reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;
reserve
  B,A,M for BinOp of D,
  F,G for D* -valued FinSequence,
  f for FinSequence of D,
  d,d1,d2 for Element of D;
reserve
  F,G for non-empty non empty FinSequence of D*,
  f for non empty FinSequence of D;
reserve f,g for FinSequence of D,
        a,b,c for set,
        F,F1,F2 for finite set;

theorem
  A is commutative associative having_a_unity having_an_inverseOp &
  M is associative commutative having_a_unity &
  M is_distributive_wrt A & m > 1 &
  (for d holds M.(the_unity_wrt A,d) = the_unity_wrt A)
implies
     ex E be Enumeration of bool(Seg m\{1}),
        S be Subset of doms(m,card bool(Seg m\{1})) st
       S is with_evenly_repeated_values-member &
       card bool(Seg m\{1}) |-> 1 in S &
       for CE be non-empty non empty FinSequence of D*,f st
          CE = SignGenOp(f,A,bool(Seg m\{1})) * E & len f = m
          for Sce be Element of Fin dom App CE st Sce=S holds
             SignGenOp(f,M,A,Seg m\{1}) = A $$ (Sce,M "**" App CE)
proof
  set I=the_inverseOp_wrt A;
  assume that
A1: A is commutative associative having_a_unity having_an_inverseOp &
  M is associative commutative having_a_unity &
  M is_distributive_wrt A and
A2: m > 1 and
A3: for d holds M.(the_unity_wrt A,d) = the_unity_wrt A;
  defpred P[Nat] means
  ex E be Enumeration of bool(Seg $1\{1}),
     S be Subset of doms($1,card bool(Seg $1\{1})) st
       S is with_evenly_repeated_values-member &
       card bool(Seg $1\{1}) |-> 1 in S &
       for CE be non-empty non empty FinSequence of D*,f st
          CE = SignGenOp(f,A,bool(Seg $1\{1})) * E & len f = $1
          for Sce be Element of Fin dom App CE st Sce=S holds
             SignGenOp(f,M,A,Seg $1\{1}) = A $$ (Sce,M "**" App CE);
A4:P[2]
  proof
    set B=bool(Seg 2\{1});
A5: Seg 2\{1} = {2} by ZFMISC_1:17,FINSEQ_1:2;
    then B = { {}, {2} } by ZFMISC_1:24;
    then
A6:card B=2 by CARD_2:57;
    set S = { <*1,1*>, <*2,2*>};
    S c= doms(2,card B)
    proof
      1 in Seg 2 & 2 in Seg 2;
      then
      reconsider O=1,T=2 as Element of Seg 2;
      let x;
      assume x in S;
      then
A7: x=<*O,O*> or x= <*T,T*> by TARSKI:def 2;
      then reconsider x as FinSequence of Seg 2;
      len x =2 by A7,CARD_1:def 7;
      then x is Element of (card B)-tuples_on Seg 2 by A6,FINSEQ_2:92;
      hence thesis;
    end;
    then reconsider S = { <*1,1*>, <*2,2*>} as Subset of doms(2,card B);
    take E = the Enumeration of bool(Seg 2\{1});
    take S;
    thus S is with_evenly_repeated_values-member;
    A8: card bool(Seg 2\{1}) = card bool(Seg (1+1) \{1})
    .= 2 * card bool(Seg 1 \{1}) by Th9;
    Seg (1+0)\{1} = {} by FINSEQ_1:2;
    then card bool (Seg (1+0)\{1}) = 1 by ZFMISC_1:1,CARD_1:30;
    then card bool(Seg 2\{1}) |-> 1 = <*1,1*> by A8,FINSEQ_2:61;
    hence card bool(Seg 2\{1}) |-> 1 in S by TARSKI:def 2;
    let CE be non-empty non empty FinSequence of D*,f such that
A9:   CE = SignGenOp(f,A,B) * E & len f = 2;
    let Sce be Element of Fin dom App CE such that
A10: Sce=S;
    consider Sd be Element of Fin dom App CE such that
A11:   Sd = { <*1,1*>, <*2,2*>} &
    SignGenOp(f,M,A,{2}) = A $$ (Sd,M "**" App CE) by A1,A5,A9,Th141;
    thus SignGenOp(f,M,A,Seg 2\{1}) = A $$ (Sce,M "**" App CE)
      by ZFMISC_1:17,FINSEQ_1:2,A10,A11;
  end;
A12: for j being Nat st 2 <= j holds P[j] implies P[j+1]
  proof
    let j be Nat;
    assume
A13:2 <= j;
    set F=bool(Seg j\{1}),Dj=doms(j,card F);
    set ccc = card F |-> 1;
    assume P[j];
    then consider E be Enumeration of F,S be Subset of doms(j,card F) such that
A14: S is with_evenly_repeated_values-member and
A15: ccc in S and
A16:for CE be non-empty non empty FinSequence of D*,f st
      CE = SignGenOp(f,A,bool(Seg j\{1})) * E & len f = j
      for Sce be Element of Fin dom App CE st Sce=S holds
      SignGenOp(f,M,A,Seg j\{1}) = A $$ (Sce,M "**" App CE);
    reconsider m=j-1 as Nat by A13;
    reconsider Dj as non empty finite set by A13;
    set EF=Ext(F,1+m,2+m),SF=swap(F,1+m,2+m);
A17:union F= Seg j\{1}=Seg (m+1)\{1} by ZFMISC_1:81;
    then
A18: union F c= Seg (1+m) by XBOOLE_1:36;
A19: 1+m+1 > 1+m by NAT_1:13;
    then
A20: not m+2 in union F by A18,FINSEQ_1:1;
    then reconsider Ee = Ext(E,1+m,2+m) as Enumeration of EF by Th28;
    reconsider Es = Swap(E,1+m,2+m) as Enumeration of SF by A20,Th27;
    reconsider Ees=Ee^Es as Enumeration of EF\/SF by Th79,A20,A19,Th22;
    defpred P[object,object] means
    for f,g be object st f in Dj & g in Dj & $1= [f,g] holds $2={f,g};
A21: for x st x in [:Dj,Dj:] ex y st P[x,y]
    proof
      let x such that
A22:x in [:Dj,Dj:];
      consider x1,x2 be object such that
A23:x1 in Dj & x2 in Dj & [x1,x2] =x by A22,ZFMISC_1:def 2;
      take X={x1,x2};
      let f,g be object such that
A24: f in Dj & g in Dj & x = [f,g];
      f=x1 & g = x2 by A23,A24,XTUPLE_0:1;
      hence thesis;
    end;
    consider h be Function such that
A25: dom h = [:Dj,Dj:] &
    for x st x in [:Dj,Dj:] holds P[x,h.x] from CLASSES1:sch 1(A21);
    set H=h|[:S,S:];
A26: [:S,S:] c= [:Dj,Dj:] by ZFMISC_1:96;
