUSUUNXMXOKJZEZUUNXOXQKJZEZVGZUVCXLXOVHEZUUSUWFXLXMXNUVE
-      XLXNUUIVIZWDZUUSUWGUWFUUNXMXQXOVJVKVLXLUWCUVCUWEXLUWCWOZUVGUUNXMTJZYAHZUV
-      AUVBUWJUVGUWKCHZUWLUWJXIXJUWCUVGUWMOXIXJXKUWCVMZXIXJXKUWCVNZXLUWCVOABCUUN
-      WFQUWJXJXKUWKFXNKJEZUWLUWMOUWOXIXJXKUWCVQZXLXNVHEZUWCUWPUWHUWCUWRUWPUUNXM
-      XNWGVKVLACDUWKVPQWHUWJYGXKUVLUVMXLYGUWCUVNRUWQXLUWBUVKUUNXLUWBUVOUVKXLUUE
-      UWBUVOVSZUUFUWSXMFVRHVBXMFXOWIWJWKSUVTWCWEUWAQUWJXIYQUUNXMUUAKJZEZUVBUWLO
-      ZUWNXLYQUWCYSRXLUWBUWTUUNXLUWBUURUWTXLXQXOVRHZEZUWBUURVSXLXOUXCEZUUKUXDXL
-      UWGUXEUWIXOVTSUULXPXOXOWAUQXOXMXQWBSXLUUAXQXMKXLUUAUUCXQUUDUUMWHPZWCWEABY
-      AUUNWFZQVEXLUWEWOZUUNYITJZDHZUWLUVAUVBUXHUXJUWKXNTJZDHZUWLUXHUXIUXKDUXHUX
-      IUUNXOTJZUXKXLUXIUXMOUWEXLYIXOUUNTYKPRUXHUUNXMXNUWEUUNXEEXLUWEUUNUUNXOXQW
-      LWMWNXLXMXEEUWEUUGRXLXNXEEUWEUUJRWPWHWQUXHXJXKUWKXNUUBKJEZUWLUXLOXIXJXKUW
-      EVNXIXJXKUWEVQZUXHUUNXOUUCKJZEZUVDUWRUUBVHEZUXNXLUWEUXQXLUWDUXPUUNXLXQUUC
-      XOKUUMPWRWSXLUVDUWEUVERXLUWRUWEUWHRXLUXRUWEXLXNXPUWHXLXPUULVIWDRUUNXMXNUU
-      BWTXAACDUWKWFQWHUXHYGXKUUNYIYJKJZEZUVAUXJOXLYGUWEUVNRUXOXLUXTUWEXLUXSUWDU
-      UNXLYIXOYJXQKYKYLXBWRXFAXSDUUNWFQUXHXIYQUXAUXBXIXJXKUWEVMXLYQUWEYSRXLUWDU
-      WTUUNXLUWDUURUWTXLUVQUWDUURVSUVSXOXMXQWISUXFWCWEUXGQVEXCXDXCXDXG $.
+      ( wcel cc0 chash cfv caddc co cfzo cconcat wfn wceq oveq2d syl2anc adantr
+      syl syl3anc cmin vx cword w3a ccatcl stoic3 ccatlen 3adant3 oveq1d fneq2d
+      wrdfn eqtrd mpbid simp1 3adant1 3ad2ant1 nn0cnd 3ad2ant2 3ad2ant3 addassd
+      cn0 lencl 3eqtr4d cv wo cz nn0zd fzospliti ex mpan9 wa simp2 id syl2an3an
+      ccatval1 simpl3 cuz uzidd uzaddcl fzoss2 sseqtr4d sselda zaddcld ccatval2
+      wss simpl2 fzosubel3 eqtr4d fzoss1 nn0uz eleq2s simpl1 cc elfzoelz adantl
+      zcnd subsub4d fveq2d eleq2d biimpa 3jca fzosubel2 oveq12d biimpar eqfnfvd
+      jaodan syldan ) BAUBZEZCXGEZDXGEZUCZUAFBGHZCGHZIJZDGHZIJZKJZBCLJZDLJZBCDL
+      JZLJZXKXSFXSGHZKJZMZXSXQMXKXSXGEZYDXHXIXRXGEZXJYEABCUDZAXRDUDUEAXSUJRXKYC
+      XQXSXKYBXPFKXKYBXRGHZXOIJZXPXHXIYFXJYBYINYGAAXRDUFUEXKYHXNXOIXHXIYHXNNXJA
+      ABCUFUGZUHZUKOUIULXKYAFYAGHZKJZMZYAXQMXKYAXGEZYNXKXHXTXGEZYOXHXIXJUMZXIXJ
+      YPXHACDUDUNZABXTUDPAYAUJRXKYMXQYAXKYLXPFKXKXLXTGHZIJZXLXMXOIJZIJZYLXPXKYS
+      UUAXLIXIXJYSUUANXHAACDUFUNOZXKXHYPYLYTNYQYRAABXTUFPXKXLXMXOXKXLXHXIXLUTEZ
+      XJABVAUOZUPZXKXMXIXHXMUTEZXJACVAUQZUPZXKXOXJXHXOUTEZXIADVAURZUPUSZVBOUIUL
+      XKUAVCZXQEZUUMFXLKJZEZUUMXLXPKJZEZVDZUUMXSHZUUMYAHZNZXKXLVEEZUUNUUSXKXLUU
+      EVFZUUNUVCUUSUUMFXPXLVGVHVIXKUUPUVBUURXKUUPVJZUUMXRHZUUMBHZUUTUVAXKXHXIUU
+      PUUPUVFUVGNYQXHXIXJVKZUUPVLZABCUUMVNVMUVEYFXJUUMFYHKJZEZUUTUVFNZXKYFUUPXH
+      XIYFXJYGUGZQXHXIXJUUPVOXKUUOUVJUUMXKUUOFXNKJZUVJXKXNXLVPHZEZUUOUVNWDXKXLU
+      VOEUUGUVPXKXLUVDVQUUHXMXLXLVRPZXLFXNVSRXKYHXNFKYJOZVTWAAXRDUUMVNZSXKXHYPU
+      UPUUPUVAUVGNYQYRUVIABXTUUMVNVMVBXKUURUUMXLXNKJZEZUUMXNXPKJZEZVDZUVBXKXNVE
+      EZUURUWDXKXLXMUVDXKXMUUHVFZWBZUURUWEUWDUUMXLXPXNVGVHVIXKUWAUVBUWCXKUWAVJZ
+      UVFUUMXLTJZXTHZUUTUVAUWHUVFUWICHZUWJXKXHXIUWAUWAUVFUWKNYQUVHUWAVLABCUUMWC
+      VMUWHXIXJUWIFXMKJEZUWJUWKNXHXIXJUWAWEXHXIXJUWAVOZXKXMVEEZUWAUWLUWFUWAUWNU
+      WLUUMXLXMWFVHVIACDUWIVNSWGUWHYFXJUVKUVLXKYFUWAUVMQUWMXKUVTUVJUUMXKUVTUVNU
+      VJXKUUDUVTUVNWDZUUEUWOXLFVPHUTXLFXNWHWIWJRUVRVTWAUVSSUWHXHYPUUMXLYTKJZEZU
+      VAUWJNZXHXIXJUWAWKXKYPUWAYRQXKUVTUWPUUMXKUVTUUQUWPXKXPXNVPHZEZUVTUUQWDXKX
+      NUWSEUUJUWTXKXNUWGVQUUKXOXNXNVRPXNXLXPVSRXKYTXPXLKXKYTUUBXPUUCUULWGOZVTWA
+      ABXTUUMWCZSVBXKUWCVJZUUMYHTJZDHZUWJUUTUVAUXCUXEUWIXMTJZDHZUWJUXCUXDUXFDUX
+      CUXDUUMXNTJZUXFXKUXDUXHNUWCXKYHXNUUMTYJOQUXCUUMXLXMUWCUUMWLEXKUWCUUMUUMXN
+      XPWMWOWNXKXLWLEUWCUUFQXKXMWLEUWCUUIQWPWGWQUXCXIXJUWIXMUUAKJEZUWJUXGNXHXIX
+      JUWCWEXHXIXJUWCVOZUXCUUMXNUUBKJZEZUVCUWNUUAVEEZUCZUXIXKUWCUXLXKUWBUXKUUMX
+      KXPUUBXNKUULOWRWSXKUXNUWCXKUVCUWNUXMUVDUWFXKXMXOUWFXKXOUUKVFWBWTQUUMXLXMU
+      UAXAPACDUWIWCSWGUXCYFXJUUMYHYIKJZEZUUTUXENXKYFUWCUVMQUXJXKUXPUWCXKUXOUWBU
+      UMXKYHXNYIXPKYJYKXBWRXCAXRDUUMWCSUXCXHYPUWQUWRXHXIXJUWCWKXKYPUWCYRQXKUWBU
+      WPUUMXKUWBUUQUWPXKUVPUWBUUQWDUVQXNXLXPWHRUXAVTWAUXBSVBXEXFXEXFXD $.
 
     $( The range of a concatenated word.  (Contributed by Stefan O'Rear,
        15-Aug-2015.) $)
@@ -149924,18 +149922,18 @@ computer programs (as last() or lastChar()), the terminology used for
      shortened by AV, 1-May-2020.) $)
   lswccatn0lsw $p |- ( ( A e. Word V /\ B e. Word V /\ B =/= (/) )
                        -> ( lastS ` ( A ++ B ) ) = ( lastS ` B ) ) $=
-    ( wcel w3a cconcat co clsw cfv chash c1 cmin cvv wceq lsw wa oveq1d 3adant3
-    cz eqtrd cword c0 wne ovex mp1i caddc cfzo ccatlen clt lencl nn0zd 3ad2ant1
-    wbr cn lennncl 3adant1 simpl nnz adantl zaddcld crp zre nnrp ltaddrp syl2an
-    cr 3jca syl2anc fzolb sylibr fzoend syl eqeltrd ccatval2 syld3an3 cc nn0cnd
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-    WLVRXSVSXQXRUQVTXSXTWLKLWSWLWAQTVETRWBTTWGWFWNWONWHWGWOWNBWEOWCWDT $.
+    ( wcel w3a cconcat co chash cfv c1 cmin clsw wa oveq1d 3adant3 lencl syl2an
+    wceq cz eqtrd cword c0 wne caddc ccatlen clt wbr cn nn0zd lennncl simpl nnz
+    cfzo zaddcl sylan2 cr crp zre nnrp ltaddrp 3jca 3impb sylibr fzoend eqeltrd
+    fzolb syl ccatval2 syld3an3 nn0cnd addcl 1cnd sub32d pncan2 fveq2d cvv ovex
+    cc lsw mp1i 3ad2ant2 3eqtr4d ) ACUAZDZBWCDZBUBUCZEZABFGZHIZJKGZWHIZBHIZJKGZ
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+    VSVTWEWDWPWNRWFBWCVSWAWB $.
 
   $( The last symbol of a word concatenated with the empty word is the last
      symbol of the word.  (Contributed by AV, 22-Oct-2018.)  (Proof shortened
@@ -150209,9 +150207,9 @@ computer programs (as last() or lastChar()), the terminology used for
      4-Mar-2022.) $)
   ccatws1len $p |- ( W e. Word V
                       -> ( # ` ( W ++ <" X "> ) ) = ( ( # ` W ) + 1 ) ) $=
-    ( cword wcel cs1 cconcat co chash cfv caddc wceq wrdv s1cli ccatlen sylancl
-    c1 cvv s1len a1i oveq2d eqtrd ) BADEZBCFZGHIJZBIJZUDIJZKHZUFQKHUCBRDZEUDUIE
-    UEUHLABMCNRBUDOPUCUGQUFKUGQLUCCSTUAUB $.
+    ( cword wcel cs1 cconcat co chash cfv caddc c1 cvv wceq s1cli ccatlen mpan2
+    s1len oveq2i syl6eq ) BADEZBCFZGHIJZBIJZUBIJZKHZUDLKHUAUBMDEUCUFNCOAMBUBPQU
+    ELUDKCRST $.
 
   $( The length of a word is ` N ` iff the length of the concatenation of the
      word with a singleton word is ` N + 1 ` .  (Contributed by AV,
@@ -150233,12 +150231,11 @@ computer programs (as last() or lastChar()), the terminology used for
     NOUCUEQEUGUERUCUEBCPSUETUAUB $.
 
   $( The length of the concatenation of two singleton words.  (Contributed by
-     Alexander van der Vekens, 22-Sep-2018.) $)
-  ccat2s1len $p |- ( ( X e. V /\ Y e. V )
-                     -> ( # ` ( <" X "> ++ <" Y "> ) ) = 2 ) $=
-    ( wcel cs1 cword cconcat co chash cfv c2 wceq s1cl wa caddc ccatlen oveq12i
-    c1 s1len 1p1e2 eqtri syl6eq syl2an ) BADBEZAFZDZCEZUEDZUDUGGHIJZKLCADBAMCAM
-    UFUHNUIUDIJZUGIJZOHZKAUDUGPULRROHKUJRUKROBSCSQTUAUBUC $.
+     Alexander van der Vekens, 22-Sep-2018.)  (Revised by JJ, 14-Jan-2024.) $)
+  ccat2s1len $p |- ( # ` ( <" X "> ++ <" Y "> ) ) = 2 $=
+    ( cs1 cvv cword wcel cconcat co chash cfv c2 wceq s1cli wa caddc ccatlen c1
+    s1len oveq12i 1p1e2 eqtri syl6eq mp2an ) ACZDEZFZBCZUEFZUDUGGHIJZKLAMBMUFUH
+    NUIUDIJZUGIJZOHZKDDUDUGPULQQOHKUJQUKQOARBRSTUAUBUC $.
 
   $( The concatenation of a word with two singleton words is a word.
      (Contributed by Alexander van der Vekens, 22-Sep-2018.) $)
@@ -150855,34 +150852,34 @@ computer programs (as last() or lastChar()), the terminology used for
       ( ( S substr <. X , Y >. ) ++ ( S substr <. Y , Z >. ) ) =
         ( S substr <. X , Z >. ) ) $=
       ( wcel cc0 cfz co chash cfv wa cmin cfzo adantr syl2anc caddc wceq oveq2d
-      syl3anc vx cword w3a cop csubstr cconcat wfn wf swrdcl ccatcl ffn ccatlen
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-      3syl simpld syl npncan3d eqtrd fneq2d mpbid cv wo simpr zsubcld fzospliti
-      eleq2d biimpar ccatval1 simpll simplr1 swrdfv syl31anc ccatval2 fzosubel3
-      simplr2 simplr3 eqeltrd oveq1d cc elfzoelz adantl subcld subadd23d nncand
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-      VSXKUULUUQUUOOXDXFXIXCUUQWFUVJABCEUUPWAWBWTXA $.
+      syl vx cword w3a cop csubstr cconcat wfn swrdcl ccatcl wrdfn simpl simpr1
+      ccatlen simpr2 simpr3 fzass4 biimpri swrdlen syl3anc 3adant3r1 oveq12d cz
+      simpld elfzelz zcnd npncan3d 3eqtrd mpbid cv wo zsubcld anim1ci fzospliti
+      fneq2d ad2antrr eleq2d biimpar ccatval1 simpll simplr1 simpr swrdfv eqtrd
+      syl31anc ccatval2 simplr2 simplr3 fzosubel3 oveq1d elfzoelz adantl subcld
+      eqeltrd cc subadd23d nncand fveq2d jaodan syldan eqtr4d eqfnfvd ) BAUBZFZ
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+      XKUUNUUSUUQOXDXFXIXCUUSWGXKUUSWAABCEUURWBWDWTXA $.
   $}
 
   ${
@@ -150891,20 +150888,19 @@ computer programs (as last() or lastChar()), the terminology used for
        Carneiro, 27-Sep-2015.) $)
     swrdccat2 $p |- ( ( S e. Word B /\ T e. Word B ) ->
  ( ( S ++ T ) substr <. ( # ` S ) , ( ( # ` S ) + ( # ` T ) ) >. ) = T ) $=
-      ( vk wcel wa cc0 chash cfv cfzo co caddc wfn wf cfz cuz cn0 adantr oveq2d
-      wceq cword cconcat cop csubstr ccatcl swrdcl wrdf ffn 4syl lencl syl6eleq
-      cmin nn0uz cz nn0zd syl adantl uzaddcl syl2anc elfzuzb sylanbrc nn0addcld
-      uzid ccatlen eleqtrrd swrdlen syl3anc nn0cnd pncan2d eqtrd fneq2d ffnd cv
-      mpbid eleq2d biimpar swrdfv syl31anc ccatval3 3expa eqfnfvd ) BAUAZEZCWBE
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-      XSXTTABCXOVSVTVJWA $.
+      ( vk wcel wa cc0 chash cfv cfzo co caddc wfn wrdfn cfz wceq syl2an oveq2d
+      cuz cn0 cword cconcat cop csubstr ccatcl swrdcl 3syl lencl nn0uz syl6eleq
+      cmin adantr nn0zd uzidd uzaddcl elfzuzb sylanbrc nn0addcl ccatlen swrdlen
+      eleqtrrd syl3anc cc nn0cnd pncan2 eqtrd fneq2d mpbid adantl cv w3a eleq2d
+      3jca biimpar swrdfv syl2an2r ccatval3 3expa eqfnfvd ) BAUAZEZCVTEZFZDGCHI
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+      WEXLWCWOWDGJXKRVLVNAWFWGWHXLVOVPWAWBXMXOXPPABCXLVQVRVFVS $.
   $}
 
 
@@ -151276,31 +151272,30 @@ computer programs (as last() or lastChar()), the terminology used for
     ccatpfx $p |- ( ( S e. Word A /\ Y e. ( 0 ... Z )
                        /\ Z e. ( 0 ... ( # ` S ) ) )
         -> ( ( S prefix Y ) ++ ( S substr <. Y , Z >. ) ) = ( S prefix Z ) ) $=
-      ( wcel cc0 cfz co chash cfzo syl2anc caddc wceq cmin wa zcnd eqtrd adantr
-      cfv syl3anc vx cword w3a cop csubstr cconcat wfn wf pfxcl 3ad2ant1 swrdcl
-      cpfx ccatcl wrdf 3syl ccatlen simp1 fzass4 biimpri simpld 3adant1 swrdlen
-      pfxlen oveq12d cc cz elfzelz ad2antrl 3impb ad2antll pncan3d oveq2d mpbid
-      ffn fneq2d pfxfn 3adant2 cv wo 3ad2ant2 fzospliti eleq2d biimpar ccatval1
-      simpr pfxfv ccatval2 anim1ci fzosubel syl wb subidd eqcomd oveq1d eqeltrd
-      mpbird swrdfv syldan elfzoelz adantl npcand fveq2d 3eqtrd jaodan eqfnfvd
-      simpl3 eqtr4d ) BAUBZEZCFDGHEZDFBISZGHZEZUCZUAFDJHZBCULHZBCDUDUEHZUFHZBDU
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-      NYRUUMUUBRXAQXBXCXDWRUUFXIXMUUEUUIUUHMXNXIUUEYMRXIXJXMUUEXFUUPUUDDABWFTXG
-      XE $.
+      ( wcel cc0 cfz co chash cfv wceq wa cfzo adantr caddc cmin cc id ad2antrr
+      wfn vx cword cpfx cop csubstr cconcat pfxcl swrdcl ccatcl syl2anc ccatlen
+      wrdfn syl fzass4 biimpri simpld pfxlen swrdlen 3expb oveq12d elfzelz zcnd
+      sylan2 pncan3 syl2an adantl 3eqtrd oveq2d mpbid pfxfn adantrl cv ad2antrl
+      fneq2d wo fzospliti syl2anr eleq2d biimpar ccatval1 simpl pfxfv syl2an3an
+      syl3anc eqtrd ccatval2 w3a fzosubel subidd oveq1d eqeltrd swrdfv syl2an2r
+      cz wb elfzoelz npcan fveq2d jaodan syldan 3expa adantlrl eqtr4d eqfnfvd
+      3impb ) BAUBZEZCFDGHEZDFBIJZGHZEZBCUCHZBCDUDUEHZUFHZBDUCHZKXGXHXKLZLZUAFD
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+      KXGXPWAXPYMXGYOVFUUIRUUCCABWBWCWEXQUUKLZUUEUUCYGPHZXMJZUVACOHZBJZUUFUUTYC
+      YDUUCYGYIMHZEZUUEUVBKXGYCXPUUKYESXGYDXPUUKYFSXQUVFUUKXQUVEUUJUUCXQYGCYIDM
+      YPXQYIYKDYQUUBWEUTVRVSAXLXMUUCWFWDXQXGXHXKWGZUUKUVAFYJMHZEUVBUVDKXGXHXKUV
+      GUVGRUSUUTUVAUUCCPHZUVHXQUVAUVIKUUKXQYGCUUCPYPVHZNUUTUVICCPHZYJMHZEZUVIUV
+      HEZUUKUUKUUNUVMXQUUKRUUOUUCCDCWHVQXQUVMUVNWOZUUKXHUVOXGXKXHUVLUVHUVIXHUVK
+      FYJMXHCUUAWIWJVRVMNVIWKABCDUVAWLWMUUTUVCUUCBUUTUVCUVICOHZUUCXQUVCUVPKUUKX
+      QUVAUVICOUVJWJNUUKUUCQEYSUVPUUCKXQUUKUUCUUCCDWPVBXHYSXGXKUUAVMUUCCWQVQWEW
+      RVGWSWTXGXKUUDUUGUUFKZXHXGXKUUDUVQUUCDABWBXAXBXCXDXE $.
   $}
 
   ${
@@ -151309,16 +151304,16 @@ computer programs (as last() or lastChar()), the terminology used for
        Carneiro, 27-Sep-2015.)  (Revised by AV, 6-May-2020.) $)
     pfxccat1 $p |- ( ( S e. Word B /\ T e. Word B )
                       -> ( ( S ++ T ) prefix ( # ` S ) ) = S ) $=
-      ( vk wcel wa chash cfv cc0 cfzo cfz wceq cn0 lencl adantr syl2anc wfn syl
-      co eqtrd cword cconcat cpfx cres ccatcl caddc anim12i sylib elfz0add sylc
-      nn0fz0 ccatlen oveq2d eleqtrrd pfxres wss ccatvalfn cz nn0zd uzid uzaddcl
-      cuz syl2an fzoss2 fnssres wrdfn cv fvres adantl ccatval1 3expa eqfnfvd )
-      BAUAZEZCVMEZFZBCUBSZBGHZUCSZVQIVRJSZUDZBVPVQVMEVRIVQGHZKSZEVSWALABCUEVPVR
-      IVRCGHZUFSZKSZWCVPVRMEZWDMEZFVRIVRKSEZVRWFEVNWGVOWHABNZACNZUGVNWIVOVNWGWI
-      WJVRUKUHOVRWDVRUIUJVPWBWEIKABCULUMUNAVQVRUOPVPDVTWABVPVQIWEJSZQVTWLUPZWAV
-      TQBCAUQVPWEVRVBHZEZWMVNVRWNEZWHWOVOVNVRUREWPVNVRWJUSVRUTRWKWDVRVRVAVCVRIW
-      EVDRWLVTVQVEPVNBVTQVOABVFOVPDVGZVTEZFWQWAHZWQVQHZWQBHZWRWSWTLVPWQVTVQVHVI
-      VNVOWRWTXALABCWQVJVKTVLT $.
+      ( vk cword wcel wa cconcat co chash cfv cpfx cc0 cfzo cfz wceq cn0 adantr
+      lencl eqtrd cres ccatcl caddc anim12i nn0fz0 elfz0add sylc ccatlen oveq2d
+      sylib eleqtrrd pfxres syl2anc ccatvalfn cuz wss nn0zd uzidd syl2an fzoss2
+      uzaddcl syl fnssresd wfn wrdfn cv fvres adantl ccatval1 3expa eqfnfvd ) B
+      AEZFZCVLFZGZBCHIZBJKZLIZVPMVQNIZUAZBVOVPVLFVQMVPJKZOIZFVRVTPABCUBVOVQMVQC
+      JKZUCIZOIZWBVOVQQFZWCQFZGVQMVQOIFZVQWEFVMWFVNWGABSZACSZUDVMWHVNVMWFWHWIVQ
+      UEUJRVQWCVQUFUGVOWAWDMOAABCUHUIUKAVPVQULUMVODVSVTBVOMWDNIZVSVPBCAUNVOWDVQ
+      UOKZFZVSWKUPVMVQWLFWGWMVNVMVQVMVQWIUQURWJWCVQVQVAUSVQMWDUTVBVCVMBVSVDVNAB
+      VERVODVFZVSFZGWNVTKZWNVPKZWNBKZWOWPWQPVOWNVSVPVGVHVMVNWOWQWRPABCWNVIVJTVK
+      T $.
   $}
 
   $( The prefix of length one of a nonempty word expressed as a singleton word.
@@ -151525,9 +151520,9 @@ computer programs (as last() or lastChar()), the terminology used for
                             -> ( # ` ( ( W substr <. M , ( # ` W ) >. )
                                      ++ ( W prefix M ) ) ) = ( # ` W ) ) $=
     ( cword wcel cc0 chash cfv cfz co wa csubstr cpfx cconcat caddc wceq swrdcl
-    cop pfxcl jca adantr ccatlen syl addlenrevpfx eqtrd ) CBDZEZAFCGHZIJEZKZCAU
-    HRLJZCAMJZNJGHZUKGHULGHOJZUHUJUKUFEZULUFEZKZUMUNPUGUQUIUGUOUPBCAUHQBCASTUAB
-    UKULUBUCABCUDUE $.
+    cop pfxcl ccatlen syl2anc adantr addlenrevpfx eqtrd ) CBDZEZAFCGHZIJEZKCAUG
+    RLJZCAMJZNJGHZUIGHUJGHOJZUGUFUKULPZUHUFUIUEEUJUEEUMBCAUGQBCASBBUIUJTUAUBABC
+    UCUD $.
 