A27: dom H = [:S,S:] by A25,RELAT_1:62,ZFMISC_1:96;
    reconsider R=rng H as finite set;
    set C = canFS R;
A28:rng C = R by FUNCT_2:def 3;
    defpred I[Nat] means $1 <= len C implies
    for EC be Element of Fin [:Dj,Dj:] st EC = H"rng (C|$1) holds
    ex D12 be Subset of doms(m+2,card F+card F) st
    ([ccc,ccc] in EC implies ccc^ccc in D12) &
    D12 is with_evenly_repeated_values-member &
    for CE1,CE2 be FinSequence of D*,f,d1,d2 st
    len f = m &
    CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
    CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E
    for M1 be Function of Dj,D, M2 be Function of Dj,D st
    M1=M"**"App CE1 & M2=M"**"App CE2
    for CFes be non-empty non empty FinSequence of D* st
    CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
    Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees
    for S12 be Element of Fin dom App CFes st S12=D12
    holds
    A $$ (EC,M*( M1, M2)) =
    A$$(S12,M"**"App CFes) &
    for h be FinSequence st h in S12
    ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in EC &
    for i st i in dom (s1^s2) holds
    ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
    ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i);
A29: for i st I[i] holds I[i+1]
    proof
      let i;
      assume
A30: I[i];
      set i1=i+1;
      assume
A31: i1 <= len C;
      let HC be Element of Fin [:Dj,Dj:] such that
A32: HC = H"rng (C|i1);
A33: i1 in dom C by A31,FINSEQ_3:25,NAT_1:11;
      then C|i1 = (C|i)^<*C.i1*> by FINSEQ_5:10;
      then rng (C|i1) = rng (C|i)\/ rng <*C.i1*> by FINSEQ_1:31;
      then
A34: rng (C|i1) = rng (C|i)\/ { C.i1 } by FINSEQ_1:39;
A35: C.i1 in rng C by A33,FUNCT_1:def 3;
A36: H"rng (C|i1) = H"rng (C|i) \/ H"{C.i1} by A34,RELAT_1:140;
      then H"rng (C|i) c= HC c= [:Dj,Dj:] by A32,FINSUB_1:def 5,XBOOLE_1:7;
      then H"rng (C|i) c= [:Dj,Dj:];
      then reconsider HC1 = H"rng (C|i) as Element of Fin [:Dj,Dj:]
        by FINSUB_1:def 5;
      consider D12 be Subset of doms(m+2,card F+card F) such that
A37:   ([ccc,ccc] in HC1 implies ccc^ccc in D12) and
A38:   D12 is with_evenly_repeated_values-member and
A39:   for CE1,CE2 be FinSequence of D*,f,d1,d2 st
      len f = m &
      CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
      CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E
      for M1 be Function of Dj,D, M2 be Function of Dj,D st
      M1=M"**"App CE1 & M2=M"**"App CE2
      for CFes be non-empty non empty FinSequence of D* st
      CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
      Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees
      for S12 be Element of Fin dom App CFes st S12=D12
      holds
        A $$ (HC1,M*( M1, M2)) = A$$(S12,M"**"App CFes) &
        for h be FinSequence st h in S12
        ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in HC1 &
        for i st i in dom (s1^s2) holds
        ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
        ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i) by A30,A31,NAT_1:13;
      H"{C.i1}<>{} by A35,FUNCT_1:72;
      then consider s12 be object such that
A40:  s12 in H"{C.i1} by XBOOLE_0:def 1;
A41:  s12 in dom H & H.s12 in {C.i1} by A40,FUNCT_1:def 7;
      then consider s1,s2 be object such that
A42:  s1 in S & s2 in S & s12=[s1,s2] by ZFMISC_1:def 2;
      reconsider s1,s2 as Element of (card F)-tuples_on Seg j by A42;
      s12 in dom h & h.s12 = H.s12 by A41,RELAT_1:57,FUNCT_1:47;
      then
A43:  C.i1 = H.s12 = {s1,s2} by A42,A41,A25,TARSKI:def 1;
      reconsider s12 as Element of [:Dj,Dj:] by A41,RELAT_1:57,A25;
A44:  not C.i1 in rng (C|i)
      proof
        assume C.i1 in rng(C|i);
        then consider x such that
A45:    x in dom (C|i) & C.i1 = (C|i).x by FUNCT_1:def 3;
A46:    x in dom C & x in Seg i & (C|i).x = C.x by A45,RELAT_1:57,FUNCT_1:47;
        then x=i1 by A45,A33,FUNCT_1:def 4;
        then i1 <=i by A46,FINSEQ_1:1;
        hence thesis by NAT_1:13;
      end;
      per cases;
      suppose
A47:    s1=s2;
        H"{C.i1} c= {[s1,s2]}
        proof
          let y;
          assume y in H"{C.i1};
          then
A48:      y in dom H & H.y in {C.i1} by FUNCT_1:def 7;
          consider t1,t2 be object such that
A49:      t1 in S & t2 in S & y=[t1,t2] by A48,ZFMISC_1:def 2;
          y in dom h & h.y = H.y by A48,RELAT_1:57,FUNCT_1:47;
          then C.i1 = H.y = {t1,t2} by A25,A48,A49,TARSKI:def 1;
          then {s1}={s1,s1}={t1,t2} by A43,A47,ENUMSET1:29;
          then t2=s1=t1 by ZFMISC_1:4;
          hence thesis by TARSKI:def 1,A47,A49;
        end;
        then
A50: H"{C.i1} = {s12} by A42,A40,ZFMISC_1:33;
        consider D1 be Subset of doms(m+2,card F+card F) such that
A51:     card (s1"{1+m})=0 implies s1^s2 in D1 and
A52:    D1 is with_evenly_repeated_values-member and
A53:    for CE1,CE2 be FinSequence of D*, f,d1,d2 st len f = m &
        CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
        CE2 = SignGenOp(f^<*A.( (the_inverseOp_wrt A).d1,d2)*>,A,F) * E
        for CFes be non-empty non empty FinSequence of D* st
        CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
        Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees
        for Sd be Element of Fin dom App CFes st D1 = Sd holds
        M.((M "**" App CE1).s1,(M "**" App CE2).s2)= A$$(Sd,M"**"App CFes) &
        for h be FinSequence,i st h in Sd & i in dom h holds
        ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
        ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i)
        by A47,A1,A18,Th126,A42,A14,Th40;
        reconsider DD=D12\/D1 as Subset of doms(m+2,card F+card F);
        take DD;
        thus [ccc,ccc] in HC implies ccc^ccc in DD
        proof
          assume [ccc,ccc] in HC;
          then per cases by XBOOLE_0:def 3,A50,A36,A32;
          suppose [ccc,ccc] in HC1;
            hence thesis by A37,XBOOLE_0:def 3;
          end;
          suppose [ccc,ccc] in {s12};
            then [ccc,ccc] =s12 by TARSKI:def 1;
            then
A54:          ccc = s1 & ccc = s2 by A42,XTUPLE_0:1;
            1+0<>1+m by A13;
            then not 1 in {1+m} by TARSKI:def 1;
            then ccc"{1+m} = {} by FUNCOP_1:16;
            hence thesis by A54,A51,XBOOLE_0:def 3;
          end;
        end;
        thus DD is with_evenly_repeated_values-member by A38,A52;
        let CE1,CE2 be FinSequence of D*,f,d1,d2 such that
A55:    len f = m and
A56:    CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
        CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E;
        let M1 be Function of Dj,D, M2 be Function of Dj,D such that
A57: M1=M"**"App CE1 & M2=M"**"App CE2;
        let CFes be non-empty non empty FinSequence of D* such that
A58:     CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
        Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees;
        let SS be Element of Fin dom App CFes such that
A59:    SS=DD;
        D1 c= DD & D12 c= DD & DD c= dom App CFes
        by A59,XBOOLE_1:7,FINSUB_1:def 5;
        then D1 c= dom App CFes & D12 c= dom App CFes;
        then reconsider S1=D1,S12=D12 as Element of Fin dom App CFes
          by FINSUB_1:def 5;
A60:    A $$ (HC1,M*( M1, M2)) = A$$(S12,M"**"App CFes) &
          for h be FinSequence st h in S12
          ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in HC1 &
          for i st i in dom (s1^s2) holds
          ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i) by A39,A55,A56,A57,A58;
A61:    M.((M "**" App CE1).s1,(M "**" App CE2).s2)
          = A$$(S1,M"**"App CFes) &
        for h be FinSequence,i st h in S1 & i in dom h holds
          ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i) by A53,A55,A56,A58;
        not s12 in HC1 by A44,A43,FUNCT_1:def 7;
        then
A62:    HC1 misses {.s12.} by ZFMISC_1:50;
A63:    HC = HC1\/{.s12.} by A50,A34,RELAT_1:140,A32;
A64:    A $$ (HC,M*( M1, M2)) =
        A.(A $$ (HC1,M*( M1, M2)),A $$ ({.s12.},M*( M1, M2)))
        by A1,A62,SETWOP_2:4,A50,A36,A32;
        (M*( M1, M2)).(s1,s2) = (M*( M1, M2)).s12 by A42,BINOP_1:def 1;
        then M.(M1.s1,M2.s2) = (M*( M1, M2)).s12 by FINSEQOP:81;
        then
A65:    A$$(S1,M"**"App CFes) =
        A $$ ({.s12.},M*( M1, M2)) by A57,A61,A1,SETWISEO:17;
        CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
        Ext(F,1+m,2+m)\/swap(F,1+m,2+m) ) * Ees by A58,A55;
        then len CFes = len Ees by CARD_1:def 7;
        then
A66:     len CFes = len Ee+len Es & len Ee = len E = len Es
        by FINSEQ_1:22,CARD_1:def 7;
A67:    len s1 = card F = len s2 by CARD_1:def 7,A13;
A68:    S12 misses S1
        proof
          assume S12 meets S1;
          then consider x such that
A69:      x in S12 & x in S1 by XBOOLE_0:3;
          reconsider x as FinSequence by A69;
          consider d1,d2 be FinSequence such that
A70:      d1 in Dj & d2 in Dj & [d1,d2] in HC1 and
A71:      for i st i in dom (d1^d2) holds
          ((d1^d2).i = 1+len f implies x.i in {1+len f,2+len f}) &
          ((d1^d2).i <> 1+len f implies x.i =(d1^d2).i)
          by A69,A39,A55,A56,A57,A58;
          reconsider d1,d2 as Element of (card F)-tuples_on Seg j by A70;
A72:      len d1 = card F =len d2 & card F = len E by CARD_1:def 7,A13;
          S1 c= dom App CFes by FINSUB_1:def 5;
          then x in dom App CFes by A69;
          then
A73:      len x = len CFes by Th47;
          len (s1^s2) = len x = len (s2^s1) by CARD_1:def 7,A72, A73,A66;
          then
A74:      dom (s1^s2)= dom x = dom (s2^s1) by FINSEQ_3:30;
A75:      for i st 1<=i & i <= card F holds s1.i=d1.i
          proof
            let i such that
A76:        1<=i & i <= card F;
A77:        i in dom s1 c= dom (s1^s2) by A76,A67,FINSEQ_1:26,FINSEQ_3:25;
            then
A78:        (s1^s2).i in rng (s1^s2) by FUNCT_1:def 3;
A79:        (s1^s2).i=s1.i by FINSEQ_1:def 7,A77;
A80:        i in dom d1 c= dom (d1^d2) by A76,A72,FINSEQ_1:26,FINSEQ_3:25;
            then
A81:        (d1^d2).i in rng (d1^d2) by FUNCT_1:def 3;
A82:        (d1^d2).i=d1.i by FINSEQ_1:def 7,A80;
A83:        ((s1^s2).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s1^s2).i <> 1+len f implies x.i =(s1^s2).i)