   $( Reconstruct a nonempty word from its prefix and last symbol.  (Contributed
      by Alexander van der Vekens, 5-Aug-2018.)  (Revised by AV, 9-May-2020.) $)
@@ -151582,14 +151577,14 @@ computer programs (as last() or lastChar()), the terminology used for
     ( C e. Word X /\ D e. Word X ) /\ ( # ` A ) = ( # ` C ) ) ->
       ( ( A ++ B ) = ( C ++ D ) <-> ( A = C /\ B = D ) ) ) $=
     ( wcel wa chash cfv wceq cconcat co cpfx pfxccat1 caddc cop csubstr ccatlen
-    syl 3eqtr3d cword oveq1 3ad2ant1 simp3 oveq2d 3ad2ant2 eqtrd eqeq12d syl5ib
-    w3a simpr simpl3 fveq2d simpl1 simpl2 opeq12d oveq12d swrdccat2 jcad oveq12
-    ex impbid1 ) AEUAZFBVCFGZCVCFDVCFGZAHIZCHIZJZUJZABKLZCDKLZJZACJZBDJZGVIVLVM
-    VNVLVJVFMLZVKVFMLZJVIVMVJVKVFMUBVIVOAVPCVDVEVOAJVHEABNUCVIVPVKVGMLZCVIVFVGV
-    KMVDVEVHUDUEVEVDVQCJVHECDNUFUGUHUIVIVLVNVIVLGZVJVFVFBHIOLZPZQLZVKVGVGDHIOLZ
-    PZQLZBDVRVJVKVTWCQVIVLUKZVRVFVGVSWBVDVEVHVLULVRVJHIZVKHIZVSWBVRVJVKHWEUMVRV
-    DWFVSJVDVEVHVLUNZEABRSVRVEWGWBJVDVEVHVLUOZECDRSTUPUQVRVDWABJWHEABURSVRVEWDD
-    JWIECDURSTVAUSACBDKUTVB $.
+    syl 3eqtr3d cword w3a wi oveq1 oveq2 sylan9eqr eqeqan12d syl5ib 3impb simpr
+    simpl3 fveq2d simpl1 simpl2 opeq12d oveq12d swrdccat2 jcad oveq12 impbid1
+    ex ) AEUAZFBVBFGZCVBFDVBFGZAHIZCHIZJZUBZABKLZCDKLZJZACJZBDJZGVHVKVLVMVCVDVG
+    VKVLUCVKVIVEMLZVJVEMLZJVCVDVGGZGVLVIVJVEMUDVCVPVNAVOCEABNVGVDVOVJVFMLCVEVFV
+    JMUEECDNUFUGUHUIVHVKVMVHVKGZVIVEVEBHIOLZPZQLZVJVFVFDHIOLZPZQLZBDVQVIVJVSWBQ
+    VHVKUJZVQVEVFVRWAVCVDVGVKUKVQVIHIZVJHIZVRWAVQVIVJHWDULVQVCWEVRJVCVDVGVKUMZE
+    EABRSVQVDWFWAJVCVDVGVKUNZEECDRSTUOUPVQVCVTBJWGEABUQSVQVDWCDJWHECDUQSTVAURAC
+    BDKUSUT $.
 
   $( An ~ opth -like theorem for recovering the two halves of a concatenated
      word.  (Contributed by Mario Carneiro, 1-Oct-2015.) $)
@@ -151601,10 +151596,10 @@ computer programs (as last() or lastChar()), the terminology used for
     simp2l simp2r bitrd syl5ib ccatopth biimpd 3expia com23 3adant3 mpdd oveq12
     impbid1 ) AEUAZFZBVCFZGZCVCFZDVCFZGZBHIZDHIZJZKZABLMZCDLMZJZACJBDJGZVMVPAHI
     ZCHIZJZVQVPVNHIZVOHIZJZVMVTVNVOHUBVMWCVRVKNMZVSVKNMZJVTVMWAWDWBWEVMWAVRVJNM
-    ZWDVFVIWAWFJVLEABOUCVMVJVKVRNVFVIVLUDUEUFVIVFWBWEJVLECDOUGUHVMVRVSVKVMVRVMV
-    DVRPFVDVEVIVLUIEAQRSVMVSVMVGVSPFVFVGVHVLUKECQRSVMVKVMVHVKPFVFVGVHVLULEDQRSU
-    JUMUNVFVIVPVTVQTTVLVFVIGVTVPVQVFVIVTVPVQTVFVIVTKVPVQABCDEUOUPUQURUSUTACBDLV
-    AVB $.
+    ZWDVFVIWAWFJVLEEABOUCVMVJVKVRNVFVIVLUDUEUFVIVFWBWEJVLEECDOUGUHVMVRVSVKVMVRV
+    MVDVRPFVDVEVIVLUIEAQRSVMVSVMVGVSPFVFVGVHVLUKECQRSVMVKVMVHVKPFVFVGVHVLULEDQR
+    SUJUMUNVFVIVPVTVQTTVLVFVIGVTVPVQVFVIVTVPVQTVFVIVTKVPVQABCDEUOUPUQURUSUTACBD
+    LVAVB $.
 
   $( Concatenation of words is left-cancellative.  (Contributed by Mario
      Carneiro, 2-Oct-2015.) $)
@@ -151807,10 +151802,10 @@ computer programs (as last() or lastChar()), the terminology used for
           /\ ( M e. ( 0 ... N ) /\ N e. ( 0 ... ( ( # ` A ) + ( # ` B ) ) ) ) )
            -> ( ( A ++ B ) substr <. M , N >. ) Fn ( 0 ..^ ( N - M ) ) ) $=
     ( cword wcel wa cc0 cfz chash cfv caddc cconcat cop csubstr cmin cfzo wfn
-    co ccatcl adantr simprl ccatlen eqcomd oveq2d eleq2d biimpcd adantl syl3anc
-    wi impcom swrdvalfn ) AEFZGBUNGHZCIDJTGZDIAKLBKLMTZJTZGZHZHABNTZUNGZUPDIVAK
-    LZJTZGZVACDOPTIDCQTRTSUOVBUTEABUAUBUOUPUSUCUTUOVEUSUOVEUKUPUOUSVEUOURVDDUOU
-    QVCIJUOVCUQEABUDUEUFUGUHUIULVACDEUMUJ $.
+    co ccatcl adantr simprl ccatlen oveq2d biimpar adantrl swrdvalfn syl3anc
+    eleq2d ) AEFZGBUKGHZCIDJTGZDIAKLBKLMTZJTZGZHZHABNTZUKGZUMDIURKLZJTZGZURCDOP
+    TIDCQTRTSULUSUQEABUAUBULUMUPUCULUPVBUMULVBUPULVAUODULUTUNIJEEABUDUEUJUFUGUR
+    CDEUHUI $.
 
   ${
     $d A k $.  $d B k $.  $d M k $.  $d N k $.  $d V k $.
@@ -151909,52 +151904,48 @@ computer programs (as last() or lastChar()), the terminology used for
                   -> ( ( M e. ( L ... N ) /\ N e. ( L ... ( L + ( # ` B ) ) ) )
                            -> ( ( A ++ B ) substr <. M , N >. )
                               = ( B substr <. ( M - L ) , ( N - L ) >. ) ) ) $=
-      ( wcel wa cfz co cfv cmin wceq cc0 wi syl adantr wbr adantl cword cconcat
-      vk chash caddc cop csubstr cfzo wfn wb oveq1 eleq2d oveq12d anbi12d ax-mp
-      id cn0 lencl cuz wss elnn0uz biimpi fzss1 sseld anim12d syl5bi swrdccatfn
-      imp syldan cc w3a cz cle elfz2 3anim123i 3comr sylbi oveq2d fneq2d mpbird
-      zcn nnncan2 simpr elfzmlbm ad2antrl elfzmlbp nn0zd syl11 impcom swrdvalfn
-      ex syl3anc cv clt cif simpl elfzelz zaddcl expcom elfzoelz impel sylanbrc
-      df-3an ccatsymb wn cr zre anim12i elnn0z 0red anim1i ancoms anim2i le2add
-      syl2an2 recn addid2d breq1d sylibd readdcl lenltd expd com12 mpan9 eqcomi
-      breq2i notbii syl6ibr com23 3adant2 elfzonn0 iffalsed ad2antrr addsubassd
-      oveq2 eqeq1d syl5ib fveq2d 3eqtrd swrdfv ccatcl eqeltrid ccatlen fzmmmeqm
-      3syl biimpd syl31anc 3jca sylan 3eqtr4d eqfnfvd ) AFUAZHZBUULHZIZDCEJKZHZ
-      ECCBUDLZUEKZJKZHZIZABUBKZDEUFUGKZBDCMKZECMKZUFUGKZNUUOUVBIZUCOUVFUVEMKZUH
-      KZUVDUVGUVHUVDUVJUIUVDOEDMKZUHKZUIZUUOUVBDOEJKZHZEOAUDLZUURUEKZJKZHZIZUVM
-      UUOUVBUVTUVBDUVPEJKZHZEUVPUVQJKZHZIZUUOUVTCUVPNZUVBUWEUJGUWFUUQUWBUVAUWDU
-      WFUUPUWADCUVPEJUKULUWFUUTUWCEUWFCUVPUUSUVQJUWFUPCUVPUURUEUKUMULZUNUOUUMUW
-      EUVTPZUUNUUMUVPUQHZUWHFAURZUWIUWBUVOUWDUVSUWIUWAUVNDUWIUVPOUSLZHZUWAUVNUT
-      UWIUWLUVPVAVBZUVPOEVCQVDUWIUWCUVREUWIUWLUWCUVRUTZUWMUVPOUVQVCZQVDVEQRVFVH
-      ABDEFVGVIUVHUVJUVLUVDUVHUVIUVKOUHUVBUVIUVKNZUUOUVBEVJHZDVJHZCVJHZVKZUWPUU
-      QUWTUVAUUQCVLHZEVLHZDVLHZVKZCDVMSZDEVMSZIZIZUWTDCEVNZUXDUWTUXGUXBUXCUXAUW
-      TUXBUWQUXCUWRUXAUWSEWADWAZCWAZVOVPRVQREDCWBQTVRVSVTUVHUUNUVEOUVFJKHZUVFOU
-      URJKHZUVGUVJUIUUOUUNUVBUUMUUNWCRZUUQUXLUUOUVADCEWDWEZUVBUUOUXMUVAUUOUXMPZ
-      UUQUURVLHZUVAUXMUUOUXQUVAUXMECUURWFWKZUUNUXQUUMUUNUURFBURWGZTWHTWIBUVEUVF
-      FWJWLUVHUCWMZUVJHZIZUXTDUEKZUVCLZUXTUVEUEKZBLZUXTUVDLZUXTUVGLZUYBUYDUYCUV
-      PWNSZUYCALZUYCUVPMKZBLZWOZUYLUYFUYBUUMUUNUYCVLHZVKZUYDUYMNUYBUUOUYNUYOUVH
-      UUOUYAUUOUVBWPRUVHUXTVLHZUYNUYAUUQUYPUYNPZUUOUVAUUQUXCUYQDCEWQUYPUXCUYNUX
-      TDWRWSQWEUXTOUVIWTZXAUUMUUNUYNXCXBABUYCFXDQUYBUYIUYJUYLUVHUXTUQHZUYIXEZUY
-      AUUQUYSUYTPZUUOUVAUUQUXHVUAUXIUXGUXDVUAUXEUXDVUAPUXFUXDUXEVUAUXAUXCUXEVUA
-      PUXBUXAUXCIZUYSUXEUYTVUBUYSUXEUYTPVUBUYSIUXEUYCCWNSZXEZUYTVUBCXFHZDXFHZIZ
-      UYSUXEVUDPZUXAVUEUXCVUFCXGDXGXHUYSUYPOUXTVMSZIVUGVUHPZUXTXIUYPUXTXFHZVUIV
-      UJUXTXGVUIVUKVUGVUHVUKVUGIZVUIVUHVULVUIUXEVUDVULVUIUXEIZCUYCVMSZVUDVULVUM
-      OCUEKZUYCVMSZVUNVUGOXFHZVUEIZVUKVUKVUFIZVUMVUPPVUFVUEVURVUFVUQVUEVUFXJXKX
-      LVUGVUFVUKVUEVUFWCXMZOCUXTDXNXOVULVUOCUYCVMVUEVUOCNVUKVUFVUECCXPXQWEXRXSV
-      ULCUYCVUGVUEVUKVUEVUFWPTVULVUSUYCXFHVUTUXTDXTQYAXSYBYCYBYDVQYDUYIVUCUVPCU
-      YCWNCUVPGYEYFYGYHWKYIYJYCRWIVQWEUXTUVIYKXAYLUYBUYKUYEBUVHUYPUYKUYENZUYAUU
-      QUYPVVAPZUUOUVAUUQUXHVVBUXIUXDVVBUXGUXAUXCVVBUXBVUBUYPVVAUWFVUBUYPIZVVAPG
-      VVCUYCCMKZUYENUWFVVAVVCUXTDCUYPUXTVJHVUBUXTWATVUBUWRUYPUXCUWRUXAUXJTRUXAU
-      WSUXCUYPUXKYMYNUWFVVDUYKUYECUVPUYCMYOYPYQUOWKYJRVQWEUYRXAYRYSUYBUVCUULHZU
-      VOEOUVCUDLZJKZHZUXTUVLHZUYGUYDNUUOVVEUVBUYAFABUUAYMUVHUVOUYAUVBUUOUVOUUQU
-      UOUVOPUVAUUOUUQUVOUUMUUQUVOPUUNUUMUUPUVNDUUMUWICUWKHUUPUVNUTUWJUWICUVPUWK
-      GUWMUUBCOEVCUUEVDRYCRWIRUVHVVHUYAUVBUUOVVHUVAUUOVVHPZUUQUVAUWDVVJUWFUVAUW
-      DUJGUWGUOUWDUUOVVHUWDUUOIVVHUVSUUOUWDUVSUUOUWCUVREUUOUWLUWNUUMUWLUUNUUMUW
-      IUWLUWJUWMQRUWOQVDWIUUOVVHUVSUJUWDUUOVVGUVREUUOVVFUVQOJFABUUCVRULTVTWKVQT
-      WIRUVHUYAVVIUUQUYAVVIPUUOUVAUUQUYAVVIUUQUVJUVLUXTUUQUVIUVKOUHCDEUUDVRULUU
-      FWEVHFUVCDEUXTYTUUGUVHUUNUXLUXMVKUYAUYHUYFNUVHUUNUXLUXMUXNUXOUVBUUOUXMUVA
-      UXPUUQUUOUVAUXMUUNUVAUXMPZUUMUUNUXQVVKUXSUXRQTYCTWIUUHFBUVEUVFUXTYTUUIUUJ
-      UUKWK $.
+      ( wcel wa cfz co cfv caddc cmin wceq cc0 wi sylbi ad2antrl wbr vk cconcat
+      cword chash cop csubstr cfzo wfn wb oveq1 eleq2d id oveq12d anbi12d ax-mp
+      cn0 lencl cuz elnn0uz fzss1 sseld anim12d adantr syl5bi swrdccatfn syldan
+      wss syl imp fzmmmeqm oveq2d fneq2d mpbird simplr elfzmlbm cz nn0zd adantl
+      elfzmlbp sylan adantrl swrdvalfn syl3anc cv clt cif simpll elfzelz zaddcl
+      w3a expcom elfzoelz impel df-3an sylanbrc ccatsymb wn cle elfz2 cr elnn0z
+      zre anim12i jctl le2add syl2anc recn addid2d breq1d sylibd simprl readdcl
+      0re lenltd expd com12 mpan9 breq2i notbii syl6ib ex com23 3adant2 adantrr
+      elfzonn0 iffalsed cc zcn ad2antlr ad2antrr addsubassd oveq2 eqeq1d syl5ib
+      fveq2d 3eqtrd 3syl impcom 3jca swrdfv ccatcl biimpi sselda ccatlen biimpa
+      eqeltrid ad2ant2r syl2an2r ad2ant2l 3eqtr4d eqfnfvd ) AFUCZHZBUULHZIZDCEJ
+      KZHZECCBUDLZMKZJKZHZIZABUBKZDEUEUFKZBDCNKZECNKZUEUFKZOUUOUVBIZUAPUVFUVENK
+      ZUGKZUVDUVGUVHUVDUVJUHZUVDPEDNKZUGKZUHZUUOUVBDPEJKZHZEPAUDLZUURMKZJKZHZIZ
+      UVNUUOUVBUWAUVBDUVQEJKZHZEUVQUVRJKZHZIZUUOUWACUVQOZUVBUWFUIGUWGUUQUWCUVAU
+      WEUWGUUPUWBDCUVQEJUJUKUWGUUTUWDEUWGCUVQUUSUVRJUWGULCUVQUURMUJUMUKZUNUOUUM
+      UWFUWAQZUUNUUMUVQUPHZUWIFAUQZUWJUWCUVPUWEUVTUWJUWBUVODUWJUVQPURLZHZUWBUVO
+      VGUVQUSZUVQPEUTRVAUWJUWDUVSEUWJUWMUWDUVSVGZUWNUVQPUVRUTZRVAVBVHVCVDVIABDE
+      FVEVFUUQUVKUVNUIUUOUVAUUQUVJUVMUVDUUQUVIUVLPUGCDEVJVKZVLSVMUVHUUNUVEPUVFJ
+      KHZUVFPUURJKHZUVGUVJUHUUMUUNUVBVNZUUQUWRUUOUVADCEVOSZUUOUVAUWSUUQUUOUURVP
+      HZUVAUWSUUNUXBUUMUUNUURFBUQVQZVRECUURVSZVTWABUVEUVFFWBWCUVHUAWDZUVJHZIZUX
+      EDMKZUVCLZUXEUVEMKZBLZUXEUVDLZUXEUVGLZUXGUXIUXHUVQWETZUXHALZUXHUVQNKZBLZW
+      FZUXQUXKUXGUUMUUNUXHVPHZWJZUXIUXROUXGUUOUXSUXTUUOUVBUXFWGUVHUXEVPHZUXSUXF
+      UUQUYAUXSQZUUOUVAUUQDVPHZUYBDCEWHUYAUYCUXSUXEDWIWKVHSUXEPUVIWLZWMUUMUUNUX
+      SWNWOABUXHFWPVHUXGUXNUXOUXQUVHUXEUPHZUXNWQZUXFUUQUYEUYFQZUUOUVAUUQCVPHZEV
+      PHZUYCWJZCDWRTZDEWRTZIZIZUYGDCEWSZUYJUYKUYGUYLUYJUYKUYGUYHUYCUYKUYGQUYIUY
+      HUYCIZUYEUYKUYFUYPUYEUYKUYFQUYPUYEIUYKUXHCWETZWQZUYFUYPCWTHZDWTHZIZUYEUYK
+      UYRQZUYHUYSUYCUYTCXBDXBXCUYEUYAPUXEWRTZIVUAVUBQZUXEXAUYAUXEWTHZVUCVUDUXEX
+      BVUCVUEVUAVUBVUEVUAIZVUCVUBVUFVUCUYKUYRVUFVUCUYKIZCUXHWRTZUYRVUFVUGPCMKZU
+      XHWRTZVUHVUFPWTHZUYSIZVUEUYTIZVUGVUJQUYSVULVUEUYTUYSVUKXMXDSVUEUYTVUMUYSV
+      UMULWAPCUXEDXEXFVUFVUICUXHWRUYSVUICOVUEUYTUYSCCXGXHSXIXJVUFCUXHVUEUYSUYTX
+      KVUEUYTUXHWTHUYSUXEDXLWAXNXJXOXPXOXQRXQUYQUXNCUVQUXHWEGXRXSXTYAYBYCVIYDRS
+      UXEUVIYEWMYFUXGUXPUXJBUVHUYAUXPUXJOZUXFUUQUYAVUNQZUUOUVAUUQUYNVUOUYOUYJVU
+      OUYMUYHUYCVUOUYIUYPUYAVUNUWGUYPUYAIZVUNQGVUPUXHCNKZUXJOUWGVUNVUPUXEDCUYAU
+      XEYGHUYPUXEYHVRUYCDYGHUYHUYADYHYIUYHCYGHUYCUYACYHYJYKUWGVUQUXPUXJCUVQUXHN
+      YLYMYNUOYAYCVCRSUYDWMYOYPUVHUVCUULHZUVPEPUVCUDLZJKZHZWJUXFUXEUVMHZUXLUXIO
+      UVHVURUVPVVAUUOVURUVBFABUUAVCUUMUUQUVPUUNUVAUUMUUPUVODUUMUWJCUWLHUUPUVOVG
+      UWKUWJCUVQUWLGUWJUWMUWNUUBZUUFCPEUTYQUUCUUGUUOUVAVVAUUQUVAUUOVVAUVAUWEUUO
+      VVAQUWGUVAUWEUIGUWHUOUWEUUOVVAUWEUUOIVVAUVTUUOUWEUVTUUOUWDUVSEUUMUWOUUNUU
+      MUWJUWMUWOUWKVVCUWPYQVCVAYRUUOVVAUVTUIUWEUUOVUTUVSEUUOVUSUVRPJFFABUUDVKUK
+      VRVMYARYRWAYSUVHUXFVVBUUQUXFVVBUIUUOUVAUUQUVJUVMUXEUWQUKSUUEFUVCDEUXEYTUU
+      HUVHUUNUWRUWSWJUXFUXMUXKOUVHUUNUWRUWSUWTUXAUUNUVAUWSUUMUUQUUNUXBUVAUWSUXC
+      UXDVTUUIYSFBUVEUVFUXEYTVTUUJUUKYA $.
 