            by A74,A69,A77,A53,A55,A56,A58;
            per cases;
            suppose
A84:          (d1^d2).i = 1+len f;
              then
A85:          x.i in {1+len f,2+len f} by A80,A71;
              reconsider d=(s1^s2).i as Nat;
              rng (s1^s2) c= Seg j by RELAT_1:def 19;
              then d <=m+1 by A78,FINSEQ_1:1;
              then
A86:          d <m+1+1 by NAT_1:13;
              (s1^s2).i = 1+len f by A83,A85,TARSKI:def 2,A86,A55;
              hence thesis by A84,A79,FINSEQ_1:def 7,A80;
            end;
            suppose
A87:          (d1^d2).i <> 1+len f;
              then
A88:          (d1^d2).i = x.i by A80,A71;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A81,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not d in {1+len f,2+len f} by A55,A87,TARSKI:def 2;
              hence thesis by A88,A79,A82,A74,A69,A77,A53,A55,A56,A58;
            end;
          end;
          for k st 1<=k & k <= card F holds s2.k=d2.k
          proof
            let k such that
A89:        1<=k & k <= card F;
            set i = k+card F;
A90:        k in dom s2 by A89,A67,FINSEQ_3:25;
            then
A91:        i in dom (s1^s2) by A67,FINSEQ_1:28;
            then
A92:        (s1^s2).i in rng (s1^s2) by FUNCT_1:def 3;
A93:        (s1^s2).i=s2.k by FINSEQ_1:def 7,A90,A67;
A94:         k in dom d2 by A89,A72,FINSEQ_3:25;
            then
A95:        i in dom (d1^d2) by A72,FINSEQ_1:28;
            then
A96:        (d1^d2).i in rng (d1^d2) by FUNCT_1:def 3;
A97:        (d1^d2).i=d2.k by FINSEQ_1:def 7,A94,A72;
A98:        ((s1^s2).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s1^s2).i <> 1+len f implies x.i =(s1^s2).i)
            by A74,A69,A91,A53,A55,A56,A58;
            per cases;
            suppose
A99:          (d1^d2).i = 1+len f;
              then
A100:         x.i in {1+len f,2+len f} by A95,A71;
              reconsider d=(s1^s2).i as Nat;
              rng (s1^s2) c= Seg j by RELAT_1:def 19;
              then d<=j by A92,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              hence thesis by A99,A93,A97,A98,A100,TARSKI:def 2,A55;
            end;
            suppose
A101:         (d1^d2).i <> 1+len f;
              then
A102:         (d1^d2).i = x.i by A95,A71;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A96,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not d in {1+len f,2+len f} by A55,A101,TARSKI:def 2;
              hence thesis by A102,A93,A97,A74,A69,A91,A53,A55,A56,A58;
            end;
          end;
          then
          d1=s1 & d2=s2 by A75,CARD_1:def 7,A13,A72;
          hence thesis by A42,A44,A43,FUNCT_1:def 7,A70;
        end;
        thus A $$ (HC,M*( M1, M2)) =
        A$$(SS,M"**"App CFes) by A59,A64,A65,A60,A1,A68,SETWOP_2:4;
        let h be FinSequence such that
A103:   h in SS;
        per cases by A103,A59,XBOOLE_0:def 3;
        suppose h in S12;
          then consider s1,s2 be FinSequence such that
A104:     s1 in Dj & s2 in Dj & [s1,s2] in HC1 &
          for i st i in dom (s1^s2) holds
          ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i) by A39,A55,A56,A57,A58;
          take s1,s2;
          thus thesis by A104,A63,XBOOLE_0:def 3;
        end;
        suppose
A105:     h in S1;
          take s1,s2;
          S1 c= dom App CFes by FINSUB_1:def 5;
          then h in dom App CFes by A105;
          then
A106:     len h = len CFes by Th47;
          len (s1^s2) = len s1 +len s2 & len E=card F
          by FINSEQ_1:22,CARD_1:def 7;
          then
A107:     dom (s1^s2)= dom h  by A67, A106,A66,FINSEQ_3:30;
          s12 in {.s12.} by TARSKI:def 1;
          hence thesis by A107,A42,A105,A53,A55,A56,A58,A63,XBOOLE_0:def 3;
        end;
      end;
      suppose
A108:   s1<>s2;
A109:   [s2,s1] in dom H by A27,A42,ZFMISC_1:def 2;
        then [s2,s1] in dom h & h.[s2,s1] = H.[s2,s1] by RELAT_1:57,FUNCT_1:47;
        then
A110:   H.[s2,s1] = {s2,s1} by A25;
        then
A111:   [s2,s1] in H"{C.i1} by A41,A43,A109,FUNCT_1:def 7;
        reconsider s21=[s2,s1] as Element of [:Dj,Dj:] by ZFMISC_1:def 2;
A112:   [s1,s2]<>[s2,s1] by A108,XTUPLE_0:1;
        then
A113:   not [s1,s2] in {[s2,s1]} & not [s2,s1] in {[s1,s2]} by TARSKI:def 1;
        H"{C.i1} c= {[s1,s2],[s2,s1]}
        proof
          let y;
          assume y in H"{C.i1};
          then
A115:     y in dom H & H.y in {C.i1} by FUNCT_1:def 7;
          then consider t1,t2 be object such that
A116:     t1 in S & t2 in S & y=[t1,t2] by ZFMISC_1:def 2;
          y in dom h & h.y = H.y by A115,RELAT_1:57,FUNCT_1:47;
          then
A117:     C.i1 = H.y = {t1,t2} by A25,A115,A116,TARSKI:def 1;
A118:     t1<>t2
          proof
            assume t1=t2;
            then {t1}={t1,t2} by ENUMSET1:29;
            then s1=t1=s2 by ZFMISC_1:4,A117,A43;
            hence thesis by A108;
          end;
          t1 in {s1,s2} & t2 in {s1,s2} by A43,A117,TARSKI:def 2;
          then (t1=s1 or t1=s2) & (t2=s1 or t2=s2) by TARSKI:def 2;
          hence thesis by TARSKI:def 2,A118,A116;
        end;
        then