     $( Lemma for ~ pfxccatin12lem2 and ~ pfxccatin12lem3 .  (Contributed by AV,
        30-Mar-2018.)  (Revised by AV, 27-May-2018.) $)
@@ -151962,12 +151953,12 @@ computer programs (as last() or lastChar()), the terminology used for
                   /\ ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + ( # ` B ) ) ) ) )
                              -> ( ( A ++ B ) e. Word V /\ M e. ( 0 ... N )
                                 /\ N e. ( 0 ... ( # ` ( A ++ B ) ) ) ) ) $=
-      ( cword wcel wa cc0 cfz co chash cfv caddc cconcat ccatcl adantr adantl
-      elfz0fzfz0 cuz wss elfzuz2 fzss1 sseld impr wb ccatlen wceq eqcomi oveq1d
-      syl a1i eqtrd oveq2d eleq2d mpbird 3jca ) AFHZIBUTIJZDKCLMIZECCBNOZPMZLMZ
-      IZJZJZABQMZUTIZDKELMIZEKVINOZLMZIZVAVJVGFABRSVGVKVACDEVDUATVHVNEKVDLMZIZV
-      AVBVFVPVAVBJZVEVOEVQCKUBOIZVEVOUCVBVRVADKCUDTCKVDUEUMUFUGVAVNVPUHVGVAVMVO
-      EVAVLVDKLVAVLANOZVCPMVDFABUIVAVSCVCPVSCUJVACVSGUKUNULUOUPUQSURUS $.
+      ( cword wcel wa cc0 cfz chash cfv caddc cconcat ccatcl adantr elfz0fzfz0
+      co adantl cuz wss elfzuz2 syl sselda ccatlen oveq1i syl6eqr oveq2d eleq2d
+      fzss1 syl5ibr imp 3jca ) AFHZIBUPIJZDKCLTIZECCBMNZOTZLTZIJZJABPTZUPIZDKEL
+      TIZEKVCMNZLTZIZUQVDVBFABQRVBVEUQCDEUTSUAUQVBVHVBVHUQEKUTLTZIURVAVIEURCKUB
+      NIVAVIUCDKCUDCKUTULUEUFUQVGVIEUQVFUTKLUQVFAMNZUSOTUTFFABUGCVJUSOGUHUIUJUK
+      UMUNUO $.
 
     $( Lemma 2 for ~ pfxccatin12 .  (Contributed by AV, 30-Mar-2018.)  (Revised
        by AV, 9-May-2020.) $)
@@ -152057,42 +152048,34 @@ computer programs (as last() or lastChar()), the terminology used for
         -> ( ( M e. ( 0 ... L ) /\ N e. ( L ... ( L + ( # ` B ) ) ) )
              -> ( ( A ++ B ) substr <. M , N >. )
                 = ( ( A substr <. M , L >. ) ++ ( B prefix ( N - L ) ) ) ) ) $=
-      ( wcel wa cc0 cfz co cfv caddc wceq cfzo adantl syl w3a ad2antrl vk cword
-      chash cconcat cop csubstr cmin cpfx wfn ccatcl elfz0fzfz0 cuz wss elfzuz2
-      adantr fzss1 ccatlen eqcomi oveq1i syl6eq oveq2d sseqtr4d sseld swrdvalfn
-      impr syl3anc swrdcl pfxcl anim12i ccatvalfn simpll simprl elnn0uz eluzfz2
-      cn0 lencl sylbi eqeltrid ad2antrr swrdlen simplr cz nn0zd elfzmlbp pfxlen
-      simpr syl2anc oveq12d cc cle wbr wi elfz2nn0 nn0cn zcn 3jca elfzelz syl11
-      ex 3adant3 imp npncan3 eqtr2d fneq2d mpbird anim2i ancomd pfxccatin12lem3
-      sylc simpl nn0fz0 sylib eleqtrrd df-3an sylanbrc ccatval1 pfxccatin12lem2
-      eqtr4d elfzuz eluzelz ancoms syl5com impcom pfxccatin12lem4 jca pm2.61ian
-      cv wn ccatval2 eqfnfvd ) AFUBZHZBYKHZIZDJCKLHZECCBUCMZNLZKLZHZIZABUDLZDEU
-      EUFLZADCUEUFLZBECUGLZUHLZUDLZOYNYTIZUAJEDUGLZPLZUUBUUFUUGUUAYKHZDJEKLHZEJ
-      UUAUCMZKLZHZUUBUUIUIYNUUJYTFABUJUOYTUUKYNCDEYQUKQYNYOYSUUNYNYOIZYRUUMEUUO
-      YRJYQKLZUUMUUOCJULMZHZYRUUPUMYOUURYNDJCUNQCJYQUPRUUOUULYQJKYNUULYQOYOYNUU
-      LAUCMZYPNLYQFABUQUUSCYPNCUUSGURUSUTUOVAVBVCVEUUADEFVDVFUUGUUFUUIUIUUFJUUC
-      UCMZUUEUCMZNLZPLZUIZUUGUUCYKHZUUEYKHZIZUVDYNUVGYTYLUVEYMUVFFADCVGFBUUDVHV
-      IUOZUUCUUEFVJRUUGUUIUVCUUFUUGUUHUVBJPUUGUVBCDUGLZUUDNLZUUHUUGUUTUVIUVAUUD
-      NUUGYLYOCJUUSKLZHZUUTUVIOZYLYMYTVKZYNYOYSVLZYLUVLYMYTYLUUSVOHZUVLFAVPZUVP
-      CUUSUVKGUVPUUSUUQHUUSUVKHZUUSVMJUUSVNVQVRRVSFADCVTZVFUUGYMUUDJYPKLHZUVAUU
-      DOZYLYMYTWAUUGYPWBHZYSIUVTYNUWBYTYSYMUWBYLYMYPFBVPWCQZYOYSWFVIECYPWDZRFBU
-      UDWEZWGWHUUGCWIHZDWIHZEWIHZSZUVJUUHOYTUWIYNYOYSUWIYODVOHZCVOHZDCWJWKZSZYS
-      UWIWLZDCWMZUWJUWKUWNUWLEWBHZUWJUWKIZUWIYSUWPUWQUWIUWPUWQIUWFUWGUWHUWQUWFU
-      WPUWKUWFUWJCWNQQUWJUWGUWPUWKDWNTUWPUWHUWQEWOUOWPWSECYQWQWRWTVQXAQCDEXBRXC
-      VAXDXEUAYGZJUVIPLZHZUUGUWRUUIHZIZUWRUUBMZUWRUUFMZOUWTUXBIZUXCUWRUUCMZUXDU
-      XEUUGUXAUWTIUXCUXFOUWTUUGUXAVLUXEUWTUXAUXBUXAUWTUUGUXAWFZXFXGABUWRCDEFGXH
-      XIUXEUVEUVFUWRJUUTPLZHZSZUXDUXFOUXEUVGUXIUXJUUGUVGUWTUXAUVHTUXEUWRUWSUXHU
-      WTUXBXJUXEUUTUVIJPUXEYLYOUVLSZUVMUUGUXKUWTUXAUUGYLYOUVLUVNUVOYLUVLYMYTYLC
-      UUSUVKGYLUVPUVRUVQUUSXKXLVRVSZWPTUVSRVAXMUVEUVFUXIXNXOFUUCUUEUWRXPRXRUWTY
-      HZUXBIZUXCUWRUUTUGLUUEMZUXDUXNUUGUXAUXMIZUXCUXOOUXMUUGUXAVLUXNUXMUXAUXBUX
-      AUXMUXGXFXGZABUWRCDEFGXQXIUXNUVEUVFUWRUUTUVBPLZHZSZUXDUXOOUXNUVGUXSUXTUUG
-      UVGUXMUXAUVHTUXNUWRUVIUVJPLZUXRUXNUWKUWJUWPSZUXPUWRUYAHUUGUYBUXMUXAYTUYBY
-      NYSYOUYBYSECULMHZYOUYBWLECYQXSUYCUWPYOUYBCEXTYOUWMUWPUYBWLZUWOUWJUWKUYDUW
-      LUWKUWJUYDUWKUWJIZUWPUYBUYEUWPIUWKUWJUWPUWKUWJUWPVKUWKUWJUWPWAUYEUWPWFWPW
-      SYAWTVQYBRYCQTUXQUWRCDEYDXIUUGUXRUYAOUXMUXAUUGUUTUVIUVBUVJPUUGYLYOUVLUVMU
-      VNUVOUXLUVSVFZUUGUUTUVIUVAUUDNUYFUUGYMUVTIZUWAYTYNUYGYSYNUYGWLYOYSYNUYGYS
-      YNIZYMUVTYNYMYSYLYMWFQUYHUWBYSUVTYNUWBYSUWCQYSYNXJUWDWGYEWSQYCUWERWHWHTXM
-      UVEUVFUXSXNXOFUUCUUEUWRYIRXRYFYJWS $.
+      ( wcel wa cc0 cfz co chash cfv caddc wceq cfzo w3a syl ad2antrl cword cop
+      cconcat csubstr cmin cpfx wfn pfxccatin12lem2c swrdvalfn swrdcl ccatvalfn
+      vk pfxcl syl2an adantr simpll simprl lencl nn0fz0 sylib eqeltrid ad2antrr
+      cn0 swrdlen syl3anc simpr cz nn0zd adantl elfzmlbp sylan syl2an2r adantrl
+      pfxlen oveq12d cc cle wbr wi elfz2nn0 nn0cn ad2antll zcn ex elfzelz syl11
+      3jca 3adant3 sylbi npncan3 eqtr2d oveq2d fneq2d mpbird cv pfxccatin12lem3
+      anim2i ancomd sylc anim12i simpl eleqtrrd df-3an sylanbrc ccatval1 eqtr4d
+      imp wn pfxccatin12lem2 cuz elfzuz eluzelz id 3expia ancoms syl5com impcom
+      pfxccatin12lem4 ccatval2 pm2.61ian eqfnfvd ) AFUAZHZBYBHZIZDJCKLHZECCBMNZ
+      OLZKLHZIZABUCLZDEUBUDLZADCUBUDLZBECUELZUFLZUCLZPYEYJIZULJEDUELZQLZYLYPYQY
+      KYBHDJEKLHEJYKMNKLHRYLYSUGABCDEFGUHYKDEFUISYQYPYSUGYPJYMMNZYOMNZOLZQLZUGZ
+      YEUUDYJYCYMYBHZYOYBHZUUDYDFADCUJZFBYNUMZYMYOFUKUNUOYQYSUUCYPYQYRUUBJQYQUU
+      BCDUELZYNOLZYRYQYTUUIUUAYNOYQYCYFCJAMNZKLZHZYTUUIPYCYDYJUPYEYFYIUQYCUUMYD
+      YJYCCUUKUULGYCUUKVCHUUKUULHFAURUUKUSUTVAVBFADCVDVEZYEYIUUAYNPZYFYEYDYIYNJ
+      YGKLHZUUOYCYDVFYEYGVGHZYIUUPYDUUQYCYDYGFBURVHVIECYGVJVKFBYNVNVLVMVOZYJUUJ
+      YRPZYEYJCVPHZDVPHZEVPHZRZUUSYFYIUVCYFDVCHZCVCHZDCVQVRZRZYIUVCVSZDCVTZUVDU
+      VEUVHUVFEVGHZUVDUVEIZUVCYIUVJUVKUVCUVJUVKIUUTUVAUVBUVEUUTUVJUVDCWAWBUVDUV
+      AUVJUVEDWATUVJUVBUVKEWCUOWGWDECYHWEWFWHWIXGCDEWJSVIWKWLWMWNULWOZJUUIQLZHZ
+      YQUVLYSHZIZUVLYLNZUVLYPNZPUVNUVPIZUVQUVLYMNZUVRUVSYQUVOUVNIUVQUVTPUVNYQUV
+      OUQUVSUVNUVOUVPUVOUVNYQUVOVFZWQWRABUVLCDEFGWPWSUVSUUEUUFUVLJYTQLZHZRZUVRU
+      VTPUVSUUEUUFIZUWCUWDYQUWEUVNUVOYEUWEYJYCUUEYDUUFUUGUUHWTUOZTUVSUVLUVMUWBU
+      VNUVPXAYQUWBUVMPUVNUVOYQYTUUIJQUUNWLTXBUUEUUFUWCXCXDFYMYOUVLXESXFUVNXHZUV
+      PIZUVQUVLYTUELYONZUVRUWHYQUVOUWGIZUVQUWIPUWGYQUVOUQUWHUWGUVOUVPUVOUWGUWAW
+      QWRZABUVLCDEFGXIWSUWHUUEUUFUVLYTUUBQLZHZRZUVRUWIPUWHUWEUWMUWNYQUWEUWGUVOU
+      WFTUWHUVLUUIUUJQLZUWLUWHUVEUVDUVJRZUWJUVLUWOHYQUWPUWGUVOYJUWPYEYIYFUWPYIE
+      CXJNHZYFUWPVSECYHXKUWQUVJYFUWPCEXLYFUVGUVJUWPVSZUVIUVDUVEUWRUVFUVEUVDUWRU
+      VEUVDUVJUWPUWPXMXNXOWHWIXPSXQVITUWKUVLCDEXRWSYQUWLUWOPUWGUVOYQYTUUIUUBUUJ
+      QUUNUURVOTXBUUEUUFUWMXCXDFYMYOUVLXSSXFXTYAWD $.
 
     $( The subword of a concatenation is either a subword of the first
        concatenated word or a subword of the second concatenated word or a
@@ -152528,18 +152511,18 @@ Splicing words (substring replacement)
     spllen $p |- ( ph -> ( # ` ( S splice <. F , T , R >. ) ) =
           ( ( # ` S ) + ( ( # ` R ) - ( T - F ) ) ) ) $=
       ( co chash cfv caddc cmin wcel cc0 wceq syl syl2anc cotp csplice cpfx cop
-      cconcat csubstr cword cfz splval syl13anc fveq2d pfxcl ccatcl ccatlen cn0
-      swrdcl nn0cnd cz elfzelz zcnd addcld elfzel2 addsub12d cuz elfzuz elfzuz3
-      lencl uztrn elfzuzb sylanbrc pfxlen oveq1d addcomd 3eqtrd elfzuz2 eluzfz2
-      swrdlen syl3anc oveq12d subsub3d oveq2d 3eqtr4d ) ADFECUAUBKZLMDFUCKZCUEK
-      ZDEDLMZUDUFKZUEKZLMZWELMZWGLMZNKZWFCLMZEFOKOKZNKZAWCWHLADBUGZPZFQEUHKZPZE
-      QWFUHKZPZCWPPZWCWHRGHIJCDEFWPWRWTWPUIUJUKAWEWPPZWGWPPZWIWLRAWDWPPZXBXCAWQ
-      XEGBDFULSZJBWDCUMTAWQXDGBDEWFUPSBWEWGUNTAWMFNKZWFEOKZNKWFXGEOKZNKWLWOAXGW
-      FEAWMFAWMAXBWMUOPJBCVGSUQZAFAWSFURPHFQEUSSUTZVAAWFAXAWFURPIEQWFVBSUTAEAXA
-      EURPIEQWFUSSUTZVCAWJXGWKXHNAWJWDLMZWMNKZFWMNKXGAXEXBWJXNRXFJBWDCUNTAXMFWM
-      NAWQFWTPZXMFRGAFQVDMZPZWFFVDMZPZXOAWSXQHFQEVESAWFEVDMPZEXRPZXSAXAXTIEQWFV
-      FSAWSYAHFQEVFSEWFFVHTFQWFVIVJBDFVKTVLAFWMXKXJVMVNAWQXAWFWTPZWKXHRGIAWFXPP
-      ZYBAXAYCIEQWFVOSQWFVPSBDEWFVQVRVSAWNXIWFNAWMEFXJXLXKVTWAWBVN $.
+      cconcat csubstr cword cfz splval syl13anc fveq2d ccatcl swrdcl ccatlen cc
+      pfxcl lencl nn0cnd elfzelz zcnd addcld elfzel2 addsub12d cuz elfzuz uztrn
+      elfzuz3 elfzuzb pfxlen oveq1d addcomd 3eqtrd elfzuz2 eluzfz2 3syl swrdlen
+      sylanbrc syl3anc oveq12d subsub3d oveq2d 3eqtr4d ) ADFECUAUBKZLMDFUCKZCUE
+      KZDEDLMZUDUFKZUEKZLMZWELMZWGLMZNKZWFCLMZEFOKOKZNKZAWCWHLADBUGZPZFQEUHKZPZ
+      EQWFUHKZPZCWPPZWCWHRGHIJCDEFWPWRWTWPUIUJUKAWEWPPZWGWPPZWIWLRAWDWPPZXBXCAW
+      QXEGBDFUPSZJBWDCULTAWQXDGBDEWFUMSBBWEWGUNTAWMFNKZWFEOKZNKWFXGEOKZNKWLWOAX
+      GWFEAWMFAXBWMUOPJXBWMBCUQURSZAWSFUOPHWSFFQEUSUTSZVAAXAWFUOPIXAWFEQWFVBUTS
+      AXAEUOPIXAEEQWFUSUTSZVCAWJXGWKXHNAWJWDLMZWMNKZFWMNKXGAXEXBWJXNRXFJBBWDCUN
+      TAXMFWMNAWQFWTPZXMFRGAFQVDMZPZWFFVDMZPZXOAWSXQHFQEVESAWFEVDMPZEXRPZXSAXAX
+      TIEQWFVGSAWSYAHFQEVGSEWFFVFTFQWFVHVQBDFVITVJAFWMXKXJVKVLAWQXAWFWTPZWKXHRG
+      IAXAWFXPPYBIEQWFVMQWFVNVOBDEWFVPVRVSAWNXIWFNAWMEFXJXLXKVTWAWBVL $.
 
     ${
       splfv1.x $e |- ( ph -> X e. ( 0 ..^ F ) ) $.
@@ -152549,16 +152532,16 @@ Splicing words (substring replacement)
             ( S ` X ) ) $=
         ( co cfv chash wcel cc0 wceq cfzo syl2anc cotp csplice cpfx cconcat cop
         csubstr cword cfz splval syl13anc fveq1d pfxcl syl ccatcl caddc cuz wss
-        swrdcl cn0 cz elfzelz uzid 3syl cfn wrdfin hashcl uzaddcl fzoss2 sseldd
-        ccatlen wa fzass4 bicomi simplbi pfxlen oveq1d oveq2d eleqtrrd ccatval1
-        eqtrd syl3anc pfxfv 3eqtrd ) AGDFECUAUBMZNGDFUCMZCUDMZDEDONZUEUFMZUDMZN
-        ZGWFNZGDNZAGWDWIADBUGZPZFQEUHMZPZEQWGUHMZPZCWMPZWDWIRHIJKCDEFWMWOWQWMUI
-        UJUKAWFWMPZWHWMPZGQWFONZSMZPWJWKRAWEWMPZWSWTAWNXDHBDFULUMZKBWECUNTAWNXA
-        HBDEWGURUMAGQFCONZUOMZSMZXCAQFSMZXHGAXGFUPNZPZXIXHUQAFXJPZXFUSPZXKAWPFU
-        TPXLIFQEVAFVBVCAWSCVDPXMKBCVECVFVCXFFFVGTFQXGVHUMLVIAXBXGQSAXBWEONZXFUO
-        MZXGAXDWSXBXORXEKBWECVJTAXNFXFUOAWNFWQPZXNFRHAWPWRXPIJWPWRVKZXPEFWGUHMP
-        ZXPXRVKXQQFEWGVLVMVNTZBDFVOTZVPVTVQVRBWFWHGVSWAAWKGWENZWLAXDWSGQXNSMZPW
-        KYARXEKAGXIYBLAXNFQSXTVQVRBWECGVSWAAWNXPGXIPYAWLRHXSLGFBDWBWAVTWC $.
+        swrdcl cn0 elfzelz uzid 3syl lencl uzaddcl fzoss2 sseldd ccatlen fzass4
+        cz wa biimpri simpld pfxlen oveq1d eqtrd oveq2d eleqtrrd ccatval1 pfxfv
+        syl3anc 3eqtrd ) AGDFECUAUBMZNGDFUCMZCUDMZDEDONZUEUFMZUDMZNZGWDNZGDNZAG
+        WBWGADBUGZPZFQEUHMZPZEQWEUHMZPZCWKPZWBWGRHIJKCDEFWKWMWOWKUIUJUKAWDWKPZW
+        FWKPZGQWDONZSMZPWHWIRAWCWKPZWQWRAWLXBHBDFULUMZKBWCCUNTAWLWSHBDEWEURUMAG
+        QFCONZUOMZSMZXAAQFSMZXFGAXEFUPNZPZXGXFUQAFXHPZXDUSPZXIAWNFVIPXJIFQEUTFV
+        AVBAWQXKKBCVCUMXDFFVDTFQXEVEUMLVFAWTXEQSAWTWCONZXDUOMZXEAXBWQWTXMRXCKBB
+        WCCVGTAXLFXDUOAWLFWOPZXLFRHAWNWPXNIJWNWPVJZXNEFWEUHMPZXNXPVJXOQFEWEVHVK
+        VLTZBDFVMTZVNVOVPVQBWDWFGVRVTAWIGWCNZWJAXBWQGQXLSMZPWIXSRXCKAGXGXTLAXLF
+        QSXRVPVQBWCCGVRVTAWLXNGXGPXSWJRHXQLGFBDVSVTVOWA $.
     $}
 
     ${
@@ -152569,20 +152552,20 @@ Splicing words (substring replacement)
       splfv2a $p |- ( ph -> ( ( S splice <. F , T , R >. ) ` ( F + X ) ) =
             ( R ` X ) ) $=
         ( caddc co cfv wcel cc0 wceq cn0 syl cotp csplice chash cconcat csubstr
-        cpfx cop cword cfz splval syl13anc elfznn0 nn0cnd cfzo cz elfzoelz zcnd
-        addcomd cuz nn0uz syl6eleq elfzuz3 uztrn syl2anc sylanbrc pfxlen oveq2d
-        elfzuzb eqtr4d fveq12d pfxcl ccatcl swrdcl 0nn0 nn0addcl sylancr fzoss1
-        wss eleq2s ccatlen oveq1d wrdfin hashcl 3eqtrd sseqtr4d fzoaddel sseldd
-        cfn nn0zd eqeltrd ccatval1 syl3anc ccatval3 ) AFGMNZDFECUAUBNZOGDFUFNZU
-        COZMNZWPCUDNZDEDUCOZUGUENZUDNZOZWRWSOZGCOZAWNWRWOXBADBUHZPZFQEUINZPZEQW
-        TUINZPZCXFPZWOXBRHIJKCDEFXFXHXJXFUJUKAWNGFMNZWRAFGAFAXIFSPZIFEULTZUMZAG
-        AGQCUCOZUNNPZGUOPLGQXQUPTUQURAWQFGMAXGFXJPZWQFRHAFQUSOZPWTFUSOZPZXSAFSX
-        TXOUTVAAWTEUSOPZEYAPZYBAXKYCJEQWTVBTAXIYDIFQEVBTEWTFVCVDFQWTVHVEBDFVFVD
-        ZVGZVIVJAWSXFPZXAXFPZWRQWSUCOZUNNZPXCXDRAWPXFPZXLYGAXGYKHBDFVKTZKBWPCVL
-        VDAXGYHHBDEWTVMTAWRXMYJYFAQFMNZXQFMNZUNNZYJXMAYOQYNUNNZYJAYMSPZYOYPVRZA
-        QSPXNYQVNXOQFVOVPYRYMXTSYMQYNVQUTVSTAYIYNQUNAYIWQXQMNZFXQMNYNAYKXLYIYSR
-        YLKBWPCVTVDAWQFXQMYEWAAFXQXPAXQACWHPZXQSPAXLYTKBCWBTCWCTUMURWDVGWEAXRFU
-        OPXMYOPLAFXOWIGQXQFWFVDWGWJBWSXAWRWKWLAYKXLXRXDXERYLKLBWPCGWMWLWD $.
+        cpfx cop cword cfz splval syl13anc elfznn0 nn0cnd cfzo elfzonn0 addcomd
+        cuz nn0uz syl6eleq elfzuz3 uztrn syl2anc elfzuzb sylanbrc pfxlen oveq2d
+        eqtr4d fveq12d pfxcl ccatcl swrdcl 0nn0 nn0addcl sylancr fzoss1 ccatlen
+        wss eleq2s oveq1d lencl 3eqtrd sseqtr4d nn0zd fzoaddel eqeltrd ccatval1
+        cz sseldd syl3anc ccatval3 ) AFGMNZDFECUAUBNZOGDFUFNZUCOZMNZWMCUDNZDEDU
+        COZUGUENZUDNZOZWOWPOZGCOZAWKWOWLWSADBUHZPZFQEUINZPZEQWQUINZPZCXCPZWLWSR
+        HIJKCDEFXCXEXGXCUJUKAWKGFMNZWOAFGAFAXFFSPZIFEULTZUMZAGAGQCUCOZUNNPZGSPL
+        GXNUOTUMUPAWNFGMAXDFXGPZWNFRHAFQUQOZPWQFUQOZPZXPAFSXQXLURUSAWQEUQOPZEXR
+        PZXSAXHXTJEQWQUTTAXFYAIFQEUTTEWQFVAVBFQWQVCVDBDFVEVBZVFZVGVHAWPXCPZWRXC
+        PZWOQWPUCOZUNNZPWTXARAWMXCPZXIYDAXDYHHBDFVITZKBWMCVJVBAXDYEHBDEWQVKTAWO
+        XJYGYCAQFMNZXNFMNZUNNZYGXJAYLQYKUNNZYGAYJSPZYLYMVQZAQSPXKYNVLXLQFVMVNYO
+        YJXQSYJQYKVOURVRTAYFYKQUNAYFWNXNMNZFXNMNYKAYHXIYFYPRYIKBBWMCVPVBAWNFXNM
+        YBVSAFXNXMAXNAXIXNSPKBCVTTUMUPWAVFWBAXOFWGPXJYLPLAFXLWCGQXNFWDVBWHWEBWP
+        WRWOWFWIAYHXIXOXAXBRYIKLBWMCGWJWIWA $.
     $}
   $}
 
@@ -152601,24 +152584,24 @@ Splicing words (substring replacement)
       ( S splice <. F , T , R >. ) = ( ( A ++ R ) ++ C ) ) $=
       ( co wcel cn0 wceq cotp csplice cpfx cconcat chash cfv cop csubstr ccatcl
       cword syl2anc eqeltrd lencl syl caddc nn0addcld splval syl13anc cc0 nn0uz
-      wa cfz cuz syl6eleq cz nn0zd uzid uzaddcl elfzuzb sylanbrc ccatlen fveq2d
-      oveq1d 3eqtr4d ccatpfx syl3anc eluzfz2 pfxid 3eqtrd wb pfxcl swrdcl eqtrd
-      pfxlen ccatopth syl221anc mpbid simpld uztrn simprd oveq12d ) AFHGEUAUBQZ
-      FHUCQZEUDQZFGFUEUFZUGUHQZUDQZBEUDQZDUDQAFIUJZRZHSRGSREWSRWLWQTAFBCUDQZDUD
-      QZWSNAXAWSRZDWSRZXBWSRABWSRZCWSRZXCJKIBCUIUKZLIXADUIUKULZAHBUEUFZSOAXEXIS
-      RJIBUMUNULZAGHCUEUFZUOQZSPAHXKXJAXFXKSRZKICUMUNZUPULZMEFGHWSSSWSUQURAWNWR
-      WPDUDAWMBEUDAWMBTZFHGUGUHQZCTZAWMXQUDQZXATZXPXRVAZAXSFGUCQZXAAWTHUSGVBQRZ
-      GUSWOVBQZRZXSYBTXHAHUSVCUFZRZGHVCUFZRZYCAHSYFXJUTVDZAGXLYHPAHYHRZXMXLYHRA
-      HVERYKAHXJVFHVGUNXNXKHHVHUKULZHUSGVIVJAGYFRWOGVCUFZRZYEAGSYFXOUTVDAWOGDUE
-      UFZUOQZYMAXBUEUFZXAUEUFZYOUOQZWOYPAXCXDYQYSTXGLIXADVKUKAFXBUENVLAGYRYOUOA
-      XLXIXKUOQZGYRAHXIXKUOOVMPAXEXFYRYTTJKIBCVKUKVNZVMVNAGYMRZYOSRZYPYMRAGVERU
-      UBAGXOVFGVGUNAXDUUCLIDUMUNYOGGVHUKULZGUSWOVIVJZIFHGVOVPAYBXATZWPDTZAYBWPU
-      DQZXBTZUUFUUGVAZAUUHFWOUCQZFXBAWTYEWOYDRZUUHUUKTXHUUEAWOYFRUULAWOSYFAWTWO
-      SRXHIFUMUNUTVDUSWOVQUNIFGWOVOVPAWTUUKFTXHIFVRUNNVSAYBWSRZWPWSRZXCXDYBUEUF
-      ZYRTUUIUUJVTAWTUUMXHIFGWAUNAWTUUNXHIFGWOWBUNXGLAUUOGYRAWTYEUUOGTXHUUEIFGW
-      DUKUUAWCYBWPXADIWEWFWGZWHWCAWMWSRZXQWSRZXEXFWMUEUFZXITXTYAVTAWTUUQXHIFHWA
-      UNAWTUURXHIFHGWBUNJKAUUSHXIAWTHYDRZUUSHTXHAYGWOYHRZUUTYJAYNYIUVAUUDYLGWOH
-      WIUKHUSWOVIVJIFHWDUKOWCWMXQBCIWEWFWGWHVMAUUFUUGUUPWJWKWC $.
+      wa cfz cuz syl6eleq nn0zd uzaddcl elfzuzb sylanbrc ccatlen fveq2d 3eqtr4d
+      uzidd oveq1d ccatpfx syl3anc pfxid 3eqtrd wb pfxcl swrdcl pfxlen ccatopth
+      eluzfz2 eqtrd syl221anc mpbid simpld uztrn simprd oveq12d ) AFHGEUAUBQZFH
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+      WDWLXPBCIWBWEWFWGVMAUUCUUDUUMWIWJWD $.
   $}
 
 
@@ -152709,49 +152692,48 @@ Splicing words (substring replacement)
     revccat $p |- ( ( S e. Word A /\ T e. Word A ) ->
         ( reverse ` ( S ++ T ) ) =
           ( ( reverse ` T ) ++ ( reverse ` S ) ) ) $=
-      ( wcel cc0 chash cfv caddc co cfzo wceq syl cc eqtrd oveq2d syl2anr cz c1
-      cmin adantr vx cword wa cconcat creverse wfn ccatcl revcl wrdf ffn revlen
-      wf 4syl ccatlen lencl nn0cnd addcom syl2an mpbid 3syl oveqan12rd cv wo id
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-      LYMUVCXRVRWIXSXTYA $.
+      ( wcel cc0 chash cfv caddc co cfzo wceq cc oveq2d syl2anr eqtrd adantl c1
+      syl cmin ad2antlr vx cword cconcat creverse wfn ccatcl revcl wrdfn revlen
+      wa 3syl ccatlen lencl nn0cnd addcom syl2an 3eqtrd fneq2d mpbid oveqan12rd
+      cv wo cz nn0zd fzospliti simpll simplr cfz fzoval eleq2d biimpa fznn0sub2
+      id eleqtrrd ccatval3 syl3anc oveq1d adantr 1cnd addsubd peano2zm ad2antrr
+      zcnd elfzoelz fveq2d revfv adantll 3eqtr4d cuz wss uzaddcl eqeltrd fzoss2
+      cn0 uzidd sselda syl2an2r biimpar ccatval1 subsub3d zaddcl fzrev2i subidd
+      3eqtr4rd addcl sub32d pncan2 oveq12d eqtr4d 3eltr4d simpl fzosubel3 nn0uz
+      fzoss1 eleq2s sseqtr4d ccatval2 jaodan syldan eqfnfvd ) BAUBZDZCYADZUJZUA
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+      FUVBYDUUQYEUUSYGJUXDUVAXHVJWRAYKYLUVBXQVPWHXRXSXT $.
 
     $( Reversal is an involution on words.  (Contributed by Mario Carneiro,
        1-Oct-2015.) $)
@@ -153251,19 +153233,18 @@ because the border cases ( ` M = N ` , ` -. ( M ..^ N ) C_ ( 0 ..^ L ) `
      16-Oct-2022.) $)
   cshwlen $p |- ( ( W e. Word V /\ N e. ZZ )
                   -> ( # ` ( W cyclShift N ) ) = ( # ` W ) ) $=
-    ( wcel wa ccsh co chash cfv wi c0 3eqtrd fveq2d caddc adantr syl2anc ex syl
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-    cz cop cshword swrdcl pfxcl ccatlen lennncl pm3.21 com24 pm2.43i imp31 cmin
-    cc0 cfz simpl pm3.22 adantl zmodfzp1 cn0 lencl nn0fz0 sylib swrdlen syl3anc
-    cn pfxlen oveq12d nn0cnd zmodcl ancoms npcan syl2an eqtrd expcom pm2.61ine
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-    WBWNWTWCWDWERLWFWG $.
+    ( wcel wa ccsh co chash cfv wceq wi c0 fveq2d caddc adantr ex syl ancoms cc
+    nn0cnd cword cz 0csh0 oveq1 id 3eqtr4a a1d wne cmo cop csubstr cpfx cconcat
+    cshword swrdcl pfxcl ccatlen syl2anc ad2antrr cn lennncl pm3.21 com24 imp31
+    pm2.43i cmin cc0 cfz simpl zmodfzp1 adantl cn0 lencl nn0fz0 swrdlen syl3anc
+    sylib pfxlen sylan2 oveq12d zmodcl npcan syl2an 3eqtrd expcom pm2.61ine
+    eqtrd ) CBUAZDZAUBDZEZCAFGZHIZCHIZJZKCLCLJZWOWKWPWLCHWPLAFGLWLCAUCCLAFUDWPU
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   $( A cyclically shifted word is a function from a half-open range of integers
      of the same length as the word as domain to the set of symbols for the
@@ -154236,7 +154217,7 @@ the symbol at any position is repeated at multiples of L (modulo the
       cats1len $p |- ( # ` T ) = N $=
         ( chash cfv cs1 cconcat co fveq2i c1 caddc cvv wcel eqtri cword ccatlen
         wceq s1cli mp2an s1len oveq12i ) BJKAELZMNZJKZDBUIJFOUJCPQNZDUJAJKZUHJK
-        ZQNZUKARUAZSUHUOSUJUNUCGEUDRAUHUBUEULCUMPQHEUFUGTITT $.
+        ZQNZUKARUAZSUHUOSUJUNUCGEUDRRAUHUBUEULCUMPQHEUFUGTITT $.
     $}
 
     ${
@@ -217415,43 +217396,43 @@ proposition to be be proved (the first four hypotheses tell its values
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     $( Valuation of a pair in a monoid.  (Contributed by Stefan O'Rear,
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@@ -226006,22 +225987,22 @@ operation is a permutation group (group consisting of permutations), see
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   $}
 
   ${
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     $( The inverse of the reverse of a word composed with the word relates to
        the identity.  (This provides an explicit expression for the
@@ -229983,16 +229964,16 @@ U C_ ( T .(+) U ) ) $=
        (Contributed by Mario Carneiro, 1-Oct-2015.) $)
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-          WXPXSVUNUXLTUEUHUWHWOXNXIVIUYSUEWXINUWIUWJXTWXGWXJUYSVLZWXLXSZXSZBCDE
-          FGHIWXJJKLMNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMAUFWIZNVKZVTVKVWJVTVKUWKU
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-          RVAYTAVUTWXDWXRVBYTAVWOWXDWXRVCYTAVXCWXDWXRVDYTAVWJVWLVMWXDWXRVEYTAVW
-          KVWTVMWXDWXRVFYTAUYCUWNWXDWXRVGYTAWXDWXRUWPWXGWXQWXLUWQWXSWXIWXKWXGWX
-          QWXLUWRUWSUWTUXAUXBUXCUXDUXEUXDUXE $.
+          eqtr4i eqtr3d s111 fveq2d efgmnvl s1eqd oveq2d elfzelz zcnd 1cnd df-2
+          addassd eleq2d wfo efgsfo cvv efgrcl foelrn sylancr clt wbr wi simplr
+          wrex wn simprl simprr eqcomd efgredlemd rexlimddv ex sylbid jaod syl5
+          ) KLVJVKVLKLVMZKLVNVOVPZVJVKVLZWCAVQHVKZVQIVKZVMZLKVRAUXEUXJUXGAUXEUX
+          JAUXEUAHVKZKVSVPZUXKKUXKVTVKZWAWBVPZWDVPZUBIVKZLVSVPZUXPLUXPVTVKZWAWB
+          VPZWDVPZVMZAUYAUXKUXPVMZVGAUXOUXKUXTUXPAUXOUXKUXMVSVPZUXKAUXKTUUAUUBZ
+          WEZVLZKVQUXMWFVPZVLZUXMUYGVLZUXOUYCVMAUEUYEUXKUEUYEUUCVKUYEUHUYEUUJUU
+          DZAVQHVTVKZWGVPZUEUAHAHUEWEZVLZUYLUEHWHAHUYMUUEUUFZAHNUUKZVLZHUYMUYOU
+          UGZVLZUOUYQUYSUXHJVLVHWPZHVKUYTVNWIVPZHVKOVKWJVLVHVNUYKWGVPWKBCDEFGJM
+          NOVHQRSHTUCUEUHUIUJUKULUMWLWMWNWOZUEHWQWNAUAVQUYKVNWIVPZWFVPZUYLAVQVU
+          CWGVPZVUDUAVQVUCWRAUAVUCVNWIVPZVUEUSAVUCWSVLZVUFVUEVLAVUGIVTVKZVNWIVP
+          ZWSVLZABCDEFGHIJMNOQRSTUCUEUFUGUHUIUJUKULUMUNUOUPUQURUUHZWTVUCXAWNXBX
+          CAUYKXDVLUYLVUDVMAUYKAUYNUYKXEVLVUBUEHXFWNXHVQUYKXGWNXIXJZXCZVAAUXMVQ
+          VJVKZVLUYIAUXMXEVUNAUYFUXMXEVLVUMUYDUXKXFWNXKXLVQUXMXMWNUYDUXKKUXMUUI
+          XNAUYFUYCUXKVMVUMUYDUXKUULWNXQAUXTUXPUXRVSVPZUXPAUXPUYEVLZLVQUXRWFVPZ
+          VLZUXRVUQVLZUXTVUOVMAUEUYEUXPUYJAVQVUHWGVPZUEUBIAIUYMVLZVUTUEIWHAIUYM
+          UYOAIUYPVLZIUYRVLZUPVVBVVCUXIJVLUYTIVKVUAIVKOVKWJVLVHVNVUHWGVPWKBCDEF
+          GJMNOVHQRSITUCUEUHUIUJUKULUMWLWMWNWOZUEIWQWNAUBVQVUIWFVPZVUTAVQVUIWGV
+          PZVVEUBVQVUIWRAUBVUIVNWIVPZVVFUTAVUJVVGVVFVLAVUGVUJVUKXOVUIXAWNXBXCAV
+          UHXDVLVUTVVEVMAVUHAVVAVUHXEVLVVDUEIXFWNXHVQVUHXGWNXIXJZXCZVBAUXRVUNVL
+          VUSAUXRXEVUNAVUPUXRXEVLVVIUYDUXPXFWNXKXLVQUXRXMWNUYDUXPLUXRUUIXNAVUPV
+          UOUXPVMVVIUYDUXPUULWNXQXPUUMZAUXEYGZUXLUXQUXNUXSWDVVKUXLUXQVMZPPUCVKZ
+          UUNZUDUDUCVKZUUNZVMZVVKUXLVVNWDVPZUXQVVPWDVPZVMZVVLVVQYGZVVKVVTUXNUXS
+          VMZVVKVVRUXNWDVPZVVSUXSWDVPZVMZVVTVWBYGZAVWEUXEAHNVKZINVKZVWCVWDUQAVW
+          GKPUXKOVKVPZUXKKKVVNUUOUUPVPZVWCVEAUXKUEVLZUYHPUYDVLZVWIVWJVMVULVAVCC
+          DEFPMOSTUCKUEUXKUHUIUJUKUUQXNAVWKUYHUYHVVNUYEVLZVWJVWCVMVULVAVAAPVVMU
+          YDVCAVWLVVMUYDVLVCUYDUYDPUCCDTUCUJUURZUUSWNZUVAZVVNUXKKKUEUYGUYGUYEUU
+          TUVBXRAVWHLUDUXPOVKVPZUXPLLVVPUUOUUPVPZVWDVFAUXPUEVLZVURUDUYDVLZVWQVW
+          RVMVVHVBVDCDEFUDMOSTUCLUEUXPUHUIUJUKUUQXNAVWSVURVURVVPUYEVLZVWRVWDVMV
+          VHVBVBAUDVVOUYDVDAVWTVVOUYDVLZVDUYDUYDUDUCVWNUUSWNZUVAZVVPUXPLLUEVUQV
+          UQUYEUUTUVBXRUVCZXSVVKVVRUYEVLZUXNUYEVLZVVSUYEVLZUXSUYEVLZVVRVTVKZVVS
+          VTVKZVMVWEVWFYHVVKUXLUYEVLZVWMVXFAVXLUXEAUYFVXLVUMUYDUXKKUVDWNZXSZAVW
+          MUXEVWPXSZUYDUXLVVNXTYAAVXGUXEAUYFVXGVUMUYDUXKKUXMYBWNZXSVVKUXQUYEVLZ
+          VXAVXHAVXQUXEAVUPVXQVVIUYDUXPLUVDWNZXSZAVXAUXEVXDXSZUYDUXQVVPXTYAAVXI
+          UXEAVUPVXIVVIUYDUXPLUXRYBWNZXSVVKUXLVTVKZVVNVTVKZVOVPZUXQVTVKZVVPVTVK
+          ZVOVPZVXJVXKVVKVYBVYEVYCVYFVOAVYBVYEVMZUXEAVYBKVYELAUYFUYHVYBKVMVUMVA
+          UYDUXKKUVEYAZAVUPVURVYELVMVVIVBUYDUXPLUVEYAZXPUVFZVYCVYFVMVVKVYCYIVYF
+          PVVMUVGUDVVOUVGUVKYCYDVVKVXLVWMVXJVYDVMVXNVXOUYDUYDUXLVVNYEYAVVKVXQVX
+          AVXKVYGVMVXSVXTUYDUYDUXQVVPYEYAYFVVRUXNVVSUXSUYDYJYKYLZWTVVKVXLVWMVXQ
+          VXAVYHVVTVWAYHVXNVXOVXSVXTVYKUXLVVNUXQVVPUYDYJYKYLWTVVKVVTVWBVYLXOYDU
+          VHUVIUXGKUXFVMZKUXFVNVOVPZVJVKZVLZWCAUXJUXFKVRAVYMUXJVYPAVYMUXJAVYMUY
+          AVVJAVYMYGZUXOUXQUDYRZWDVPZUXNWDVPZUXQVYRUXNWDVPZWDVPZUXTVYQUXLVYSUXN
+          WDVYQUXLVYSVMZPYRZVVOYRZVMZVYQUXLWUDWDVPZVYSWUEWDVPZVMZWUCWUFYGZVYQWU
+          IVVMYRZUXNWDVPZUXSVMZVYQWUGWULWDVPZWUHUXSWDVPZVMZWUIWUMYGZAWUPVYMAWUG
+          WUKWDVPZUXNWDVPZWUNWUOAWUGUYEVLZWUKUYEVLZVXGWUSWUNVMAVXLWUDUYEVLZWUTV
+          XMAPUYDVCYMZUYDUXLWUDXTYAZAVVMUYDVWOYMZVXPUYDWUGWUKUXNYNXNAVWCVWDWUSW
+          UOVXEAWURVVRUXNWDAWURUXLWUDWUKWDVPZWDVPZVVRAVXLWVBWVAWURWVGVMVXMWVCWV
+          EUYDUXLWUDWUKYNXNVVNWVFUXLWDPVVMUVJYOYPYQAWUHVVSUXSWDAWUHUXQVYRWUEWDV
+          PZWDVPZVVSAVXQVYRUYEVLZWUEUYEVLZWUHWVIVMVXRAUDUYDVDYMZAVVOUYDVXCYMZUY
+          DUXQVYRWUEYNXNVVPWVHUXQWDUDVVOUVJYOYPYQYFUVLXSVYQWUTWULUYEVLZWUHUYEVL
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+          GVYMVXPXSZUYDWUKUXNXTYAVYQVYSUYEVLZWVKWVOAWVSVYMAVXQWVJWVSVXRWVLUYDUX
+          QVYRXTYAXSZAWVKVYMWVMXSZUYDVYSWUEXTYAAVXIVYMVYAXSVYQVYBWUDVTVKZVOVPZV
+          YSVTVKZWUEVTVKZVOVPZWVPWVQVYQVYBWWDWWBWWEVOAVYBWWDVMZVYMAVYBKWWDUXFVY
+          IAWWDVYEVYRVTVKZVOVPZUXFAVXQWVJWWDWWIVMVXRWVLUYDUYDUXQVYRYEYAAVYELWWH
+          VNVOVYJWWHVNVMAUDYSYCYDXQXPUVFZWWBWWEVMVYQWWBVNWWEPYSVVOYSUVKYCYDVYQV
+          XLWVBWVPWWCVMAVXLVYMVXMXSZAWVBVYMWVCXSZUYDUYDUXLWUDYEYAVYQWVSWVKWVQWW
+          FVMWVTWWAUYDUYDVYSWUEYEYAYFWUGWULWUHUXSUYDYJYKYLZWTVYQVXLWVBWVSWVKWWG
+          WUIWUJYHWWKWWLWVTWWAWWJUXLWUDVYSWUEUYDYJYKYLZWTYQVYQVXQWVJVXGVYTWUBVM
+          AVXQVYMVXRXSAWVJVYMWVLXSWVRUYDUXQVYRUXNYNXNVYQWUAUXSUXQWDVYQWULWUAUXS
+          VYQWUKVYRUXNWDVYQVVMUDVYQVVMVVOUCVKZUDVYQPVVOUCVYQWUFPVVOVMZVYQWUCWUF
+          WWNXOAWUFWWPYHZVYMAVWLVXBWWQVCVXCUYDPVVOUVMYAXSYLUVNAWWOUDVMZVYMAVWTW
+          WRVDCDUDTUCUJUVOWNXSXQUVPYQVYQWUIWUMWWMXOUVLUVQXRUVHUVIAVYPKLYIVOVPZV
+          JVKZVLZUXJAVYOWWTKAVYNWWSVJAVYNLVNVNVOVPZVOVPWWSALVNVNALAVURLXDVLVBLV
+          QUXRUVRWNUVSAUVTZWXCUWBYIWXBLVOUWAYOYPUVNUWCAWXAUXJAWXAYGZUXQUXKWWSUX
+          MWAWBVPZWDVPZVIWPZNVKZVMZUXJVIUYPAWXIVIUYPUWNZWXAAUYPUENUWDWXFUEVLWXJ
+          BCDEFGJMNOQRSTUCUEUHUIUJUKULUMUWEAWXFUYEUEAVXQWXEUYEVLZWXFUYEVLVXRAUY
+          FWXKVUMUYDUXKWWSUXMYBWNUYDUXQWXEXTYAATUWFVLZUEUYEVMZAVWKWXLWXMYGVULUX
+          KTUEUHUWGWNXOXIVIUYPUEWXFNUWHUWIXSWXDWXGUYPVLZWXIYGZYGZBCDEFGHIWXGJKL
+          MNOPQRSTUAUBUCUDUEUFUGUHUIUJUKULUMAUFWPZNVKZVTVKVWGVTVKUWJUWKWXRUGWPZ
+          NVKVMVQWXQVKVQWXSVKVMUWLUWLUGUYPWKUFUYPWKWXAWXOUNYTAUYQWXAWXOUOYTAVVB
+          WXAWXOUPYTAVWGVWHVMWXAWXOUQYTAUXJUWOWXAWXOURYTUSUTAUYHWXAWXOVAYTAVURW
+          XAWXOVBYTAVWLWXAWXOVCYTAVWTWXAWXOVDYTAVWGVWIVMWXAWXOVEYTAVWHVWQVMWXAW
+          XOVFYTAUYBUWOWXAWXOVGYTAWXAWXOUWMWXDWXNWXIUWPWXPWXFWXHWXDWXNWXIUWQUWR
+          UWSUWTUXAUXBUXCUXDUXCUXD $.
 