A119:   H"{C.i1} = {[s1,s2],[s2,s1]} by A42,A40,A111,A113,ZFMISC_1:36;
        consider D1,D2 be Subset of doms(m+2,card F+card F) such that
A120:   D1 is with_evenly_repeated_values-member &
          D2 is with_evenly_repeated_values-member and
A121:   for CE1,CE2 be FinSequence of D*,f,d1,d2 st
          len f = m &
          CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
          CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E
          for CFes be non-empty non empty FinSequence of D* st
          CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
          Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees
          for S1,S2 be Element of Fin dom App CFes st
          S1=D1 & S2=D2 holds
          S1 misses S2 &
          A.(M.((M"**"App CE1).s1,(M"**"App CE2).s2),
          M.((M"**"App CE1).s2,(M"**"App CE2).s1))= A$$(S1\/S2,M"**"App CFes) &
          (for h be FinSequence,i st h in S1 & i in dom (s1^s2) holds
          ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i)) &
          (for h be FinSequence,i st h in S2 & i in dom (s2^s1) holds
          ((s2^s1).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s2^s1).i <> 1+len f implies h.i =(s2^s1).i))
          by Th139,A108,A1,A17,XBOOLE_1:36,A42,A14;
        reconsider DD=D12\/(D1\/D2) as Subset of doms(m+2,card F+card F);
        take DD;
        thus [ccc,ccc] in HC implies ccc^ccc in DD
        proof
          assume [ccc,ccc] in HC;
          then [ccc,ccc] in HC1 \/ H"{C.i1} by A34,RELAT_1:140,A32;
          then per cases by XBOOLE_0:def 3,A119;
          suppose [ccc,ccc] in HC1;
            hence thesis by A37,XBOOLE_0:def 3;
          end;
          suppose [ccc,ccc] in {[s1,s2],[s2,s1]};
            then [ccc,ccc] =[s1,s2] or [ccc,ccc] =[s2,s1] by TARSKI:def 2;
            then ccc = s1 & ccc = s2 by XTUPLE_0:1;
            hence thesis by A108;
          end;
        end;
        thus DD is with_evenly_repeated_values-member by A38,A120;
        let CE1,CE2 be FinSequence of D*,f,d1,d2 such that
A122:   len f = m and
A123:   CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
        CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E;
        let M1 be Function of Dj,D, M2 be Function of Dj,D such that
A124:   M1=M"**"App CE1 & M2=M"**"App CE2;
        let CFes be non-empty non empty FinSequence of D* such that
A125:   CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
        Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees;
        let SS be Element of Fin dom App CFes such that
A126:   SS=DD;
        D1\/(D2\/D12) = DD =D2\/(D1\/D12) by XBOOLE_1:4;
        then D1 c= DD & D2 c= DD & D12 c= DD & DD c= dom App CFes
        by A126,XBOOLE_1:7,FINSUB_1:def 5;
        then D1 c= dom App CFes & D2 c= dom App CFes & D12 c= dom App CFes;
        then reconsider S1=D1,S2=D2,S12=D12 as Element of Fin dom App CFes
        by FINSUB_1:def 5;
A127:   A $$ (HC1,M*( M1, M2)) = A$$(S12,M"**"App CFes) &
        for h be FinSequence st h in S12
        ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in HC1 &
        for i st i in dom (s1^s2) holds
        ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
        ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i)
        by A39,A122,A123,A124,A125;
A128:   S1=D1 & S2=D2;
        then
A129:   S1 misses S2 &
        A.(M.((M"**"App CE1).s1,(M"**"App CE2).s2),
        M.((M"**"App CE1).s2,(M"**"App CE2).s1))= A$$(S1\/S2,M"**"App CFes) &
        (for h be FinSequence,i st h in S1 & i in dom (s1^s2) holds
        ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
        ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i)) &
        (for h be FinSequence,i st h in S2 & i in dom (s2^s1) holds
        ((s2^s1).i = 1+len f implies h.i in {1+len f,2+len f}) &
        ((s2^s1).i <> 1+len f implies h.i =(s2^s1).i)) by A121,A122,A123,A125;
A130:   not s12 in HC1 by A44,A43,FUNCT_1:def 7;
A131:   not s21 in HC1 by A44,A43,A110,FUNCT_1:def 7;
A132:   HC1 misses {.s12,s21.} by A131,A130,ZFMISC_1:51;
A133:   HC = HC1\/{.s12,s21.} by A34,RELAT_1:140,A32,A119,A42;
A134:   A $$ (HC,M*( M1, M2)) =
        A.(A $$ (HC1,M*( M1, M2)),A $$ ({.s12,s21.},M*( M1 , M2)))
        by A1,A132,SETWOP_2:4,A36,A32,A119,A42;
        (M*( M1 , M2)).(s1,s2) = (M*( M1, M2)).s12 &
        (M*( M1 , M2)).(s2,s1) = (M*( M1, M2)).s21 by A42,BINOP_1:def 1;
        then
A135:   M.(M1.s1,M2.s2) = (M*( M1, M2)).s12 &
        M.(M1.s2,M2.s1) = (M*( M1, M2)).s21 by FINSEQOP:81;
A136:   A$$(S1\/S2,M"**"App CFes) = A $$ ({.s12,s21.},M*( M1,M2))
        by A129,A135,A124,A1,A112,A42,SETWOP_2:1;
A137:   S12 misses S1
        proof
          assume S12 meets S1;
          then consider x such that
A138:     x in S12 & x in S1 by XBOOLE_0:3;
          reconsider x as FinSequence by A138;
          consider d1,d2 be FinSequence such that
A139:     d1 in Dj & d2 in Dj & [d1,d2] in HC1 and
A140:     for i st i in dom (d1^d2) holds
          ((d1^d2).i = 1+len f implies x.i in {1+len f,2+len f}) &
          ((d1^d2).i <> 1+len f implies x.i =(d1^d2).i)