         $( The reduced word that forms the base of the sequence in ~ efgsval is
            uniquely determined, given the ending representation.  (Contributed
@@ -230891,28 +230871,28 @@ an extension of the previous (inserting an element and its inverse at
         ZUQZVWIVXEVYOWSVXBVWKVXDXFYIUTZUQZVYPVXDYIUTZUQVYRVWKVXCVYSUQVVOYJUQZVY
         TVWKVXCYJVYSVWKVVBVXCYJUQVYAVURGYKYBZYLYMVWHWUBVUQVWIVVOVVSUVEWCZVVOXFV
         XCYNWQVWKVYPUYSXGUTZHXGUTZXTVLZWUAVWKVUTVVAVYPWUGWSVUQVUTVWJVVGXSVYGVUR
-        UYSHYOWQVWKWUEWUAUQWUFYJUQZWUGWUAUQVWKWUEVXCVVSXTVLZWUAVWKVVBVVCWUEWUIW
-        SVYAVYBVURGUXQYOWQVWKVVSVXCXTVLZVVOVXCXTVLZYIUTZWUIWUAVWKVVSVVOYIUTUQZV
-        XCUVFUQWUJWULUQVWHWUMVUQVWIVVOXFVVSUVGWCVWKVXCWUCUVJVXCVVOVVSUVHWQVWKVV
-        SVXCVWKVVSVWKVVCVVSYJUQVYBVURUXQYKYBZYPVWKVXCWUCYPZYQVWKWUKVXDYIVWKVVOV
-        XCVWKVVOWUDYPWUOYQUVKUVLYRVWKVVAWUHVYGVURHYKYBWUFVXDWUEYNWQYRVXDXFVYPUV
-        MUVNZVWRBCDEVVPJLQRTVXDUAUYTUBUCUDUEYEYDVWKVXGYSVXNVXIUYTVXDVXDVURVYDYS
-        VUSUQVWKVURUWHVOVWKVXQVVAVXNVUSUQVYFVYGVURVXKHWPWQVYEVWKVXGYSVKVLZVXNVK
-        VLVXGVXNVKVLZVXGVXKVKVLZHVKVLZUYTVWKWUQVXGVXNVKVWKVXRWUQVXGWSVYDVURVXGU
-        VOYBYFVWKVXRVXQVVAWUTWURWSVYDVYFVYGVURVXGVXKHYCYDVWKWUSUYSHVKVWKWUSGVXF
-        VXKVKVLZVKVLZUYSVWKVVBVXTVXQWUSWVBWSVYAVYCVYFVURGVXFVXKYCYDVWKWVAUXQGVK
-        VWKWVAUXQVVSYAVLZUXQVWKVVCVWHVVSVVTUQZWVAWVCWSVYBVWQVWKVVSVYSUQWVDVWKVV
-        SYJVYSWUNYLYMXFVVSUVPYBVURUXQVVOVVSUVQYDVWKVVCWVCUXQWSVYBVURUXQUVRYBYGY
-        HYGYFUVSVWKVXGXGUTZVXCVXFXGUTZXTVLZVXDVWKVVBVXTWVEWVGWSVYAVYCVURGVXFYOW
-        QVWKWVFVVOVXCXTVWKVVCVWHWVFVVOWSVYBVWQVURUXQVVOUVTWQYHUWAVWKVXDYSXGUTZX
-        TVLVXDXFXTVLVXDWVHXFVXDXTUWIUWBVWKVXDVWKVXDVWKVXCVVOWUCWUDUWCYPUWDUWEUW
-        FYGUWGVWKVWTVYQVURWGZXIZVYRVWIVXEVXAUQVWKVWSWVIUAVWTXMZWVJVXBVWSVWTUMUP
-        VYQVURUYTVWFXLVLXOWSWVKBCDEJLQRTUAUYTUMUPUBUCUDUEXPXBWVIUAVWTXQXRWUPVWR
-        VYQVURVXDVVPVWTUWJYDYRBCDEUYTVWNJLQRTUAUBUCUDUEUWKWQABCDEFGHIJKLMNOPQRS
-        TUAUXQVVQUBUCUDUEUFUGUHVPWBVWBUXQVVMSVMVVRVWLVVMUXQSUMYTUJYTUWRVVMVVQUX
-        QSUWLUWMUWNUWOUWPUWQUWSUXNSWTUQZUYFUYGUYJURZXNUXNUYGUAWTUQWVLVVLUAVUSWI
-        UBUWTUASWTUXAUXBUYCWVMUISWTUXOSWSZUXPUYGUYBUYJUAUXOSUXCWVNUYAUYIUJUAWVN
-        UXTUYHUXSUXOSUXQUXHUXDUXEUXFUXIYBUXGSUYDUXJYBUXK $.
+        VURUYSHYOWQVWKWUEWUAUQWUFYJUQZWUGWUAUQVWKWUEVXCVVSXTVLZWUAVWKVVBVVCWUEW
+        UIWSVYAVYBVURVURGUXQYOWQVWKVVSVXCXTVLZVVOVXCXTVLZYIUTZWUIWUAVWKVVSVVOYI
+        UTUQZVXCUVFUQWUJWULUQVWHWUMVUQVWIVVOXFVVSUVGWCVWKVXCWUCUVJVXCVVOVVSUVHW
+        QVWKVVSVXCVWKVVSVWKVVCVVSYJUQVYBVURUXQYKYBZYPVWKVXCWUCYPZYQVWKWUKVXDYIV
+        WKVVOVXCVWKVVOWUDYPWUOYQUVKUVLYRVWKVVAWUHVYGVURHYKYBWUFVXDWUEYNWQYRVXDX
+        FVYPUVMUVNZVWRBCDEVVPJLQRTVXDUAUYTUBUCUDUEYEYDVWKVXGYSVXNVXIUYTVXDVXDVU
+        RVYDYSVUSUQVWKVURUWHVOVWKVXQVVAVXNVUSUQVYFVYGVURVXKHWPWQVYEVWKVXGYSVKVL
+        ZVXNVKVLVXGVXNVKVLZVXGVXKVKVLZHVKVLZUYTVWKWUQVXGVXNVKVWKVXRWUQVXGWSVYDV
+        URVXGUVOYBYFVWKVXRVXQVVAWUTWURWSVYDVYFVYGVURVXGVXKHYCYDVWKWUSUYSHVKVWKW
+        USGVXFVXKVKVLZVKVLZUYSVWKVVBVXTVXQWUSWVBWSVYAVYCVYFVURGVXFVXKYCYDVWKWVA
+        UXQGVKVWKWVAUXQVVSYAVLZUXQVWKVVCVWHVVSVVTUQZWVAWVCWSVYBVWQVWKVVSVYSUQWV
+        DVWKVVSYJVYSWUNYLYMXFVVSUVPYBVURUXQVVOVVSUVQYDVWKVVCWVCUXQWSVYBVURUXQUV
+        RYBYGYHYGYFUVSVWKVXGXGUTZVXCVXFXGUTZXTVLZVXDVWKVVBVXTWVEWVGWSVYAVYCVURV
+        URGVXFYOWQVWKWVFVVOVXCXTVWKVVCVWHWVFVVOWSVYBVWQVURUXQVVOUVTWQYHUWAVWKVX
+        DYSXGUTZXTVLVXDXFXTVLVXDWVHXFVXDXTUWIUWBVWKVXDVWKVXDVWKVXCVVOWUCWUDUWCY
+        PUWDUWEUWFYGUWGVWKVWTVYQVURWGZXIZVYRVWIVXEVXAUQVWKVWSWVIUAVWTXMZWVJVXBV
+        WSVWTUMUPVYQVURUYTVWFXLVLXOWSWVKBCDEJLQRTUAUYTUMUPUBUCUDUEXPXBWVIUAVWTX
+        QXRWUPVWRVYQVURVXDVVPVWTUWJYDYRBCDEUYTVWNJLQRTUAUBUCUDUEUWKWQABCDEFGHIJ
+        KLMNOPQRSTUAUXQVVQUBUCUDUEUFUGUHVPWBVWBUXQVVMSVMVVRVWLVVMUXQSUMYTUJYTUW
+        RVVMVVQUXQSUWLUWMUWNUWOUWPUWQUWSUXNSWTUQZUYFUYGUYJURZXNUXNUYGUAWTUQWVLV
+        VLUAVUSWIUBUWTUASWTUXAUXBUYCWVMUISWTUXOSWSZUXPUYGUYBUYJUAUXOSUXCWVNUYAU
+        YIUJUAWVNUXTUYHUXSUXOSUXQUXHUXDUXEUXFUXIYBUXGSUYDUXJYBUXK $.
     $}
 
     $( Two extension sequences have related endpoints iff they have the same
@@ -237927,55 +237907,54 @@ factorization into prime power factors (even if the exponents are
           pgpfaclem1 $p |- ( ph -> E. s e. Word C
             ( G dom DProd s /\ ( G DProd s ) = U ) ) $=
             ( cword wcel cdprd cdm wbr co wceq cv wrex csn cfv csubg cress ccyg
-            cpgp crn cin wss cbs cmre cgrp cacs subggrp syl eqid subgacs acsmre
-            wa 3syl subgbas eleqtrd mrcsncl syl2anc subsubg mpbid simpld oveq1i
-            wb simprd ressabs syl5eq cycsubgcyg2 eqeltrrd cprime pgpprm subgpgp
-            brelrng syl3anc elind oveq2 eleq1d elrab2 sylanbrc cc0 chash c0g wf
-            cfzo wrdf sylancl c1 caddc cn0 cun syl6eq oveq2d eqtrd cres cop wfn
-            ffn breqtrrd cvv fvex cn 3eqtrd sneqd dprdsubg ccntz ssrab3 fzodisj
-            cats1cld fss c0 cz lencl nn0zd fzosn ineq2d syl5reqr cconcat fveq2i
-            cs1 s1cld ccatlen s1len oveq2i cuz nn0uz fzosplitsn cats1un reseq1d
-            syl6eleq wn fzonel fsnunres dprdsn sylancr ssun2 snss mpbir fnressn
-            eleqtrrid fveq1i nn0cnd addid2d eqcomd fveq2d 1nn a1i lbfzo0 sylibr
-            eqeltrid ccatval3 s1fv mp1i opeq2d ablcntzd ineq12d syl6eqr 3eqtr4d
-            incom subg0 dmdprdsplit2 clsm dprdsplit cabl lsmcom subgss sseqtr4d
-            oveq12d subglsm breq2 eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHV
-            EVFZVGZKHVEVHZIVIZKRVJZUXLVGZKUXPVEVHZIVIZWJZRUXJVKADGHPVLZMVMZVBUS
-            AUYBKVNVMZVDZKUYBVOVHZVPVQVRZVSZVDZUYBDVDZAUYDUYBIVTZAUYBLVNVMZVDZU
-            YDUYJWJZAUYKLWAVMZWBVMVDZPUYNVDZUYLALWCVDZUYKUYNWDVMVDUYOAIUYCVDZUY
-            QUEIKLUGWEWFZUYNLUYNWGZWHUYKUYNWIWKAPIUYNUNAUYRIUYNVIUEIKLUGWLWFZWM
-            ZUYKPMUYNUHWNWOAUYRUYLUYMWTUEUYBIKLUGWPWFWQZWRZAVPUYFUYEALUYBVOVHZU
-            YEVPAVUEKIVOVHZUYBVOVHZUYELVUFUYBVOUGWSAUYRUYJVUGUYEVIUEAUYDUYJVUCX
-            AZIUYBKUYCXBWOXCAUYQUYPVUEVPVDUYSVUBPUYNLMUYTUHXDWOXEZAEXFVDZUYEVPV
-            DEUYEVQVGZUYEUYFVDAEKVQVGZVUJUCEKXGWFVUIAVULUYDVUKUCVUDEUYBKXHWOEUY
-            EVQXFVPXIXJXKKSVJZVOVHZUYGVDZUYHSUYBUYCDVUMUYBVIVUNUYEUYGVUMUYBKVOX
-            LXMUAXNXOZUUDZAXPGXQVMZXTVHZVURVLZHKXPHXQVMZXTVHZKXRVMZKUUAVMZAVVBD
-            HXSZDUYCVTVVBUYCHXSAUXKVVEVUQDHYAZWFVUOSUYCDUAUUBVVBDUYCHUUEYBZAUUF
-            VUSVURVURYCYDVHZXTVHZVSVUSVUTVSXPVURVVHUUCAVVIVUTVUSAVURUUGVDVVIVUT
-            VIAVURAGUXJVDZVURYEVDUSDGUUHWFZUUIVURUUJWFUUKUULZAVVBXPVVHXTVHZVUSV
-            UTYFZAVVAVVHXPXTAVVAVURUYBUUOZXQVMZYDVHZVVHAVVAGVVOUUMVHZXQVMZVVQHV
-            VRXQVBUUNAVVJVVOUXJVDZVVSVVQVIUSAUYBDVUPUUPZDGVVOUUQWOXCVVPYCVURYDU
-            YBUURZUUSYGYHAVURXPUUTVMZVDVVMVVNVIAVURYEVWCVVKUVAUVEXPVURUVBWFYIZV
-            VDWGZVVCWGZAKGHVUSYJZUXLUTAVWGGVURUYBYKZVLZYFZVUSYJZGAHVWJVUSAHVVRV
-            WJVBAVVJUYIVVRVWJVIUSVUPGUYBDUVCWOXCUVDAGVUSYLZVURVUSVDUVFVWKGVIAVV
-            JVUSDGXSVWLUSDGYAVUSDGYMWKXPVURUVGVUSGVURUYBUVHYBYIZYNZAKVWIHVUTYJZ
-            UXLAKVWIUXLVGZKVWIVEVHZUYBVIZAVURYOVDUYDVWPVWRWJGXQYPZVUDVURUYBKYOU
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-            VOVVBVURHUVNWOAVXBVWHAVXAUYBVURAVXAXPVURYDVHZVVRVMZXPVVOVMZUYBAVXAV
-            URVVRVMVXFVURHVVRVBUVPAVURVXEVVRAVXEVURAVURAVURVVKUVQUVRUVSUVTXCAVV
-            JVVTXPXPVVPXTVHVDZVXFVXGVIUSVWAAVVPYQVDVXHAVVPYCYQVWBYCYQVDAUWAUWBU
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-            LVGVXLUYCVDVXJVWOKYTWFUWJAUYBOVSZQVLVXKVXLVSZVVCVLUQAVXOOUYBVSVXNAV
-            XKOVXLUYBAVXKKGVEVHOAVWGGKVEVWMYHVAYIZAVXLVWQUYBAVWOVWIKVEVXIYHAVWP
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-            WFUKUWLYSUWMUWPZAUXNUYBOKUWQVMZVHZUYBOFVHZIAUXNVXKVXLVXTVHOUYBVXTVH
-            ZVYAAVUSVUTVXTHKVVBVVGVVLVWDVXTWGZVXSUWRAVXKOVXLUYBVXTVXPVXQUXCAKUW
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-            UGVYDULUXDXJURYRUXTUXMUXOWJRHUXJUXPHVIZUXQUXMUXSUXOUXPHKUXLUXEVYEUX
-            RUXNIUXPHKVEXLUXFUXGUXHUXI $.
+            wa cpgp crn cin wss cbs cmre cgrp cacs subggrp eqid subgacs acsmred
+            syl subgbas eleqtrd mrcsncl syl2anc wb subsubg simpld oveq1i simprd
+            mpbid ressabs syl5eq eqeltrrd cprime pgpprm subgpgp brelrng syl3anc
+            cycsubgcyg2 elind oveq2 eleq1d sylanbrc cats1cld cc0 chash cfzo c0g
+            elrab2 wf sylancl c1 caddc cn0 cun syl6eq oveq2d eqtrd cres cop wfn
+            wrdfn breqtrrd cvv fvex mpbir cn 3eqtrd sneqd dprdsubg ccntz ssrab3
+            wrdf fss c0 fzodisj cz lencl nn0zd fzosn ineq2d syl5reqr cs1 fveq2i
+            cconcat s1cld ccatlen s1len oveq2i nn0uz fzosplitsn cats1un reseq1d
+            cuz syl6eleq wn fzonel fsnunres dprdsn sylancr ssun2 snss eleqtrrid
+            fnressn nn0cnd addid2d fveq2d fveq1i syl6reqr eqeltri ccatval3 s1fv
+            1nn lbfzo0 mp1i opeq2d ablcntzd ineq12d incom subg0 syl6eqr 3eqtr4d
+            a1i dmdprdsplit2 clsm dprdsplit oveq12d cabl lsmcom subgss sseqtr4d
+            subglsm breq2 eqeq1d anbi12d rspcev syl12anc ) AHDVCZVDZKHVEVFZVGZK
+            HVEVHZIVIZKRVJZUXJVGZKUXNVEVHZIVIZVQZRUXHVKADGHPVLZMVMZVBUSAUXTKVNV
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+            XOXQAVURVUOUXTUUMZXPVMZYCVHZVVDAVURGVVKUUOVHZXPVMZVVMHVVNXPVBUUNAVV
+            FVVKUXHVDZVVOVVMVIUSAUXTDVUMUUPZDDGVVKUUQWNXBVVLYBVUOYCUXTUURZUUSYF
+            YGAVUOXOUVDVMZVDVVIVVJVIAVUOYDVVSVVGUUTUVEXOVUOUVAWJYHZVVAWGZVUTWGZ
+            AKGHVUPYIZUXJUTAVWCGVUOUXTYJZVLZYEZVUPYIZGAHVWFVUPAHVVNVWFVBAVVFUYG
+            VVNVWFVIUSVUMGUXTDUVBWNXBUVCAGVUPYKZVUOVUPVDUVFVWGGVIAVVFVWHUSDGYLW
+            JXOVUOUVGVUPGVUOUXTUVHYAYHZYMZAKVWEHVUQYIZUXJAKVWEUXJVGZKVWEVEVHZUX
+            TVIZAVUOYNVDUYBVWLVWNVQGXPYOZVUAVUOUXTKYNUVIUVJZWQAVWKVUOVUOHVMZYJZ
+            VLZVWEAHVUSYKZVUOVUSVDVWKVWSVIAUXIVWTVUNDHYLWJAVUOVVJVUSVUOVVJVDVUQ
+            VVJWAVUQVUPUVKVUOVVJVWOUVLYPVVTUVMVUSVUOHUVNWNAVWRVWDAVWQUXTVUOAVWQ
+            XOVUOYCVHZVVNVMZXOVVKVMZUXTAVXBVUOVVNVMVWQAVXAVUOVVNAVUOAVUOVVGUVOU
+            VPUVQVUOHVVNVBUVRUVSAVVFVVPXOXOVVLXQVHVDZVXBVXCVIUSVVQVXDAVXDVVLYQV
+            DVVLYBYQVVRUWCUVTVVLUWDYPUWMDGVVKXOUWAXHUXTYNVDVXCUXTVIAUXSMYOUXTYN
+            UWBUWEYRUWFYSYHZYMZAKVWCVEVHZKVWKVEVHZKVVAVWAUBAKVWCUXJVGVXGUYAVDVW
+            JVWCKYTWJZAKVWKUXJVGVXHUYAVDVXFVWKKYTWJUWGAUXTOVTZQVLVXGVXHVTZVUTVL
+            UQAVXKOUXTVTVXJAVXGOVXHUXTAVXGKGVEVHOAVWCGKVEVWIYGVAYHZAVXHVWMUXTAV
+            WKVWEKVEVXEYGAVWLVWNVWPWSYHZUWHOUXTUWIYFAVUTQAVUTLXRVMZQAUYOVUTVXNV
+            IUEIKLVUTUGVWBUWJWJUKUWKYSUWLUWNZAUXLUXTOKUWOVMZVHZUXTOFVHZIAUXLVXG
+            VXHVXPVHOUXTVXPVHZVXQAVUPVUQVXPHKVUSVVCVVHVVTVXPWGZVXOUWPAVXGOVXHUX
+            TVXPVXLVXMUWQAKUWRVDOUYAVDUYBVXSVXQVIUBAVXGOUYAVXLVXIXCVUAVXPOUXTKV
+            XTUWSXHYRAUYOUYHOIWAVXQVXRVIUEVUEAOUYLIAOUYIVDOUYLWAUPUYLOLUYQUWTWJ
+            UYRUXAFVXPIUXTOKLUGVXTULUXBXHURYRUXRUXKUXMVQRHUXHUXNHVIZUXOUXKUXQUX
+            MUXNHKUXJUXCVYAUXPUXLIUXNHKVEXKUXDUXEUXFUXG $.
         $}
 
         $( Lemma for ~ pgpfac .  (Contributed by Mario Carneiro, 27-Apr-2016.)
@@ -263024,30 +263003,30 @@ According to Wikipedia ("Integer", 25-May-2019,
       cneg cid cdif cfn crab wfn eqid psgnfn fndm ssrab3 ressbas2 cnmsgnbas cvv
       fvexi ressplusg prex cmgp cnfldmul mgpplusg csubg cgrp psgndmsubg subggrp
       ccnfld syl cnmsgngrp a1i wral wfun fnfun funfn mpbi cgsu chash cexp cpmtr
-      wf crn cword wrex psgnvali wi cz lencl nn0zd m1expcl2 prcom adantl eleq1a
-      syl6eleq adantld rexlimdva syl5 ralrimiv ffnfv sylanbrc cconcat psgnvalii
-      ccatcl sylan2 cmnd symggrp grpmnd symgtrf sswrd sseli syl3an 3expb fveq2d
-      gsumccat caddc ccatlen oveq2d cc neg1cn ad2antll ad2antrl expaddd 3eqtr3d
-      cn0 eqtrd wb oveq12 eqeqan12d syl5ibrcom rexlimdvva anim12i reeanv sylibr
-      an4s impel isghmd ) AFNZKUABUBOZPDCEEUCZUDUDUIZUEZUUCBUFOZUGUUCDUFOQKUHZU
-      JUKUCULNZKUUFUUCEUUHKUUFUMZUNZUUCUUIQUUFAUUIBEKGUUFUOZUUIUOHUPZUUIEUQRURU
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-      XEUXKUUDWCUWPUXEUXKQUUAUVHUVAUVSXTWPYAUWQUUDUVDUWCUUDYBNUWQYCVOUWOUWCYHNU
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-      WETUWJUWSUWKUWTUXLUWIUWREUUGUVBUVRUVTUUBYKXQUUQUVEUWBUWDPYKYLYRYMYNUWNUVJ
-      UWFMUVIWHZTUWHUUTUVJUWMUXMUVLMAUVRUVHBEGUVKHWIYOUVGUWFLMUVIUVIYPYQYSYT $.
+      wf crn cword wrex psgnvali wi cz lencl nn0zd m1expcl2 prcom syl6eleq 3syl
+      eleq1a adantld rexlimiv ralrimiv sylanbrc cconcat ccatcl psgnvalii sylan2
+      syl5 ffnfv cmnd symggrp grpmnd symgtrf sswrd sseli gsumccat syl3an fveq2d
+      3expb caddc ccatlen adantl oveq2d cc neg1cn cn0 ad2antll ad2antrl expaddd
+      3eqtr3d oveq12 eqeqan12d an4s syl5ibrcom rexlimdvva anim12i reeanv sylibr
+      eqtrd wb impel isghmd ) AFNZKUABUBOZPDCEEUCZUDUDUIZUEZUUDBUFOZUGUUDDUFOQK
+      UHZUJUKUCULNZKUUGUUDEUUIKUUGUMZUNZUUDUUJQUUGAUUJBEKGUUGUOZUUJUOHUPZUUJEUQ
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+      XFUWPUXBUWQEUUBUVNUWNUXBUWQQZUUBBXINZUVNUVBUUGWGZNUWNUVRUXHNUXFUUBBVINUXG
+      ABFGXJBXKVMUVJUXHUVBUVIUUGUGUVJUXHUGUUGAUVIBUVLGUULXLUVIUUGXMRZXNUVJUXHUV
+      RUXIXNUUGUUCBUVBUVRUULUUOXOXPXRXQUWPUXEUUEUVEUWBXSSZWCSUWSUWPUXDUXJUUEWCU
+      WOUXDUXJQUUBUVIUVIUVBUVRXTYAYBUWPUUEUVEUWBUUEYCNUWPYDVOUWNUWBYENUUBUVNUVI
+      UVRWLYFUVNUVEYENUUBUWNUVPYGYHYRYIUVDUVTUVGUWDUWKUWTYSUVDUVTTZUVGUWDTUWIUW
+      RUWJUWSUXKUWHUWQEUUHUVCUVQUVSUUCYJXQUURUVFUWAUWCPYJYKYLYMYNUWMUVKUWEMUVJW
+      HZTUWGUVAUVKUWLUXLUVMMAUVQUVIBEGUVLHWIYOUVHUWELMUVJUVJYPYQYTUUA $.
   $}
 
   ${
@@ -408129,19 +408108,17 @@ in common (for j=1, ... , k).  In contrast to the definition in Aksoy et
   $( The number of vertices in a walk equals the length of the walk after it is
      "closed" (i.e. enhanced by an edge from its last vertex to its first
      vertex).  (Contributed by Alexander van der Vekens, 29-Jun-2018.)
-     (Revised by AV, 2-May-2021.) $)
-  wlklenvclwlk $p |- ( ( W e. Word ( Vtx ` G ) /\ 1 <_ ( # ` W ) )
+     (Revised by AV, 2-May-2021.)  (Revised by JJ, 14-Jan-2024.) $)
+  wlklenvclwlk $p |- ( W e. Word ( Vtx ` G )
                    -> ( <. F , ( W ++ <" ( W ` 0 ) "> ) >. e. ( Walks ` G )
                         -> ( # ` F ) = ( # ` W ) ) ) $=
-    ( cc0 cfv cs1 cconcat co wcel c1 chash wbr wa wceq wi caddc cc nn0cn adantr
-    cn0 cop cwlks cword cle df-br wlklenvp1 wlkcl wrdsymb1 s1cld ccatlen syldan
-    cvtx s1len a1i oveq2d eqtrd eqeq1d lencl adantl 1cnd addcan2d biimpd syl5bi
-    eqcom ex com23 syl sylbid com3l sylc sylbir com12 ) ACDCEZFZGHZUABUBEZIZCBU
-    LEZUCZIZJCKEZUDLZMZAKEZWANZVQAVOVPLZWCWEOZAVOVPUEWFVOKEZWDJPHZNZWDTIZWGVOAB
-    UFVOABUGWCWJWKWEWCWJWAJPHZWINZWKWEOZWCWHWLWIWCWHWAVNKEZPHZWLVTWBVNVSIWHWPNW
-    CVMVRVRCUHUIVRCVNUJUKWCWOJWAPWOJNWCVMUMUNUOUPUQVTWMWNOZWBVTWATIZWQVRCURWRWK
-    WMWEWRWKWMWEOWMWIWLNZWRWKMZWEWLWIVDWTWSWEWTWDWAJWKWDQIWRWDRUSWRWAQIWKWARSWT
-    UTVAVBVCVEVFVGSVHVIVJVKVL $.
+    ( cc0 cfv cs1 cconcat co cop cwlks wcel chash cn0 c1 caddc wceq cvtx adantr
+    wa cc cword wbr df-br wlkcl wlklenvp1 jca sylbir wb ccatws1len eqeq1d eqcom
+    syl6bb nn0cn adantl lencl nn0cnd 1cnd addcan2d biimpd sylbid expimpd syl5 )
+    ACDCEZFGHZIBJEZKZALEZMKZVDLEZVGNOHZPZSZCBQEZUAKZVGCLEZPZVFAVDVEUBZVLAVDVEUC
+    VQVHVKVDABUDVDABUEUFUGVNVHVKVPVNVHSZVKVJVONOHZPZVPVNVKVTUHVHVNVKVSVJPVTVNVI
+    VSVJVMCVCUIUJVSVJUKULRVRVTVPVRVGVONVHVGTKVNVGUMUNVNVOTKVHVNVOVMCUOUPRVRUQUR
+    USUTVAVB $.
 
   ${
     $d A a b f g p $.  $d B a b f g p $.  $d G a b f g p $.  $d V f g p $.
@@ -411919,47 +411896,46 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1)
       ( vi co wcel cfv c1 wa wi c0 cc0 cmin cfzo wceq cwwlksn cpr cconcat caddc
       clsw cs1 cvv cn0 cword w3a wwlknbp wne cv chash wral wwlknp cop csn simp1
       cun simprl cats1un syl2an opex snnz neii intnan df-ne un00 xchbinxr mpbir
-      wn eqnetrd s1cl ad2antrl ccatcl simplrl simpll adantr fzossfzop1 ad2antlr
-      a1i sseld imp wb oveq2 eleq2d adantl mpbird ccats1val1 syl3anc fzonn0p1p1
-      eqcomd eleqtrrd exp41 impcom eleq1d ralbidva biimpd ex com23 3impia oveq1
-      preq12d ad2antll nn0cn ax-1cn pncan sylancl eqtrd fveq2d fzonn0p1 3eqtr4d
-      cc lsw simpr ccats1val2 biimpcd exp4c com12 3adant3 fvoveq1 ralsng ralunb
-      fveq2 sylanbrc cfz cuz elnn0uz eluzfz2 sylbi fzelp1 fzosplit 3syl cz nn0z
-      fzosn syl raleqdv oveq1d uneq2d ccatlen simpl2 s1len nn0cnd 3eqtrd oveq2d
-      oveq12d peano2nn0 3jca expd cwwlks iswwlksn iswwlks anbi1i syl6bb sylibrd
-      jca mpcom 3impib ) BEDUAJKZAFKZBUELZAUBZCKZBAUFZUCJZEMUDJZDUAJKZDUGKZEUHK
-      ZBFUIZKZUJUVAUVBUVENZUVIOZDEFBGUKUVJUVKUVAUVOOUVMUVJUVKNZUVAUVOUVPUVANUVN
-      UVGPULZUVGUVLKZIUMZUVGLZUVSMUDJZUVGLZUBZCKZIQUVGUNLZMRJZSJZUOZUJZUWEUVHMU
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-      UWKUXDUVQUVRUWHUXDUVGBUWNAUQZURZUTZPUXBUVMUVBUVGUXGTUXCUVMUWOUXAUSZUVKUVB
-      UVEVABAFVBVCUXGPULZUXDUXIBPTZUXFPTZNZVLUXKUXJUXFPUXEUWNAVDVEVFVGUXIUXGPTU
-      XLUXGPVHBUXFVIVJVKWBVMUXBUVMUVFUVLKZUVRUXCUXHUVBUXMUVKUVEAFVNVOZFBUVFVPVC
-      UXDUWHUWDIQUVHSJZUOZUXDUXPUWDIUWTEURZUTZUOZUXDUWDIUWTUOZUWDIUXQUOZUXSUXBU
-      XCUXTUVMUWOUXAUXCUXTOUVMUWONZUXCUXAUXTUYBUXCUXAUXTOUYBUXCNZUXAUXTUYCUWSUW
-      DIUWTUYCUVSUWTKZNUWRUWCCUYCUYDUWRUWCTZUXCUYBUYDUYEOZUVNUVKUYBUYFOZUVBUVKU
-      YGOUVEUVBUVKUYBUYDUYEUVBUVKNZUYBNZUYDNZUWPUVTUWQUWBUYJUVTUWPUYJUVMUVBUVSQ
-      UWNSJZKZUVTUWPTUYHUVMUWOUYDVQZUYIUVBUYDUVBUVKUYBVRZVSZUYJUYLUVSUXOKZUYIUY
-      DUYPUVKUYDUYPOUVBUYBUVKUWTUXOUVSEVTWCWAWDUYBUYLUYPWEZUYHUYDUWOUYQUVMUWOUY
-      KUXOUVSUWNUVHQSWFZWGWHWAWIAUVSFBWJWKWMUYJUWBUWQUYJUVMUVBUWAUYKKUWBUWQTUYM
-      UYOUYJUWAUXOUYKUYDUWAUXOKUYIUVSEWLWHUYBUYKUXOTZUYHUYDUWOUYSUVMUYRWHWAWNAU
-      WAFBWJWKWMXDWOVSWPWPWDWQWRWSWTXAXBWDUXDUYAEUVGLZUVHUVGLZUBZCKZUXBUXCVUCUV
-      MUWOUXCVUCOUXAUXCUYBVUCUVNUVKUYBVUCOZUVEUVBUVKVUDOUVEUVBUVKUYBVUCUYIUVEVU
-      CUYIUVDVUBCUYIUVCUYTAVUAUYIUWNMRJZBLZEBLZUVCUYTUYIVUEEBUYIVUEUVHMRJZEUWOV
-      UEVUHTUYHUVMUWNUVHMRXCXEUVKVUHETZUVBUYBUVKEXNKMXNKZVUIEXFXGEMXHXIWAXJXKUV
-      MUVCVUFTUYHUWOBUVLXOVOUYIUVMUVBEUYKKZUYTVUGTUYHUVMUWOVAZUYNUYIVUKEUXOKZUV
-      KVUMUVBUYBEXLWAUWOVUKVUMWEUYHUVMUWOUYKUXOEUYRWGXEWIAEFBWJWKXMUYIVUAAUYIUV
-      MUVBUVHUWNTZVUAATVULUYNUYBVUNUYHUYBUWNUVHUVMUWOXPWMWHAUVHFBXQWKWMXDWQXRXS
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     $( Extension of a walk (as word) by adding an edge/vertex.  (Contributed by
        Alexander van der Vekens, 5-Aug-2018.)  (Revised by AV, 16-Apr-2021.)
@@ -411978,12 +411954,12 @@ sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1)
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   $}
 
   ${
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                           /\ ( A ` 0 ) = ( B ` 0 ) )
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+      KVKUULUUMUUNWJUUSYSVLVMVPWMUXGCUXKUXMWNXAUXBUXGCUXJUXNUWEUWTUXJUXNQZUXAUV
+      IUVTUWTVXAUWDUVIUWQUWHVXAUWSVUTUXJMUXLOKZUXNVUTUXIUXLMOVUTUXIVWAUXLUVGUWF
+      UXIVWAQUVHUWGUVGUWFHUXHVVOINUVEUVEABUUOUUPUUQVWHWEYMUVIUVQMTFGZUWOYBGZVXB
+      UXNQUWHUVGVXCUVHUVGVWEVXCUYOUVQUURUUTSUWHUWNXQGVXDUVEBXDUWNUVAUSMUVQUWOUV
+      BYDWEUVCUVDVLXMXN $.
 
     $( The concatenation of two words representing closed walks anchored at the
        same vertex represents a closed walk.  The resulting walk is a "double
@@ -414289,19 +414263,18 @@ that the word does not contain the terminating vertex p(n) of the walk,
     clwlkclwwlk2 $p |- ( ( G e. USPGraph /\ P e. Word V /\ 1 <_ ( # ` P ) )
                          -> ( E. f f ( ClWalks ` G ) ( P ++ <" ( P ` 0 ) "> )
                               <-> P e. ( ClWWalks ` G ) ) ) $=
-      ( wcel c1 chash cfv cle wbr co cpfx wceq wa 3adant1 c2 caddc cuspgr cword
-      w3a cc0 cs1 cconcat cclwwlk cv cclwlks wex lswccats1fst wrdlenccats1lenm1
-      cmin biantrurd adantr oveq2d simpl wrdsymb1 s1cld eqidd pfxccatid syl3anc
-      eqtr2d eleq1d wb simp1 simp2 ccatcl syl2anc cn0 lencl 1e2m1 a1i breq1d cr
-      clsw 2re 1red nn0re lesubaddd bitrd syl biimpa s1len oveq2i breqtrrdi jca
-      ccatlen breqtrrd clwlkclwwlk 3bitr4rd ) DUAHZAEUBZHZIAJKZLMZUCZAUDAKZUEZU
-      FNZWTJKZIUMNZONZDUGKZHZWTVPKUDWTKPZXEQZAXDHBUHWTDUIKMBUJZWQXFXEWNWPXFWLAE
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-      KXLXMWGREAWSWHWBWIWTBCDEFGWJVBWK $.
+      ( wcel c1 chash cfv cle wbr co wceq cpfx c2 wb 3adant1 caddc cuspgr cword
+      w3a cv cc0 cs1 cconcat cclwlks wex clsw cmin cclwwlk simp1 wrdsymb1 s1cld
+      wa ccatcl syldan cn0 lencl 1e2m1 breq1i cr 2re a1i nn0re lesubaddd syl5bb
+      1red syl biimpa s1len oveq2i breqtrrdi ccatlen breqtrrd wrdlenccats1lenm1
+      clwlkclwwlk syl3anc oveq2d adantr simpl pfxccatid eqtr2d eleq1d biantrurd
+      eqidd lswccats1fst bitr2d bitrd ) DUAHZAEUBZHZIAJKZLMZUCZBUDAUEAKZUFZUGNZ
+      DUHKMBUIZWSUJKUEWSKOZWSWSJKZIUKNZPNZDULKZHZUPZAXEHZWPWKWSWLHZQXBLMZWTXGRW
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+      FGVRVSWMWOXGXHRWKXLXHXFXGXLAXDXEXLXDWSWNPNZAWMXDXTOWOWMXCWNWSPWQEAVQVTWAX
+      LWMXKWNWNOXTAOWMWOWBXMXLWNWGAWRWNEWCVSWDWEXLXAXFAEWHWFWISWJ $.
   $}
 
   ${
@@ -415071,51 +415044,50 @@ of fixed length (as words) is also finite.  (Contributed by Alexander
                             { ( P ` i ) , ( P ` ( i + 1 ) ) } e. ( Edg ` G )
                             /\ { ( lastS ` P ) , ( P ` 0 ) } e. ( Edg ` G ) ) )
                      -> ( P ++ <" ( P ` 0 ) "> ) e. D ) $=
-      ( wcel cfv wceq wa c1 caddc co cc0 cmin cfzo adantr adantl wb cword chash
-      cn cvtx cpr cedg wral clsw w3a cs1 cconcat cwwlksn wne ccatws1n0 3ad2ant2
-      cv c0 simp2l fstwrdne0 s1cld 3adant3 ccatcl syl2anc csn wi simprl cn0 clt
-      cz wbr elfzonn0 nnz elfzo0 nn0re nnre peano2rem syl 3jca simpr ltm1d lttr
-      cr imp syl12anc ex impancom 3adant2 sylbi elfzo0z syl3anbrc adantlr oveq2
-      impcom ad2antll ccatval1 syl3anc elfzom1p1elfzo eleqtrrd preq12d ralbidva
-      eleq2d mpbird eqcomd eleq1d biimpcd expdcom fzo0end fvoveq1 eqcoms eqtr4d
-      3imp eqtr2d fveq2 nncn 1cnd npcand fveq2d ccatws1ls 3eqtr3rd ralsn sylibr
-      lsw cun addsubd oveq2d cuz nnm1nn0 elnn0uz sylib fzosplitsn eqtrd raleqdv
-      ovex ralunb syl6bb 3ad2ant1 mpbir2and ccatlen a1i eqid s1len oveq1d nnnn0
-      id oveq12d cwwlks iswwlksn iswwlks anbi1d bitrd lswccats1 biimpri eqeq12d
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-      ZJUWGAUWBUWCBVVEUWBJVVFUWEVVGUWFVVEUWBUHXMOVVEUWBUUOUUMGUUPUUQ $.
+      ( wcel cfv wceq wa c1 caddc co cc0 cmin cfzo wral adantr adantl cword cpr
+      cn cvtx chash cv cedg clsw w3a cs1 cconcat cwwlksn wne ccatws1n0 3ad2ant2
+      c0 simprl fstwrdne0 s1cld ccatcl syl2anc 3adant3 csn simplrl cn0 elfzonn0
+      wi cz clt wbr nnz elfzo0 nn0re nnre peano2rem syl 3jca ltm1d anim1ci lttr
+      cr sylc ex impancom 3adant2 sylbi impcom elfzo0z syl3anbrc adantlr eleq2d
+      wb oveq2 ad2antll mpbird ccatval1 syl3anc elfzom1p1elfzo eleqtrrd preq12d
+      eleq1d ralbidva biimprcd expdcom 3imp fzo0end syl5ibrcom imp fvoveq1 nncn
+      lsw eqtr2d 1cnd npcand fveq2d fveq2 ccatws1ls 3eqtr2rd biimpcd ovex ralsn
+      sylibr cun addsubd oveq2d nnm1nn0 elnn0uz sylib fzosplitsn raleqdv ralunb
+      cuz eqtrd syl6bb 3ad2ant1 mpbir2and ccatlen id s1len eqid oveq1d iswwlksn
+      oveq12d cwwlks nnnn0 iswwlks anbi1i lswccats1 lbfzo0 biimpri eqtr4d fveq1
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+      UWDVVDUWEVVBUWAUHXPOVVBUWAUULUUNGUUOUUP $.
 
     $d D t $.  $d G i t $.  $d G t w $.  $d N t $.
     clwwlkf1o.f $e |- F = ( t e. D |-> ( t prefix N ) ) $.
@@ -415293,31 +415265,31 @@ the set of closed walks of a fixed length represented by walks (as
                           <-> ( W ++ <" ( W ` 0 ) "> ) e. ( WWalks ` G ) ) ) $=
       ( vi wcel c0 wa cc0 cfv co c1 caddc cpr cmin cfzo wral wceq syl adantr cv
       cword wne cs1 cconcat cedg chash clsw cwwlks cclwwlk simpl fstwrdne s1cld
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+      csn ccatlen s1len oveq2i syl6eq oveq1d cc lencl nn0cnd 1cnd addsubd eqtrd
+      jca oveq2d raleqdv cun cuz cn0 cn lennncl nnm1nn0 sylib fzosplitsn ralunb
+      elnn0uz syl6bb nn0zd elfzom1elfzo sylan ccats1val1 syl3anc elfzom1elp1fzo
+      cz preq12d eleq1d ralbidva ovex fveq2 fvoveq1 ralsn fzo0end eqtr4d npcan1
+      fveq2d eqidd ccats1val2 syl5bb anbi12d 3bitrd ccat0 necon3d adantld mpcom
+      lsw syl6bi wb cvv wrdv s1cli ccatalpha sylancl mpbir2and w3a eqid iswwlks
+      df-3an bitri a1i mpbirand isclwwlk 3anass baib 3bitr4rd ) CBUBZFZCGUCZHZE
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+      UUILOWJYKUUJRZYRUWPYSUWRYOUWNYQUWOYKUUJYNWKYKUUJLYNMWLWGWHWMYJUWPUUNYSYJU
+      WNUUMUWOYLYJUWNUUJCJZUUMYJYHUWFUUJUWGFZUWNUWSRUVKUVLYJUWBUWTUWCUUIWNSYLUU
+      JBCWCWDYHUUMUWSRYICYGXGTWOYJUWOUUIYNJZYLYJUUSUUIYNYHUUSUUIRZYIYHUVOUXBUVQ
+      UUIWPSTWQYJYHUWFUUIUUIRUXAYLRUVKUVLYJUUIWRYLUUIBCWSWDVEWGWHWTXAXBYJUUQYNG
+      UCZYNYGFZHZUUDYJUXCUXDUVJYJUXCUVNUVJYIUXCYHUVJYNGCGUVJYNGRCGRZYMGRZHUXFBC
+      YMXCUXFUXGUKXHXDXEXFYJUXDYHUVIUVKUVMYHUXDUVJXIZYIYHCXJUBZFYMUXIFUXHBCXKYL
+      XLCYMBXMXNTXOVFUUQUXEUUDHZXIYJUUQUXCUXDUUDXPUXJEYSABYNDYSXQZXRUXCUXDUUDXS
+      XTYAYBUURYJUUPUURYJUULUUOXPYJUUPHEYSABCDUXKYCYJUULUUOYDXTYEYF $.
 
     $( A word over vertices represents a closed walk of a fixed length ` N `
        greater than zero iff the word concatenated with its first symbol
@@ -415580,8 +415552,8 @@ closed walk as word of the same length (in an undirected graph).
     id ccatlen syl2an oveqan12d eqtrd 3adant3 sylanbrc ) ADCFGHZBECFGHZIAJIBJKZ
     RABLGZCUAJZHZUSMJZDENGZKZUSVCCFGHUPAUTHZAMJZDKZOZUQBUTHZBMJZEKZOZURURVACDAP
     CEBPURUBVHVEVLVIURURVAVEVGSVIVKSURUIABCUCUDUEUPUQVDURUPUQOVBVFVJNGZVCUPACUF
-    JZUGZHBVOHVBVMKUQCDVNAVNUHZQCEVNBVPQVNABUJUKUPUQVFDVJENCDATCEBTULUMUNCVCUSP
-    UO $.
+    JZUGZHBVOHVBVMKUQCDVNAVNUHZQCEVNBVPQVNVNABUJUKUPUQVFDVJENCDATCEBTULUMUNCVCU
+    SPUO $.
 