          by A138,A39,A122,A123,A124,A125;
          reconsider d1,d2 as Element of (card F)-tuples_on Seg j by A139;
A141:     len s1 = card F = len s2 by CARD_1:def 7,A13;
A142:     len d1 = card F =len d2 by CARD_1:def 7,A13;
          for i st 1<=i & i <= card F holds s1.i=d1.i
          proof
            let i such that
A143:       1<=i & i <= card F;
A144:       i in dom s1 c= dom (s1^s2)
            by A143,A141,FINSEQ_1:26,FINSEQ_3:25;
            then
A145:       (s1^s2).i in rng (s1^s2) by FUNCT_1:def 3;
A146:       (s1^s2).i=s1.i by FINSEQ_1:def 7,A144;
A147:       i in dom d1 c= dom (d1^d2)
            by A143,A142,FINSEQ_1:26,FINSEQ_3:25;
A148:       (d1^d2).i in rng (d1^d2) by A147,FUNCT_1:def 3;
A149:       (d1^d2).i=d1.i by FINSEQ_1:def 7,A147;
A150:       ((s1^s2).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s1^s2).i <> 1+len f implies x.i =(s1^s2).i)
            by A138,A144,A121,A122,A123,A125,A128;
            per cases;
            suppose
A151:         (d1^d2).i = 1+len f;
              then
A152:         x.i in {1+len f,2+len f} by A147,A140;
              reconsider d=(s1^s2).i as Nat;
              rng (s1^s2) c= Seg j by RELAT_1:def 19;
              then d<=j by A145,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              hence thesis by A151,A146,A149,A150,A152,TARSKI:def 2,A122;
            end;
            suppose
A153:         (d1^d2).i <> 1+len f;
              then
A154:         (d1^d2).i = x.i by A147,A140;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A148,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not x.i in {1+len f,2+len f} by A122,A154,A153,TARSKI:def 2;
              hence thesis by A154,A146,A149,A138,A144,A129;
            end;
          end;
          then
A155:     s1=d1 by A141,CARD_1:def 7,A13;
          for k st 1<=k & k <= card F holds s2.k=d2.k
          proof
            let k such that
A156:       1<=k & k <= card F;
            set i = k+card F;
A157:       k in dom s2 by A156,A141,FINSEQ_3:25;
            then
A158:       i in dom (s1^s2) by A141,FINSEQ_1:28;
            then
A159:       (s1^s2).i in rng (s1^s2) by FUNCT_1:def 3;
A160:       (s1^s2).i=s2.k by FINSEQ_1:def 7,A157,A141;
A161:       k in dom d2 by A156,A142,FINSEQ_3:25;
            then
A162:       i in dom (d1^d2) by A142,FINSEQ_1:28;
            then
A163:       (d1^d2).i in rng (d1^d2) by FUNCT_1:def 3;
A164:       (d1^d2).i=d2.k by FINSEQ_1:def 7,A161,A142;
A165:       ((s1^s2).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s1^s2).i <> 1+len f implies x.i =(s1^s2).i)
            by A138,A158,A121,A122,A123,A125,A128;
            per cases;
            suppose
A166:         (d1^d2).i = 1+len f;
              then
A167:         x.i in {1+len f,2+len f} by A162,A140;
              reconsider d=(s1^s2).i as Nat;
              rng (s1^s2) c= Seg j by RELAT_1:def 19;
              then d<=j by A159,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              hence thesis by A166,A160,A164,A122,A165,A167,TARSKI:def 2;
            end;
            suppose
A168:         (d1^d2).i <> 1+len f;
              then
A169:         (d1^d2).i = x.i by A162,A140;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A163,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not d in {1+len f,2+len f} by A122,A168,TARSKI:def 2;
              hence thesis by A169,A160,A164,A158,A138,A129;
            end;
          end;
          then s2=d2 by A141,CARD_1:def 7,A13;
          hence thesis by A44,A43,FUNCT_1:def 7,A139,A155,A42;
        end;
        S12 misses S2
        proof
          assume S12 meets S2;
          then consider x such that
A170:     x in S12 & x in S2 by XBOOLE_0:3;
          reconsider x as FinSequence by A170;
          consider d1,d2 be FinSequence such that
A171:     d1 in Dj & d2 in Dj & [d1,d2] in HC1 and
A172:     for i st i in dom (d1^d2) holds
          ((d1^d2).i = 1+len f implies x.i in {1+len f,2+len f}) &
          ((d1^d2).i <> 1+len f implies x.i =(d1^d2).i)
          by A170,A39,A122,A123,A124,A125;
          reconsider d1,d2 as Element of (card F)-tuples_on Seg j by A171;
A173:     len s1 = card F = len s2 by CARD_1:def 7,A13;
A174:     len d1 = card F =len d2 by CARD_1:def 7,A13;
          for i st 1<=i & i <= card F holds s2.i=d1.i
          proof
            let i such that
A175:       1<=i & i <= card F;
A176:       i in dom s2 c= dom (s2^s1) by A175,A173,FINSEQ_1:26,FINSEQ_3:25;
            then
A177:       (s2^s1).i in rng (s2^s1) by FUNCT_1:def 3;
A178:       (s2^s1).i=s2.i by FINSEQ_1:def 7,A176;
A179:       i in dom d1 c= dom (d1^d2) by A175,A174,FINSEQ_1:26,FINSEQ_3:25;
            then
A180:       (d1^d2).i in rng (d1^d2) by FUNCT_1:def 3;
A181:       (d1^d2).i=d1.i by FINSEQ_1:def 7,A179;
A182:       ((s2^s1).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s2^s1).i <> 1+len f implies x.i =(s2^s1).i)
            by A170,A176,A128,A121,A122,A123,A125;
            per cases;
            suppose
A183:         (d1^d2).i = 1+len f;
              then
A184:         x.i in {1+len f,2+len f} by A179,A172;