   ${
     $d G i $.  $d N i $.  $d W i $.
@@ -464623,46 +464595,46 @@ its graph has a given second element (that is, function value).
     $( Conditions for a concatenation to be injective.  (Contributed by Thierry
        Arnoux, 11-Dec-2023.) $)
     ccatf1 $p |- ( ph -> ( A ++ B ) : dom ( A ++ B ) -1-1-> S ) $=
-      ( co cfv wceq wcel syl2anc syl wa ad5antr syl3anc vi vj cconcat cdm wf cv
-      wi wral wf1 cc0 chash cfzo cword ccatcl wrdf ffdmd simpllr ccatval1 simpr
-      simplr 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi ad3antrrr r19.21bi mpd c0
-      crn wfun f1fun eleqtrrd fvelrn cmin caddc ccatlen oveq2d eleqtrd ccatval2
-      cin cz cn0 lencl nn0zd fzosubel3 eqeltrd elind wn noel pm2.21dd wo eleq2d
-      a1i adantr fzospliti ad5ant13 mpjaodan 3eqtr3rd fzossz sseldi zcnd nn0cnd
-      cc f1veqaeq anassrs imp syl1111anc subcan2d ad2antrr ex anasss ralrimivva
-      sylanbrc ) ABCUCLZUDZDXPUEUAUFZXPMZUBUFZXPMZNZXRXTNZUGZUBXQUHUAXQUHXQDXPU
-      IAUJXPUKMZULLZDXPAXPDUMZOZYFDXPUEABYGOZCYGOZYHGHDBCUNPZDXPUOQUPAYDUAUBXQX
-      QAXRXQOZXTXQOZYDAYLRZYMRZYBYCYOYBRZXRUJBUKMZULLZOZYCXRYQYEULLZOZYPYSRZXTY
-      ROZYCXTYTOZUUBUUCRZXRBMZXTBMZNZYCUUEXSYAUUFUUGYOYBYSUUCUQUUEYIYJYSXSUUFNZ
-      AYIYLYMYBYSUUCGSZAYJYLYMYBYSUUCHSZYPYSUUCUTDBCXRURZTUUEYIYJUUCYAUUGNZUUJU
-      UKUUBUUCUSDBCXTURZTVAUUBUUHYCUGZUBYRYPUUOUBYRUHZUAYRAUUPUAYRUHZYLYMYBAYRD
-      BUIZUUQABUDZYRNZUUSDBUIZUURAYIUUTGDBVBZQZIUUTUVAUURUUSYRDBVCVDPUURYRDBUEU
-      UQUAUBYRDBVEVFQVGVHVHVIUUBUUDRZUUFVJOZYCUVDUUFBVKZCVKZWBZVJUVDUVFUVGUUFUV
-      DBVLZXRUUSOUUFUVFOAUVIYLYMYBYSUUDAUVAUVIIUUSDBVMQZSUVDXRYRUUSYPYSUUDUTZUV
-      DYIUUTAYIYLYMYBYSUUDGSZUVBQVNXRBVOPUVDUUFXTYQVPLZCMZUVGUVDXSYAUUFUVNYOYBY
-      SUUDUQUVDYIYJYSUUIUVLAYJYLYMYBYSUUDHSZUVKUULTUVDYIYJXTYQYQCUKMZVQLZULLZOZ
-      YAUVNNZUVLUVOUVDXTYTUVRUUBUUDUSAYTUVRNZYLYMYBYSUUDAYEUVQYQULAYIYJYEUVQNGH
-      DBCVRPVSZSVTZDBCXTWAZTVAUVDCVLZUVMCUDZOZUVNUVGOAUWEYLYMYBYSUUDAUWFDCUIZUW
-      EJUWFDCVMQZSUVDUVMUJUVPULLZUWFUVDUVSUVPWCOZUVMUWJOZUWCAUWKYLYMYBYSUUDAUVP
-      AYJUVPWDOHDCWEQWFZSXTYQUVPWGZPUVDYJUWFUWJNZUVODCVBZQVNUVMCVOPWHWIAUVHVJNZ
-      YLYMYBYSUUDKSVTUVEWJUVDUUFWKWOWLAYMUUCUUDWMZYLYBYSAYMRXTYFOZYQWCOZUWRAYMU
-      WSAXQYFXTAYHXQYFNYKDXPVBQZWNVDAUWTYMAYQAYIYQWDOGDBWEQZWFZWPXTUJYEYQWQPZWR
-      WSYPUUARZUUCYCUUDUXEUUCRZUUGVJOZYCUXFUUGUVHVJUXFUVFUVGUUGUXFUVIXTUUSOUUGU
-      VFOAUVIYLYMYBUUAUUCUVJSUXFXTYRUUSUXEUUCUSZAUUTYLYMYBUUAUUCUVCSVNXTBVOPUXF
-      UUGXRYQVPLZCMZUVGUXFXSYAUXJUUGYOYBUUAUUCUQUXFYIYJXRUVROZXSUXJNZAYIYLYMYBU
-      UAUUCGSZAYJYLYMYBUUAUUCHSZUXFXRYTUVRYPUUAUUCUTAUWAYLYMYBUUAUUCUWBSVTZDBCX
-      RWAZTUXFYIYJUUCUUMUXMUXNUXHUUNTWTUXFUWEUXIUWFOZUXJUVGOAUWEYLYMYBUUAUUCUWI
-      SUXFUXIUWJUWFUXFUXKUWKUXIUWJOZUXOAUWKYLYMYBUUAUUCUWMSXRYQUVPWGZPAUWOYLYMY
-      BUUAUUCAYJUWOHUWPQZSVNUXICVOPWHWIAUWQYLYMYBUUAUUCKSVTUXGWJUXFUUGWKWOWLUXE
-      UUDRZXRXTYQUYAXRUYAYTWCXRYQYEXAZYPUUAUUDUTZXBXCUYAXTUYAYTWCXTUYBUXEUUDUSZ
-      XBXCAYQXEOYLYMYBUUAUUDAYQUXBXDSUYAUWHUXQUWGUXJUVNNZUXIUVMNZAUWHYLYMYBUUAU
-      UDJSUYAUXIUWJUWFUYAUXKUWKUXRUYAXRYTUVRUYCAUWAYLYMYBUUAUUDUWBSZVTZAUWKYLYM
-      YBUUAUUDUWMSZUXSPAUWOYLYMYBUUAUUDUXTSZVNUYAUVMUWJUWFUYAUVSUWKUWLUYAXTYTUV
-      RUYDUYGVTZUYIUWNPUYJVNUYAXSYAUXJUVNYOYBUUAUUDUQUYAYIYJUXKUXLAYIYLYMYBUUAU
-      UDGSZAYJYLYMYBUUAUUDHSZUYHUXPTUYAYIYJUVSUVTUYLUYMUYKUWDTVAUWHUXQRUWGRUYEU
-      YFUWHUXQUWGUYEUYFUGUWFDUXIUVMCXFXGXHXIXJAYMUWRYLYBUUAUXDWRWSYNYSUUAWMZYMY
-      BYNXRYFOZUWTUYNAYLUYOAXQYFXRUXAWNVDAUWTYLUXCWPXRUJYEYQWQPXKWSXLXMXNUAUBXQ
-      DXPVEXO $.
+      ( vi co cfv wceq wcel syl2anc syl wa ad5antr vj cconcat cdm wf cv wi wral
+      weq wf1 cc0 chash cfzo cword ccatcl ffdmd simpllr simplr ccatval1 syl3anc
+      wrdf simpr 3eqtr3d wrddm f1eq2 biimpa dff13 simprbi ad3antrrr r19.21bi c0
+      mpd crn cin wfun f1fun eleqtrrd fvelrn cmin caddc ccatlen oveq2d ccatval2
+      eleqtrd cz cn0 lencl nn0zd fzosubel3 eqeltrd elind wn a1i pm2.21dd eleq2d
+      noel wo adantr fzospliti ad5ant13 mpjaodan 3eqtr3rd fzossz sseldi zcnd cc
+      nn0cnd f1veqaeq anassrs syl1111anc subcan2d ad2antrr ex anasss ralrimivva
+      imp sylanbrc ) ABCUBMZUCZDXQUDLUEZXQNZUAUEZXQNZOZLUAUHZUFZUAXRUGLXRUGXRDX
+      QUIAUJXQUKNZULMZDXQAXQDUMZPZYGDXQUDABYHPZCYHPZYIGHDBCUNQZDXQUTRUOAYELUAXR
+      XRAXSXRPZYAXRPZYEAYMSZYNSZYCYDYPYCSZXSUJBUKNZULMZPZYDXSYRYFULMZPZYQYTSZYA
+      YSPZYDYAUUAPZUUCUUDSZXSBNZYABNZOZYDUUFXTYBUUGUUHYPYCYTUUDUPUUFYJYKYTXTUUG
+      OZAYJYMYNYCYTUUDGTZAYKYMYNYCYTUUDHTZYQYTUUDUQDBCXSURZUSUUFYJYKUUDYBUUHOZU
+      UKUULUUCUUDVADBCYAURZUSVBUUCUUIYDUFZUAYSYQUUPUAYSUGZLYSAUUQLYSUGZYMYNYCAY
+      SDBUIZUURABUCZYSOZUUTDBUIZUUSAYJUVAGDBVCZRZIUVAUVBUUSUUTYSDBVDVEQUUSYSDBU
+      DUURLUAYSDBVFVGRVHVIVIVKUUCUUESZUUGVJPZYDUVEUUGBVLZCVLZVMZVJUVEUVGUVHUUGU
+      VEBVNZXSUUTPUUGUVGPAUVJYMYNYCYTUUEAUVBUVJIUUTDBVORZTUVEXSYSUUTYQYTUUEUQZU
+      VEYJUVAAYJYMYNYCYTUUEGTZUVCRVPXSBVQQUVEUUGYAYRVRMZCNZUVHUVEXTYBUUGUVOYPYC
+      YTUUEUPUVEYJYKYTUUJUVMAYKYMYNYCYTUUEHTZUVLUUMUSUVEYJYKYAYRYRCUKNZVSMZULMZ
+      PZYBUVOOZUVMUVPUVEYAUUAUVSUUCUUEVAAUUAUVSOZYMYNYCYTUUEAYFUVRYRULAYJYKYFUV
+      ROGHDDBCVTQWAZTWCZDBCYAWBZUSVBUVECVNZUVNCUCZPZUVOUVHPAUWFYMYNYCYTUUEAUWGD
+      CUIZUWFJUWGDCVORZTUVEUVNUJUVQULMZUWGUVEUVTUVQWDPZUVNUWKPZUWDAUWLYMYNYCYTU
+      UEAUVQAYKUVQWEPHDCWFRWGZTYAYRUVQWHZQUVEYKUWGUWKOZUVPDCVCZRVPUVNCVQQWIWJAU
+      VIVJOZYMYNYCYTUUEKTWCUVFWKUVEUUGWOWLWMAYNUUDUUEWPZYMYCYTAYNSYAYGPZYRWDPZU
+      WSAYNUWTAXRYGYAAYIXRYGOYLDXQVCRZWNVEAUXAYNAYRAYJYRWEPGDBWFRZWGZWQYAUJYFYR
+      WRQZWSWTYQUUBSZUUDYDUUEUXFUUDSZUUHVJPZYDUXGUUHUVIVJUXGUVGUVHUUHUXGUVJYAUU
+      TPUUHUVGPAUVJYMYNYCUUBUUDUVKTUXGYAYSUUTUXFUUDVAZAUVAYMYNYCUUBUUDUVDTVPYAB
+      VQQUXGUUHXSYRVRMZCNZUVHUXGXTYBUXKUUHYPYCUUBUUDUPUXGYJYKXSUVSPZXTUXKOZAYJY
+      MYNYCUUBUUDGTZAYKYMYNYCUUBUUDHTZUXGXSUUAUVSYQUUBUUDUQAUWBYMYNYCUUBUUDUWCT
+      WCZDBCXSWBZUSUXGYJYKUUDUUNUXNUXOUXIUUOUSXAUXGUWFUXJUWGPZUXKUVHPAUWFYMYNYC
+      UUBUUDUWJTUXGUXJUWKUWGUXGUXLUWLUXJUWKPZUXPAUWLYMYNYCUUBUUDUWNTXSYRUVQWHZQ
+      AUWPYMYNYCUUBUUDAYKUWPHUWQRZTVPUXJCVQQWIWJAUWRYMYNYCUUBUUDKTWCUXHWKUXGUUH
+      WOWLWMUXFUUESZXSYAYRUYBXSUYBUUAWDXSYRYFXBZYQUUBUUEUQZXCXDUYBYAUYBUUAWDYAU
+      YCUXFUUEVAZXCXDAYRXEPYMYNYCUUBUUEAYRUXCXFTUYBUWIUXRUWHUXKUVOOZUXJUVNOZAUW
+      IYMYNYCUUBUUEJTUYBUXJUWKUWGUYBUXLUWLUXSUYBXSUUAUVSUYDAUWBYMYNYCUUBUUEUWCT
+      ZWCZAUWLYMYNYCUUBUUEUWNTZUXTQAUWPYMYNYCUUBUUEUYATZVPUYBUVNUWKUWGUYBUVTUWL
+      UWMUYBYAUUAUVSUYEUYHWCZUYJUWOQUYKVPUYBXTYBUXKUVOYPYCUUBUUEUPUYBYJYKUXLUXM
+      AYJYMYNYCUUBUUEGTZAYKYMYNYCUUBUUEHTZUYIUXQUSUYBYJYKUVTUWAUYMUYNUYLUWEUSVB
+      UWIUXRSUWHSUYFUYGUWIUXRUWHUYFUYGUFUWGDUXJUVNCXGXHXOXIXJAYNUWSYMYCUUBUXEWS
+      WTYOYTUUBWPZYNYCYOXSYGPZUXAUYOAYMUYPAXRYGXSUXBWNVEAUXAYMUXDWQXSUJYFYRWRQX
+      KWTXLXMXNLUAXRDXQVFXP $.
   $}
 
   ${
@@ -464904,10 +464876,10 @@ Splicing words (substring replacement)
       cmin ) AHGOPZDFECUAUBPZQHDFUHPZCUCPZUDQZOPZWEDEDUDQZUEUFPZUCPZQZHWIQZHEOP
       DQZAWBWGWCWJADBUGZRZFSEUIPZRESWHUIPZRZCWNRZWCWJTIJKLCDEFWNWPWQWNUJUKAGWFH
       OAWDUDQZCUDQZOPZFXAOPWFGAWTFXAOAWOFWQRWTFTIAWPWQFAWRWHEULQRWPWQUMKESWHUNE
-      SWHUOUPJUQBDFURUSUTAWDWNRZWSWFXBTAWOXCIBDFVAVBZLBWDCVCUSNVDVEVFAWEWNRZWIW
-      NRZHSWIUDQZVGPZRWKWLTAXCWSXEXDLBWDCVHUSAWOXFIBDEWHVIVBAHSWHEWAPZVGPZXHMAX
-      GXISVGAWOWRWHWQRZXGXITIKAWOXKIWOWHVKRXKBDVJWHVLVMVBZBDEWHVNVOVEVPBWEWIHVQ
-      VOAWOWRXKHXJRWLWMTIKXLMBDEWHHVRVSVT $.
+      SWHUOUPJUQBDFURUSUTAWDWNRZWSWFXBTAWOXCIBDFVAVBZLBBWDCVCUSNVDVEVFAWEWNRZWI
+      WNRZHSWIUDQZVGPZRWKWLTAXCWSXEXDLBWDCVHUSAWOXFIBDEWHVIVBAHSWHEWAPZVGPZXHMA
+      XGXISVGAWOWRWHWQRZXGXITIKAWOXKIWOWHVKRXKBDVJWHVLVMVBZBDEWHVNVOVEVPBWEWIHV
+      QVOAWOWRXKHXJRWLWMTIKXLMBDEWHHVRVSVT $.
   $}
 
 $(
@@ -467547,28 +467519,27 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $=
 
     $( Lemma for ~ cycpmco2 .  (Contributed by Thierry Arnoux, 4-Jan-2024.) $)
     cycpmco2lem3 $p |- ( ph -> ( ( # ` U ) - 1 ) = ( # ` W ) ) $=
-      ( co wcel vw chash cfv c1 cc0 cfz cz cword cn0 cv cdm wf1 crab ssrab2 cbs
-      eqid tocycf syl fdmd eleqtrd sseldi lencl nn0fz0 biimpi 3syl elfzelz zcnd
-      1cnd caddc cmin cpfx cs1 cconcat cop csubstr cotp csplice wceq ccnv ovexd
-      cvv eqeltrid crn eldifad s1cld splval syl13anc syl5eq fveq2d pfxcl ccatcl
-      wf syl2anc swrdcl ccatlen ccatws1len cfzo wf1o wa id eqidd f1eq123d elrab
-      dmeq sylib simprd f1cnv f1of ffvelrnd wrddm fzofzp1 pfxlen oveq1d swrdlen
-      eqtrd syl3anc oveq12d 3eqtrd fz0ssnn0 peano2zd addsubassd addassd 3eqtr2d
-      nn0zd nn0cnd addcld pncan2d addcomd mvrraddd ) ADUBUCZJUBUCZUDAYKAYKUEYKU
-      FSZTZYKUGTAJBUHZTZYKUITZYMAUAUJZUKZBYQULZUAYNUMZYNJYSUAYNUNAJHUKZYTNAYTCU
-      OUCZHABITYTUUBHWLMUAUUBHBCIKLUUBUPUQURUSUTZVAZBJVBYPYMYKVCVDVEZYKUEYKVFUR
-      VGZAVHZAYJEUDYKVISZVISZEVJSZUUHYKUDVISAYJEUDVISZYKEVJSZVISZUUKYKVISZEVJSU
-      UJAYJJEVKSZFVLZVMSZJEYKVNVOSZVMSZUBUCZUUQUBUCZUURUBUCZVISZUUMADUUSUBADJEE
-      UUPVPVQSZUUSRAJUUATEWATZUVEUUPYNTZUVDUUSVRNAEGJVSZUCZUDVISZWAQAUVHUDVIVTW
-      BZUVJAFBAFBJWCZOWDWEZUUPJEEUUAWAWAYNWFWGWHWIAUUQYNTZUURYNTZUUTUVCVRAUUOYN
-      TZUVFUVMAYOUVOUUDBJEWJURZUVLBUUOUUPWKWMAYOUVNUUDBJEYKWNURBUUQUURWOWMAUVAU
-      UKUVBUULVIAUVAUUOUBUCZUDVISZUUKAUVOUVAUVRVRUVPBUUOFWPURAUVQEUDVIAYOEYLTZU
-      VQEVRUUDAEUVIYLQAUVHUEYKWQSZTUVIYLTAUVHJUKZUVTAUVKUWAGUVGAUWABJULZUVKUWAU
-      VGWRUVKUWAUVGWLAYOUWBAJYTTYOUWBWSUUCYSUWBUAJYNYQJVRZYRUWABBYQJUWCWTYQJXDU
-      WCBXAXBXCXEXFUWABJXGUVKUWAUVGXHVEPXIAYOUWAUVTVRUUDBJXJURUTUEYKUVHXKURWBZB
-      JEXLWMXMXOAYOUVSYMUVBUULVRUUDUWDUUEBJEYKXNXPXQXRAUUKYKEAUUKAEAEAYLUIEYKXS
-      UWDVAZYDXTVGUUFAEUWEYEZYAAUUNUUIEVJAEUDYKUWFUUGUUFYBXMYCAEUUHUWFAUDYKUUGU
-      UFYFYGAUDYKUUGUUFYHXRYI $.
+      ( wcel co vw chash cfv c1 cword cn0 cv cdm wf1 crab ssrab2 wf eqid tocycf
+      cbs syl fdmd eleqtrd sseldi lencl nn0cnd 1cnd caddc cmin cpfx cs1 cconcat
+      cop csubstr cotp csplice cvv wceq ovexd eqeltrid crn eldifad s1cld splval
+      syl13anc syl5eq fveq2d pfxcl ccatcl syl2anc swrdcl ccatlen ccatws1len cc0
+      ccnv cfz cfzo wf1o wa dmeq eqidd f1eq123d elrab sylib f1cnv simpl2im f1of
+      ffvelrnd wrddm fzofzp1 pfxlen oveq1d eqtrd nn0fz0 swrdlen syl3anc oveq12d
+      id 3eqtrd fz0ssnn0 nn0zd peano2zd zcnd addsubassd addassd 3eqtr2d pncan2d
+      addcld addcomd mvrraddd ) ADUBUCZJUBUCZUDAYGAJBUEZSZYGUFSZAUAUGZUHZBYKUIZ
+      UAYHUJZYHJYMUAYHUKAJHUHZYNNAYNCUOUCZHABISYNYPHULMUAYPHBCIKLYPUMUNUPUQURZU
+      SZBJUTUPZVAZAVBZAYFEUDYGVCTZVCTZEVDTZUUBYGUDVCTAYFEUDVCTZYGEVDTZVCTZUUEYG
+      VCTZEVDTUUDAYFJEVETZFVFZVGTZJEYGVHVITZVGTZUBUCZUUKUBUCZUULUBUCZVCTZUUGADU
+      UMUBADJEEUUJVJVKTZUUMRAJYOSEVLSZUUSUUJYHSZUURUUMVMNAEGJWJZUCZUDVCTZVLQAUV
+      BUDVCVNVOZUVDAFBAFBJVPZOVQVRZUUJJEEYOVLVLYHVSVTWAWBAUUKYHSZUULYHSZUUNUUQV
+      MAUUIYHSZUUTUVGAYIUVIYRBJEWCUPZUVFBUUIUUJWDWEAYIUVHYRBJEYGWFUPBBUUKUULWGW
+      EAUUOUUEUUPUUFVCAUUOUUIUBUCZUDVCTZUUEAUVIUUOUVLVMUVJBUUIFWHUPAUVKEUDVCAYI
+      EWIYGWKTZSZUVKEVMYRAEUVCUVMQAUVBWIYGWLTZSUVCUVMSAUVBJUHZUVOAUVEUVPGUVAAUV
+      EUVPUVAWMZUVEUVPUVAULAYIUVPBJUIZUVQAJYNSYIUVRWNYQYMUVRUAJYHYKJVMZYLUVPBBY
+      KJUVSXMYKJWOUVSBWPWQWRWSUVPBJWTXAUVEUVPUVAXBUPPXCAYIUVPUVOVMYRBJXDUPURWIY
+      GUVBXEUPVOZBJEXFWEXGXHAYIUVNYGUVMSZUUPUUFVMYRUVTAYJUWAYSYGXIWSBJEYGXJXKXL
+      XNAUUEYGEAUUEAEAEAUVMUFEYGXOUVTUSZXPXQXRYTAEUWBVAZXSAUUHUUCEVDAEUDYGUWCUU
+      AYTXTXGYAAEUUBUWCAUDYGUUAYTYCYBAUDYGUUAYTYDXNYE $.
 
     $( Lemma for ~ cycpmco2 .  (Contributed by Thierry Arnoux, 4-Jan-2024.) $)
     cycpmco2lem4 $p |- ( ph -> ( ( M ` W ) ` ( ( M ` <" I J "> ) ` I ) ) = ( (
@@ -467577,58 +467548,57 @@ C_ dom ( ( ( T ` { I , J } ) o. F ) \ _I ) ) $=
       wceq cs1 cotp csplice cv cdm wf1 crab ssrab2 cbs eqid tocycf fdmd eleqtrd
       syl sseldi crn eldifad s1cld splcl syl2anc eqeltrid cycpmco2f1 cmin simpr
       cycpmco2lem3 oveq2d eleqtrrd cycpmfv1 cycpmco2lem2 fveq2d cop csubstr cfz
-      cn0 lencl nn0fz0 biimpi 3syl swrdfv0 syl3anc cpfx cconcat cvv ccnv splval
-      wf ovexd syl13anc syl5eq fveq1d pfxcl ccatcl swrdcl cz fzoaddel elfzolt2b
-      1zzd ccatws1len wf1o id eqidd f1eq123d elrab3 biimpa f1cnv ffvelrnd wrddm
-      dmeq f1of fzofzp1 oveq1d eqtrd oveq12d 3eqtrd nn0cnd 3eqtr2d 3eqtr3rd clt
-      zcnd wbr nn0p1gt0 cycpmfv2 cn nn0p1nn lbfzo0 sylibr ccatval1 pfxlen nn0zd
-      ccatlen swrdlen fz0ssnn0 peano2zd elfzelz addsubassd 1cnd addassd pncan2d
-      addcld addcomd ccatval2 subidd a1i pncand eqtr2d wne breqtrrdi fzo1fzo0n0
-      elfzonn0 gt0ne0d sylanbrc elfzo1elm1fzo0 eqeltrd f1f1orn 3eqtr2rd 3eqtr4d
-      f1ocnvfv2 breqtrd eqtr3d breqtrrd eqtr4d fzne1 addid2d pfxfv0 wo mpjaodan
-      elfzr ) AFFGUBHUCUCJHUCZUCGUWAUCZFDHUCZUCZABCDEFGHIJKLMNOPQRUDAESJUEUCZUF
-      TZUGZUWBUWDUMEUWEUMZAUWGUHZEDUCZUWCUCZEUIUJTZDUCZUWDUWBUWIHBEIDKABIUGZUWG
-      MUKZADBULZUGZUWGADJEEFUNZUOUPTZUWPRAJUWPUGZUWRUWPUGZUWSUWPUGAUAUQZURZBUXB
-      USZUAUWPUTZUWPJUXDUAUWPVAAJHURZUXENAUXECVBUCZHAUWNUXEUXGHWSMUAUXGHBCIKLUX
-      GVCVDVGVEVFZVHZAFBAFBJVIZOVJVKZBUWRJEEVLVMVNZUKADURBDUSZUWGABCDEFGHIJKLMN
-      OPQRVOZUKUWIEUWFSDUEUCZUIVPTZUFTZAUWGVQZAUXQUWFUMUWGAUXPUWESUFABCDEFGHIJK
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     ${
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     ${
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@@ -490128,16 +490095,17 @@ strict in the case where the sets B(x) overlap.  (Contributed by Thierry
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       $d e f i k m n $.  $d g m F $.  $d m N $.  $d a b e g k m n T $.
       $( In case the first letter is not zero, the zero skipping sign is never
@@ -490224,25 +490192,25 @@ strict in the case where the sets B(x) overlap.  (Contributed by Thierry
       signstfvc $p |- ( ( F e. Word RR /\ G e. Word RR
         /\ N e. ( 0 ..^ ( # ` F ) ) )
         -> ( ( T ` ( F ++ G ) ) ` N ) = ( ( T ` F ) ` N ) ) $=
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     $}
 
     ${
@@ -490376,39 +490344,39 @@ strict in the case where the sets B(x) overlap.  (Contributed by Thierry
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         if ( ( ( ( T ` F ) ` ( ( # ` F ) - 1 ) ) x. K ) < 0 , 1 , 0 ) ) ) $=
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+        YDUYKUVOHYTUUBUUCUUDWKUUEUUF $.
     $}
 