              reconsider d=(s2^s1).i as Nat;
              rng (s2^s1) c= Seg j by RELAT_1:def 19;
              then d<=j by A177,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              hence thesis by A183,A178,A181,A182,A184,A122,TARSKI:def 2;
            end;
            suppose
A185:         (d1^d2).i <> 1+len f;
              then
A186:         (d1^d2).i = x.i by A179,A172;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A180,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not d in {1+len f,2+len f} by A122,A185,TARSKI:def 2;
              hence thesis by A186,A178,A181,A170,A176,A129;
            end;
          end;
          then
A187:     s2=d1 by A173,CARD_1:def 7,A13;
          for k st 1<=k & k <= card F holds s1.k=d2.k
          proof
            let k such that
A188:       1<=k & k <= card F;
            set i = k+card F;
A189:       k in dom s1 by A188,A173,FINSEQ_3:25;
            then
A190:       i in dom (s2^s1) by A173,FINSEQ_1:28;
            then
A191:       (s2^s1).i in rng (s2^s1) by FUNCT_1:def 3;
A192:       (s2^s1).i=s1.k by FINSEQ_1:def 7,A189,A173;
A193:       k in dom d2 by A188,A174,FINSEQ_3:25;
            then
A194:       i in dom (d1^d2) by A174,FINSEQ_1:28;
            then
A195:       (d1^d2).i in rng (d1^d2) by FUNCT_1:def 3;
A196:       (d1^d2).i=d2.k by FINSEQ_1:def 7,A193,A174;
A197:       ((s2^s1).i = 1+len f implies x.i in {1+len f,2+len f}) &
            ((s2^s1).i <> 1+len f implies x.i =(s2^s1).i)
            by A170,A190,A128,A121,A122,A123,A125;
            per cases;
            suppose
A198:         (d1^d2).i = 1+len f;
              then
A199:         x.i in {1+len f,2+len f} by A194,A172;
              reconsider d=(s2^s1).i as Nat;
              rng (s2^s1) c= Seg j by RELAT_1:def 19;
              then d<=j by A191,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              hence thesis by A198,A192,A196,A122,A197,A199,TARSKI:def 2;
            end;
            suppose
A200:         (d1^d2).i <> 1+len f;
              then
A201:         (d1^d2).i = x.i by A194,A172;
              reconsider d=(d1^d2).i as Nat;
              rng (d1^d2) c= Seg j by RELAT_1:def 19;
              then d<=j by A195,FINSEQ_1:1;
              then d <m+1+1 by NAT_1:13;
              then not d in {1+len f,2+len f} by A122,A200,TARSKI:def 2;
              hence thesis by A201,A192,A196,A170,A190,A129;
            end;
          end;
          then s1=d2 by A173,CARD_1:def 7,A13;
          hence thesis by A44,A43,A110,FUNCT_1:def 7,A171,A187;
        end;
        then S12 misses (S1\/S2) by A137,XBOOLE_1:70;
        hence A $$ (HC,M*( M1, M2))
        = A$$(SS,M"**"App CFes) by A134,A136,A126,A127,A1,SETWOP_2:4;
        let h be FinSequence such that
A202:   h in SS;
        per cases by A202,A126,XBOOLE_0:def 3;
        suppose h in S12;
          then consider s1,s2 be FinSequence such that
A203:     s1 in Dj & s2 in Dj & [s1,s2] in HC1 &
          for i st i in dom (s1^s2) holds
          ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
          ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i)
          by A39,A122,A123,A124,A125;
          take s1,s2;
          thus thesis by A203,A133,XBOOLE_0:def 3;
        end;
        suppose h in S1\/S2;
          then per cases by XBOOLE_0:def 3;
          suppose
A204:       h in S1;
            take s1,s2;
            s12 in {.s12,s21.} by TARSKI:def 2;
            hence thesis
            by A42,A204,A128,A121,A122,A123,A125,A133,XBOOLE_0:def 3;
          end;
          suppose
A205:       h in S2;
            take s2,s1;
            s21 in {.s12,s21.} by TARSKI:def 2;
            hence thesis by A205,A128,A121,A122,A123,A125,A133,XBOOLE_0:def 3;
          end;
        end;
      end;
    end;
A206: I[0]
    proof
      assume 0 <= len C;
      let EC be Element of Fin [:Dj,Dj:] such that
A207: EC = H"rng (C|0);
      reconsider D12 = {} as Subset of doms(m+2,card F+card F) by XBOOLE_1:2;
      take D12;
      thus [ccc,ccc] in EC implies ccc^ccc in D12 by A207;
      thus  D12 is with_evenly_repeated_values-member;
      let CE1,CE2 be FinSequence of D*,f,d1,d2 such that
      len f = m &
      CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
      CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E;
      let M1 be Function of Dj,D, M2 be Function of Dj,D such that
      M1=M"**"App CE1 & M2=M"**"App CE2;
      let CFes be non-empty non empty FinSequence of D* such that
      CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
      Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees;
      let S12 be Element of Fin dom App CFes such that
A208: S12=D12;
A209: EC={}.[:Dj,Dj:] by A207;
      S12 = {}.(dom App CFes) by A208;then
      A$$(S12,M"**"App CFes) = the_unity_wrt A =A $$ (EC,M*( M1, M2))
      by A209,A1,SETWISEO:31;
      hence thesis by A208;
    end;
A210: for i holds I[i] from NAT_1:sch 2(A206,A29);
    reconsider ss = [:S,S:] as Element of Fin [:Dj,Dj:]
    by A26,FINSUB_1:def 5;
    reconsider s = S as Element of Fin Dj by FINSUB_1:def 5;