     ${
@@ -490600,19 +490568,17 @@ strict in the case where the sets B(x) overlap.  (Contributed by Thierry
       signshf $p |- ( ( F e. Word RR /\ C e. RR+ )
         -> H : ( 0 ..^ ( ( # ` F ) + 1 ) ) --> RR ) $=
         ( cr co vx vy cword wcel crp wa cc0 chash cfv c1 caddc cfzo cs1 cconcat
-        cmul cofc cmin cof wf cv resubcl adantl 0red s1cld simpl ccatcl syl2anc
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-        AZVMVNXBUJXCXBVOXBXCXBWTHVPUDXCVQUDXTSHVRHVSVTWBWCWDWEWFWGXBUAUBXEAUOSS
-        SXHVHXMXKXLUOTSUDXBXKXLWLVBXBUGXHUHUIZULTZSXHUSZXESXHUSXBXHWSUDZYGWTXAX
-        RYHXSSHXFVFWHSXHVIVJXBYFXESXHXBYEXDUGULXBYEXCYBUKTZXDWTXAXRYEYIVKXSSHXF
-        VLWHYBUJXCUKYDWIVNWEWFWGXBUGXDULWJZXBAWTXAWKWMWNYJYJXEWOWPXESIXJRWQWR
-        $.
+        cmul cofc cmin cof wf cvv cv resubcl adantl s1cl ax-mp ccatcl mpan wrdf
+        0re syl 1cnd lencl nn0cnd wceq ccatlen s1len oveq1i syl6eq oveq2d feq2d
+        comraddd mpbid remulcl mpan2 ccatws1len ovexd rpre ofcf inidm off feq1i
+        adantr sylibr ) HSUCZUDZAUEUDZUFZUGHUHUIZUJUKTZULTZSUGUMZHUNTZHXAUNTZAU
+        OUPTZUQURTZUSWTSIUSWQUAUBWTWTWTUQSSSXBXDUTUTUAVAZSUDUBVAZSUDUFZXFXGUQTS
+        UDWQXFXGVBVCWOWTSXBUSZWPWOUGXBUHUIZULTZSXBUSZXIWOXBWNUDZXLXAWNUDZWOXMUG
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+        WOXJXAUHUIZWRUKTZUJWRUKTXNWOXJXQVNXOSSXAHVOVGXPUJWRUKUGVPVQVRWAVSVTWBWL
+        WQUAUBWTAUOSSSXCUTXHXFXGUOTSUDWQXFXGWCVCWOWTSXCUSZWPWOUGXCUHUIZULTZSXCU
+        SZXRWOXCWNUDZYAWOXNYBXOSHXAVFWDSXCVHVJWOXTWTSXCWOXSWSUGULSHUGWEVSVTWBWL
+        WQUGWSULWFZWPASUDWOAWGVCWHYCYCWTWIWJWTSIXERWKWM $.
     $}
 
     $( ` H ` , corresponding to the word ` F ` multiplied by ` ( x - C ) ` , is
@@ -494067,8 +494033,8 @@ conditions of the Five Segment Axiom ( ~ axtg5seg ).  See ~ df-ofs .
         ( clpad co cfv chash cc0 cmin cfzo csn caddc wcel nn0cnd cconcat fveq2d
         cxp lpadval cword wceq lpadlem1 ccatlen syl2anc lpadlem2 oveq1d cn0 syl
         lencl npcand 3eqtrd eqtrd ) ADBEJKLZMLNDEMLZOKZPKBQUCZEUAKZMLZDAURVBMAB
-        CDEFGHUDUBAVCVAMLZUSRKZUTUSRKDAVACUEZSEVFSZVCVEUFABCDEHUGGCVAEUHUIAVDUT
-        USRABCDEFGHIUJUKADUSADFTAUSAVGUSULSGCEUNUMTUOUPUQ $.
+        CDEFGHUDUBAVCVAMLZUSRKZUTUSRKDAVACUEZSEVFSZVCVEUFABCDEHUGGCCVAEUHUIAVDU
+        TUSRABCDEFGHIUJUKADUSADFTAUSAVGUSULSGCEUNUMTUOUPUQ $.
     $}
 
     $( Length of a left-padded word, in the general case, expressed with an
@@ -510996,56 +510962,56 @@ proper pair (of ordinal numbers) as model for a Godel-set of membership
       A. c e. ( C \ V ) ( F ` <" c "> ) = <" c "> /\
       A. x e. R A. y e. R ( F ` ( x ++ y ) ) =
         ( ( F ` x ) ++ ( F ` y ) ) ) ) ) $=
-      ( wcel cfv wceq co cvv c0 vw vr vv crn wf cv cs1 cdif wral cconcat mrsubf
-      w3a mrsubcn ralrimiva mrsubccat 3expb ralrimivva 3jca cmpt cun cfrmd ccom
+      ( vv wcel cfv wceq co c0 vw vr crn wf cv cs1 cdif wral cconcat w3a mrsubf
+      mrsubcn ralrimiva mrsubccat 3expb ralrimivva 3jca cmpt cun cfrmd cif ccom
       wa cgsu cword mrexval adantr s1eq fveq2d eqid fvex fvmpt adantl wn difun2
       eleq2i eldif bitr3i simpr2 eqeq12d rspccva sylan2br anassrs eqcomd ifeqda
-      sylan mpteq2dva coeq1d oveq2d mpteq12dv elun2 simpr1 simpr s1cld ad2antrr
-      cif wss eleqtrrd ffvelrnd sylan2 cbvmptv ssid mrsubfval sylancl cmnd cmhm
-      fmptd cvrmd cmcn fvexi cmvar unex frmdmnd ax-mp a1i eleqtrd fmpttd cplusg
-      feq23d simpr3 wb simprl simprr cbs frmdbas eqcomi frmdadd syl2anc ffvelrn
-      mpbid ad2ant2lr ad2ant2l 2ralbidva raleqbidv bitr3d chash cc0 caddc fveq2
-      raleqdv 3ad2antr1 cn0 wrd0 lencl syl nn0cnd 0cnd addid1d eleqtrrid oveq1d
-      fvoveq1 oveq2 ccatlid syl6eq rspc2va ccatlen 3eqtrrd hasheq0 sylib pm3.2i
-      syl21anc addcanad frmd0 ismhm mpbiran syl3anbrc fcompt vrmdval frmdup3lem
-      vrmdf eqtrd syl32anc 3eqtr4rd cpm cmap mrsubff ffnd cmrex elpm2r fnfvelrn
-      wfn mpanl12 eqeltrd ex impbid2 ) FIOZGEUDZOZDDGUEZJUFZUGZGPZUWKQZJCHUHZUI
-      ZAUFZBUFZUJRZGPZUWPGPZUWQGPZUJRZQZBDUIADUIZULZUWHUWIUWOUXDDEFGKLUKUWHUWMJ
-      UWNCDEFGHUWJKLMNUMUNUWHUXCABDDUWHUWPDOZUWQDOZUXCDEFGUWPUWQKLUOUPUQURUWFUX
-      EUWHUWFUXEVCZGUAHUAUFZUGZGPZUSZEPZUWGUXHUBDCHUTZVAPZUCUXNUCUFZHOZUXPUXLPZ
-      UXPUGZWPZUSZUBUFZVBZVDRZUSZUBUXNVEZUXOUCUXNUXSGPZUSZUYBVBZVDRZUSZUXMGUXHU
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-      TUYGUXHUXPUXNOZVCZUXQUXRUXSUYGUXQUXRUYGQUYPUAUXPUXKUYGHUXLUXIUXPQUXJUXSGU
-      XIUXPVHVIZUXLVJUXSGVKVLVMUYPUXQVNZVCUYGUXSUXHUYOUYRUYGUXSQZUYOUYRVCZUXHUX
-      PUWNOZUYSVUAUXPUXNHUHZOUYTVUBUWNUXPCHVOVPUXPUXNHVQVRUXHUWOVUAUYSUWFUWIUWO
-      UXDVSUWMUYSJUXPUWNUWJUXPQZUWLUYGUWKUXSVUCUWKUXSGUWJUXPVHZVIVUDVTWAWFWBWCW
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-      USVVTUXNUXOSVULVVTVJYEXNYFZVVCVJZYGYHVIVVQUWTUYFOUXAUYFOVVFUXBQVVQUWTDUYF
-      UWIUXFUWTDOUWFUXGDDUWPGYIYKVVSXPVVQUXADUYFUWIUXGUXADOUWFUXFDDUWQGYIYLVVSX
-      PUYFVVCUXNUXOUWTUXAVULVWAVWBYGYHVTYMVVNVVOVVHADUYFVVRVVNVVGBDUYFVVRYTYNYO
-      UUAYJUXHVVJYPPZYQQZVVKUXHVWCVWCYQUXHVWCUXHVVJUYFOZVWCUUBOUXHVVBTUYFOZVWEV
-      VLUXNUUCZUYFUYFTGYIXDZUXNVVJUUDUUEUUFZVWIUXHUUGUXHVWCYQYRRVWCVVJVVJUJRZYP
-      PZVWCVWCYRRZUXHVWCVWIUUHUXHVVJVWJYPUXHTDOZVWMUXDVVJVWJQZUXHTUYFDVWGUYNUUI
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-      WSUWTVVJUXAUJUWPTGYSUUJVTUWQTQZVWQVVJVWRVWJVWTVWPTGVWTVWPTTUJRZTUWQTTUJUU
-      LVWFVXATQVWGUXNTUUMXNUUNVIVWTUXAVVJVVJUJUWQTGYSWIVTUUOUVAVIUXHVWEVWEVWKVW
-      LQVWHVWHUXNVVJVVJUUPYHUUQUVBVVJSOVWDVVKYATGVKVVJSUURXNUUSVUOVUMVUMVCVVBVV
-      IVVKULVUMVUMVUTVUTUUTABUYFUYFVVCVVCUXOUXOGTTVWAVWAVWBVWBUXNUXOVULUVCZVXBU
-      VDUVEUVFUXHVUQUCUXNUXPVUPPZGPZUSZUYHUXHVVBUXNUYFVUPUEZVUQVXEQVVLVUNVXFVUS
-      VUPUXNSVUPVJZUVJXNUCGVUPUXNUYFUYFUVGXDUXHUCUXNVXDUYGUYPVXCUXSGUXHVUNUYOVX
-      CUXSQVVAUXPVUPUXNSVXGUVHWFVIWGUVKUBUYHUYFVUPGUXOUXNUXOSVULVWAVXGUVIUVLUVM
-      UXHEDHUVNRZUWAZUXLVXHOZUXMUWGOUWFVXIUXEUWFVXHDDUVOREDEFHIMLKUVPUVQVGUXHVU
-      EVUFVXJVUJVUKDSOHSOVUEVUFVCVXJDFUVRLXJVURDHHUXLSSUVSUWBXDVXHUXLEUVTYHUWCU
-      WDUWE $.
+      weq sylan mpteq2dva coeq1d oveq2d mpteq12dv wss elun2 simplr1 simpr s1cld
+      ad2antrr eleqtrrd ffvelrnd sylan2 cbvmptv ssid mrsubfval sylancl cmnd cvv
+      fmptd cmhm cvrmd cmcn fvexi cmvar frmdmnd ax-mp a1i eleqtrd fmpttd cplusg
+      unex simpr1 feq23d mpbid simpr3 simprl simprr cbs frmdbas frmdadd syl2anc
+      eqcomi ffvelrn ad2ant2lr ad2ant2l 2ralbidva chash cc0 caddc fveq2 syl6eq
+      wb raleqdv raleqbidv bitr3d 3ad2antr1 cn0 wrd0 lencl syl nn0cnd eleqtrrid
+      addid1d fvoveq1 oveq1d ccatidid rspc2va syl21anc ccatlen 3eqtrrd addcanad
+      0cnd oveq2 hasheq0 sylib pm3.2i frmd0 ismhm mpbiran syl3anbrc fcompt mpan
+      vrmdf vrmdval mpteq2ia frmdup3lem syl32anc 3eqtr4rd cpm cmap mrsubff ffnd
+      wfn cmrex elpm2r mpanl12 fnfvelrn eqeltrd ex impbid2 ) FIPZGEUCZPZDDGUDZJ
+      UEZUFZGQZUWNRZJCHUGZUHZAUEZBUEZUISZGQZUWSGQZUWTGQZUISZRZBDUHADUHZUJZUWKUW
+      LUWRUXGDEFGKLUKUWKUWPJUWQCDEFGHUWMKLMNULUMUWKUXFABDDUWKUWSDPZUWTDPZUXFDEF
+      GUWSUWTKLUNUOUPUQUWIUXHUWKUWIUXHVCZGUAHUAUEZUFZGQZURZEQZUWJUXKUBDCHUSZUTQ
+      ZOUXQOUEZHPZUXSUXOQZUXSUFZVAZURZUBUEZVBZVDSZURZUBUXQVEZUXROUXQUYBGQZURZUY
+      EVBZVDSZURZUXPGUXKUBDUYGUYIUYMUWIDUYIRZUXHCDFHINMLVFZVGZUXKUYFUYLUXRVDUXK
+      UYDUYKUYEUXKOUXQUYCUYJUXKUXSUXQPZVCZUXTUYAUYBUYJUXTUYAUYJRUYSUAUXSUXNUYJH
+      UXOUAOWFUXMUYBGUXLUXSVHVIZUXOVJUYBGVKVLVMUYSUXTVNZVCUYJUYBUXKUYRVUAUYJUYB
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+      RVUDVUBUWIUWLUWRUXGVSUWPVUBJUXSUWQJOWFZUWOUYJUWNUYBVUFUWNUYBGUWMUXSVHZVIV
+      UGVTWAWGWBWCWDWEWHWIWJWKUXKHDUXOUDZHHWLZUXPUYHRUXKOHUYJDUXOUXTUXKUYRUYJDP
+      UXSHCWMUYSDDUYBGUWLUWRUXGUWIUYRWNUYSUYBUYIDUYSUXSUXQUXKUYRWOWPUWIUYOUXHUY
+      RUYPWQZWRWSZWTUAOHUXNUYJUYTXAXGZHXBZOHCDEFUBUXOUXRHNMLKUXRVJZXCXDUXKUXRXE
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+      UXKUYIUYIGUDZUWSUWTUXRXRQZSZGQZUXCUXDVVDSZRZBUYIUHZAUYIUHZTGQZTRZVUQUXKUW
+      LVVCUWIUWLUWRUXGXTUXKDDUYIUYIGUYQUYQYAYBZUXKUXGVVJUWIUWLUWRUXGYCZUWIUWRUW
+      LUXGVVJYTUXGUWIUWLVCZVVHBDUHZADUHUXGVVJVVOVVHUXFABDDVVOUXIUXJVCZVCZVVFUXB
+      VVGUXEVVRVVEUXAGVVRUWSUYIPUWTUYIPVVEUXARVVRUWSDUYIVVOUXIUXJYDVVOUYOVVQUWI
+      UYOUWLUYPVGZVGZXPVVRUWTDUYIVVOUXIUXJYEVVTXPUYIVVDUXQUXRUWSUWTVUNUXRYFQZUY
+      IVUPVWAUYIRVVAVWAUXQUXRXFVUNVWAVJYGXNYJZVVDVJZYHYIVIVVRUXCUYIPUXDUYIPVVGU
+      XERVVRUXCDUYIUWLUXIUXCDPUWIUXJDDUWSGYKYLVVTXPVVRUXDDUYIUWLUXJUXDDPUWIUXID
+      DUWTGYKYMVVTXPUYIVVDUXQUXRUXCUXDVUNVWBVWCYHYIVTYNVVOVVPVVIADUYIVVSVVOVVHB
+      DUYIVVSUUAUUBUUCUUDYBUXKVVKYOQZYPRZVVLUXKVWDVWDYPUXKVWDUXKVVKUYIPZVWDUUEP
+      UXKVVCTUYIPVWFVVMUXQUUFZUYIUYITGYKXDZUXQVVKUUGUUHUUIZVWIUXKUUTUXKVWDYPYQS
+      VWDVVKVVKUISZYOQZVWDVWDYQSZUXKVWDVWIUUKUXKVVKVWJYOUXKTDPZVWMUXGVVKVWJRZUX
+      KTUYIDVWGUYQUUJZVWOVVNUXFVWNTUWTUISZGQZVVKUXDUISZRABTTDDUWSTRZUXBVWQUXEVW
+      RUWSTUWTGUIUULVWSUXCVVKUXDUIUWSTGYRUUMVTUWTTRZVWQVVKVWRVWJVWTVWPTGVWTVWPT
+      TUISTUWTTTUIUVAUUNYSVIVWTUXDVVKVVKUIUWTTGYRWJVTUUOUUPVIUXKVWFVWFVWKVWLRVW
+      HVWHUXQUXQVVKVVKUUQYIUURUUSVVKXFPVWEVVLYTTGVKVVKXFUVBXNUVCVUQVUOVUOVCVVCV
+      VJVVLUJVUOVUOVVBVVBUVDABUYIUYIVVDVVDUXRUXRGTTVWBVWBVWCVWCUXQUXRVUNUVEZVXA
+      UVFUVGUVHUXKVUSOUXQUXSVURQZGQZURZUYKUXKVVCUXQUYIVURUDZVUSVXDRVVMVUPVXEVVA
+      VURUXQXFVURVJZUVKXNOGVURUXQUYIUYIUVIXDOUXQVXCUYJUYRVXBUYBGVUPUYRVXBUYBRVV
+      AUXSVURUXQXFVXFUVLUVJVIUVMYSUBUYKUYIVURGUXRUXQUXRXFVUNVWBVXFUVNUVOUVPUXKE
+      DHUVQSZUWAZUXOVXGPZUXPUWJPUWIVXHUXHUWIVXGDDUVRSEDEFHIMLKUVSUVTVGUXKVUHVUI
+      VXIVULVUMDXFPHXFPVUHVUIVCVXIDFUWBLXKVUTDHHUXOXFXFUWCUWDXDVXGUXOEUWEYIUWFU
+      WGUWH $.
   $}
 
   ${
@@ -627326,13 +627292,13 @@ fixed reference functional determined by this vector (corresponding to
     ccatcan2d.c $e |- ( ph -> C e. Word V ) $.
     $( Cancellation law for concatenation.  (Contributed by SN, 6-Sep-2023.) $)
     ccatcan2d $p |- ( ph -> ( ( A ++ C ) = ( B ++ C ) <-> A = B ) ) $=
-      ( cconcat co wceq chash cfv cpfx cc wcel cn0 lencl syl adantr simpr cword
-      wa nn0cnd caddc fveq2d ccatlen syl2an2r 3eqtr3d addcan2ad oveq12d syl2anc
-      ex pfxccat1 eqeq12d sylibd oveq1 impbid1 ) ABDIJZCDIJZKZBCKZAVAUSBLMZNJZU
-      TCLMZNJZKZVBAVAVGAVAUCZUSUTVCVENAVAUAZVHVCVEDLMZAVCOPVAAVCABEUBZPZVCQPFEB
-      RSUDTAVEOPVAAVEACVKPZVEQPGECRSUDTAVJOPVAAVJADVKPZVJQPHEDRSUDTVHUSLMZUTLMZ
-      VCVJUEJZVEVJUEJZVHUSUTLVIUFAVLVAVNVOVQKFAVNVAHTZEBDUGUHAVMVAVNVPVRKGVSECD
-      UGUHUIUJUKUMAVDBVFCAVLVNVDBKFHEBDUNULAVMVNVFCKGHECDUNULUOUPBCDIUQUR $.
+      ( cconcat co wceq chash cfv cpfx cc wcel cn0 lencl adantr syl2anc ccatlen
+      wa simpr cword syl caddc fveq2 sylan9req eqtrd addcan2ad oveq12d pfxccat1
+      nn0cnd ex eqeq12d sylibd oveq1 impbid1 ) ABDIJZCDIJZKZBCKZAVAUSBLMZNJZUTC
+      LMZNJZKZVBAVAVGAVAUBZUSUTVCVENAVAUCVHVCVEDLMZAVCOPVAAVCABEUDZPZVCQPFEBRUE
+      UMSAVEOPVAAVEACVJPZVEQPGECRUEUMSAVIOPVAAVIADVJPZVIQPHEDRUEUMSVHVCVIUFJZUT
+      LMZVEVIUFJZAVAVNUSLMZVOAVKVMVQVNKFHEEBDUATUSUTLUGUHAVOVPKZVAAVLVMVRGHEECD
+      UATSUIUJUKUNAVDBVFCAVKVMVDBKFHEBDULTAVLVMVFCKGHECDULTUOUPBCDIUQUR $.
   $}
 
   ${
@@ -627571,11 +627537,11 @@ fixed reference functional determined by this vector (corresponding to
       syl caddc ccatlen cn0 cc0 wf cvv ovexd frlmbasf fnfzo0hash oveq12d 3eqtrd
       cfzo crg wa wb nn0addcld eqeltrrd frlmfzowrdb mpbir2and ) AEJUFUGZBUHZVRF
       UIUJZUKZUHZVRULUJZGUMZAEWAUHZJWAUHZWBAECUHZWEUDCVTFHLEORVTUNZUOURZAJDUHZW
-      FUEDVTFIMJPSWHUOURZVTEJUPUQAWCEULUJZJULUJZUSUGZHIUSUGZGAWEWFWCWNUMWIWKVTE
-      JUTUQAWLHWMIUSAHVAUHVBHVJUGZVTEVCZWLHUMUBAWPVDUHWGWQAVBHVJVEUDCFLWPVTVDEO
-      WHRVFUQVTEHVGUQAIVAUHVBIVJUGZVTJVCZWMIUMUCAWRVDUHWJWSAVBIVJVEUEDFMWRVTVDJ
-      PWHSVFUQVTJIVGUQVHUAVIAFVKUHGVAUHVSWBWDVLVMTAWOGVAUAAHIUBUCVNVOBVTFGVKKVR
-      NQWHVPUQVQ $.
+      FUEDVTFIMJPSWHUOURZVTEJUPUQAWCEULUJZJULUJZUSUGZHIUSUGZGAWEWFWCWNUMWIWKVTV
+      TEJUTUQAWLHWMIUSAHVAUHVBHVJUGZVTEVCZWLHUMUBAWPVDUHWGWQAVBHVJVEUDCFLWPVTVD
+      EOWHRVFUQVTEHVGUQAIVAUHVBIVJUGZVTJVCZWMIUMUCAWRVDUHWJWSAVBIVJVEUEDFMWRVTV
+      DJPWHSVFUQVTJIVGUQVHUAVIAFVKUHGVAUHVSWBWDVLVMTAWOGVAUAAHIUBUCVNVOBVTFGVKK
+      VRNQWHVPUQVQ $.
 
     $d x ph $.  $d x A $.  $d x L $.  $d x M $.  $d x N $.
     frlmvscadiccat.o $e |- O = ( .s ` W ) $.
@@ -738501,18 +738467,18 @@ Group sum operation (extension 1)
       FAEVCVJVEZYKVHSZYSYJYJVIVKYIYJYKMYIYJUUTVLZYIYKUVCVLZYIVMZVNYIUUEYJVAYIYJ
       VTHMVTHUUEYJQUVEVOYJMVPVQZWAVRYIYQROVAJZYIUUSYQRHUUTYJVHSVSWBZYIOYMWCLZAU
       UHYNYIOYNUHJZWDLZAYNTZUVKAYNTYIYNYBHZUVNYEYAUVOYHADEWEWFAYNWGSYIUVMUVKAYN
-      YIUVMOYLWDLZUVKYIUVLYLOWDYEYAUVLYLQYHADEWHWFWIYIYLWJHUVPUVKQYIYJYKUVAYIYK
-      UVCVFZWLOYLWKSWMWNWOZXBWPYIUUDYRUUGYTBYIBUAYNDOYQUVJYIUUHOYQWCLZHZIYCYDUU
-      HOYJWDLZHZUUHYNJUUHDJQYCYDYAYHUVTWQYCYDYAYHUVTWRYIUWBUVTYIUWAUVSUUHYIYJWJ
-      HUWAUVSQUVAOYJWKSZWSWTADEUUHXCXAXDYIUUGYSYJKLZBYNOYJKLZPZJYTYIYMUWDUUFUWF
-      YIUUEUWEBYNYIUUEYJUWEUVHYIYJUVEXEXFXGYIYMYKYJKLZMNLUWDYIYLUWGMNYIYJYKUVEU
-      VFXHXKYIYKYJMUVFUVEUVGXIWMXJYIBUAEYNYJOYSYIYSRUVIUVDVSWBZUVAYIUUHOYSWCLZH
-      ZIZUUHYJKLYNJZUUHEJZUWKYCYDUUHOYKWDLZHZUWLUWMQYCYDYAYHUWJWQYCYDYAYHUWJWRY
-      IUWOUWJYIUWNUWIUUHYIYKWJHUWNUWIQUVQOYKWKSZWSWTADEUUHXLXAXMXNXFXOWMYIABYNC
-      OYMUCFGUUNYIYMRUVIYIYLVBHYMRHYIYJYKUUTUVCXPYLVHSVSWBUVRXQYIUUBYRUUCYTBYIA
-      BDCOYQUCFGUUNUVJYIUWAADTZUVSADTYIYCUWQYAYCYDYHXRADWGSYIUWAUVSADUWCWNWOXQY
-      IABECOYSUCFGUUNUWHYIUWNAETZUWIAETYIYDUWRYAYCYDYHXTAEWGSYIUWNUWIAEUWPWNWOX
-      QXOXS $.
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   $}
 
   ${