    H"rng (C|len C) = dom H by A28,RELAT_1:134;
    then
A211: H"rng (C|len C) = [:S,S:] by A25,RELAT_1:62,ZFMISC_1:96;
    consider D12 be Subset of doms(m+2,card F+card F) such that
A212: ([ccc,ccc] in [:S,S:] implies ccc^ccc in D12) and
A213: D12 is with_evenly_repeated_values-member and
A214: for CE1,CE2 be FinSequence of D*,f,d1,d2 st
    len f = m & CE1 = SignGenOp(f^<*A . (d1,d2)*>,A,F) * E &
    CE2 = SignGenOp(f^<*A.( I.d1,d2)*>,A,F) * E
    for M1 be Function of Dj,D, M2 be Function of Dj,D st
    M1=M"**"App CE1 & M2=M"**"App CE2
    for CFes be non-empty non empty FinSequence of D* st
    CFes = SignGenOp(f^<*d1*>^<*d2*>,A,
    Ext(F,1+len f,2+len f)\/swap(F,1+len f,2+len f) ) * Ees
    for S12 be Element of Fin dom App CFes st S12=D12
    holds A $$ (ss,M*( M1, M2)) = A$$(S12,M"**"App CFes) &
    for h be FinSequence st h in S12
    ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in ss &
    for i st i in dom (s1^s2) holds
    ((s1^s2).i = 1+len f implies h.i in {1+len f,2+len f}) &
    ((s1^s2).i <> 1+len f implies h.i =(s1^s2).i) by A211,A210;
A215: m<>0 by A13;
    then
A216: EF\/SF = bool(Seg (j+1)\{1}) by Th39;
    reconsider EES = Ees as Enumeration of bool(Seg (j+1)\{1}) by Th39,A215;
    take EES;
A217: 2 * card F = card bool (Seg (1+j)\{1}) by A13,Th9;
    then reconsider T12=D12 as Subset of doms(j+1,card bool(Seg (j+1)\{1}));
    take T12;
    thus T12 is with_evenly_repeated_values-member by A213;
    [ccc,ccc] in [:S,S:] by A15,ZFMISC_1:def 2;
    hence card bool (Seg (j+1)\{1}) |-> 1 in T12
      by A217,A212,FINSEQ_2:123;
    let CE be non-empty non empty FinSequence of D*,f such that
A218: CE = SignGenOp(f,A,bool(Seg (j+1)\{1})) * EES & len f = j+1;
    let Tce be Element of Fin dom App CE such that
A219: Tce=T12;
    set g=f|m, d1 = f/.(m+1),d2 = f/.(j+1);
A220: m < m+1 =j < j+1 by NAT_1:13;
    1<= m+1 by NAT_1:11;
    then m+1 in dom f by A218,A220,FINSEQ_3:25;
    then
A221: f.(m+1) = d1 by PARTFUN1:def 6;
    m < len f by A220,A218,XXREAL_0:2;
    then
A222: f| (m+1) = g ^ <* d1*> by A221,FINSEQ_5:83;
    1<= j+1 by NAT_1:11;
    then j+1 in dom f by A218,FINSEQ_3:25;
    then
A223: f.(j+1) = f/.(j+1) by PARTFUN1:def 6;
    then
A224: f = g^<*d1*>^<*d2*> by A222,A218,FINSEQ_3:55;
A225: len g = m by A220,A218,XXREAL_0:2,FINSEQ_1:59;
    then
A226: len (g^<*A.(d1,d2)*>) = j = len (g^<*A.(I.d1,d2)*>) by FINSEQ_2:16;
    set CE1 = SignGenOp(g^<*A . (d1,d2)*>,A,F) * E,
    CE2 = SignGenOp(g^<*A.( I.d1,d2)*>,A,F) * E;
A227: dom App CE2 = Dj = dom App CE1 by A226,Lm3;
A228: for x st x in dom CE1 holds CE1.x is non empty
    proof
      let x;
      assume x in dom CE1;
      then CE1.x = SignGen(g^<*A.(d1,d2)*>,A,E.x) by Th80;
      hence thesis;
    end;
    reconsider CE1 as non-empty non empty FinSequence of D*
    by A228,FUNCT_1:def 9;
    for x st x in dom CE2 holds CE2.x is non empty
    proof
      let x;
      assume x in dom CE2;
      then CE2.x = SignGen(g^<*A . (I.d1,d2)*>,A,E.x) by Th80;
      hence thesis;
    end;
    then reconsider CE2 as non-empty non empty FinSequence of D*
    by FUNCT_1:def 9;
    reconsider s1=S as Element of Fin dom App CE1 by A227,FINSUB_1:def 5;
    reconsider s2=S as Element of Fin dom App CE2 by A227,FINSUB_1:def 5;
    set M1=M"**"App CE1,M2=M"**"App CE2;
    reconsider M1,M2 as Function of Dj,D by A227;
A230: CE = SignGenOp(g^<*d1*>^<*d2*>,A,
    Ext(F,1+len g,2+len g)\/swap(F,1+len g,2+len g) ) * Ees
    by A225,A223,A222,FINSEQ_3:55,A216,A218;
    M1=M"**"App CE1 & M2=M"**"App CE2
    implies for CE be non-empty non empty FinSequence of D* st
    CE = SignGenOp(g^<*d1*>^<*d2*>,A,
    Ext(F,1+len g,2+len g)\/swap(F,1+len g,2+len g) ) * Ees
    for S12 be Element of Fin dom App CE st S12=D12
    holds A $$ (ss,M*( M1, M2)) = A$$(S12,M"**"App CE) &
    for h be FinSequence st h in S12
    ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in ss &
    for i st i in dom (s1^s2) holds
    ((s1^s2).i = 1+len g implies h.i in {1+len g,2+len g}) &
    ((s1^s2).i <> 1+len g implies h.i =(s1^s2).i) by A225,A214;
    then
A231: A $$ (ss,M*( M1, M2)) = A$$(Tce,M"**"App CE) &
    for h be FinSequence st h in Tce
    ex s1,s2 be FinSequence st s1 in Dj & s2 in Dj & [s1,s2] in ss &
    for i st i in dom (s1^s2) holds
    ((s1^s2).i = 1+len g implies h.i in {1+len g,2+len g}) &
    ((s1^s2).i <> 1+len g implies h.i =(s1^s2).i) by A219,A230;
A232:  SignGenOp(g^<*A.(d1,d2)*>,M,A,Seg j\{1}) = A $$ (s1,M "**" App CE1)
    by A226,A16;
A233: SignGenOp(g^<*A.(I.d1,d2)*>,M,A,Seg j\{1}) = A $$ (s2,M "**" App CE2)
    by A226,A16;
    len g <>0 by A225,A13;
    then SignGenOp(g^<*d1*>^<*d2*>,M,A,Seg (2+len g)\{1}) =
       M. (SignGenOp(g^<*A.(d1,d2)*>,M,A,Seg (1+len g)\{1}),
           SignGenOp(g^<*A.(I.d1,d2)*>,M,A,Seg (1+len g)\{1}))
     by A1,Th130;
     hence SignGenOp(f,M,A,Seg (j+1)\{1})
     = A$$(Tce,M"**"App CE) by A224,A232,A233,A225,A227,A1,Th3,A3,A231;
   end;
A234: for i being Nat st 2 <= i holds P[i] from NAT_1:sch 8(A4,A12);
   m>=1+1 by A2,NAT_1:13;
   hence thesis by A234;
 end;
