reserve S for OrderSortedSign;
reserve S for OrderSortedSign,
  X for ManySortedSet of S,
  o for OperSymbol of S ,
  b for Element of ([:the carrier' of S,{the carrier of S}:] \/ Union (coprod X
  ))*;
reserve x for set;

theorem Th26:
  for S being locally_directed OrderSortedSign, X being non-empty
  ManySortedSet of S, R being ManySortedRelation of ParsedTermsOSA(X) holds R =
  PTCongruence(X) iff
( for s1,s2 being Element of S, x being object st x in X.s1
  holds ( s1 <= s2 implies [root-tree [x,s1],root-tree[x,s1]] in R.s2 ) &
 for y being object
 holds ( [root-tree [x,s1],y] in R.s2 or [y,root-tree [x,s1]] in R.s2)
implies s1 <= s2 & y = root-tree [x,s1] ) & for o1,o2 being OperSymbol of S, x1
  being Element of Args(o1,ParsedTermsOSA(X)), x2 being Element of Args(o2,
ParsedTermsOSA(X)), s3 being Element of S holds [Den(o1,ParsedTermsOSA(X)).x1,
Den(o2,ParsedTermsOSA(X)).x2] in R.s3 iff o1 ~= o2 & len the_arity_of o1 = len
  the_arity_of o2 & the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 &
ex w3 being Element of (the carrier of S)* st dom w3 = dom x1 & for y being Nat
  st y in dom w3 holds [x1.y,x2.y] in R.(w3/.y)
proof
  let S be locally_directed OrderSortedSign, X be non-empty ManySortedSet of S
  , R be ManySortedRelation of ParsedTermsOSA(X);
  set PTA = ParsedTermsOSA(X), SPTA = the Sorts of PTA, D = DTConOSA(X), OU =
[:the carrier' of S,{the carrier of S}:] \/ Union (coprod (X qua ManySortedSet
  of S)), C = bool [:TS(D), the carrier of S:], F = PTClasses X;
  defpred T[ ManySortedSet of the carrier of S] means
    ( for s1,s2 being Element of S, x being object st x in X.s1
 holds ( s1 <= s2 implies [root-tree [x,s1],root-tree[x,s1]] in $1.s2 ) &
for y being object holds ( [root-tree [x,s1],y]
in $1.s2 or [y,root-tree [x,s1]] in $1.s2) implies s1 <= s2 & y = root-tree [x,
  s1] );
  defpred NT[ ManySortedSet of the carrier of S] means for o1,o2 being
  OperSymbol of S, x1 being Element of Args(o1,ParsedTermsOSA(X)), x2 being
  Element of Args(o2,ParsedTermsOSA(X)), s3 being Element of S holds [Den(o1,
ParsedTermsOSA(X)).x1,Den(o2,ParsedTermsOSA(X)).x2] in $1.s3 iff o1 ~= o2 & len
  the_arity_of o1 = len the_arity_of o2 & the_result_sort_of o1 <= s3 &
the_result_sort_of o2 <= s3 & ex w3 being Element of (the carrier of S)* st dom
w3 = dom x1 & ( for y being Nat st y in dom w3 holds [x1.y,x2.y] in $1.(w3/.y)
  );
  set P = PTCongruence X;
A1: for R1,R2 be ManySortedRelation of PTA st T[R1] & NT[R1] & T[R2] & NT[R2
  ] holds R1 = R2
  proof
    let R1,R2 be ManySortedRelation of PTA;
    assume that
A2: T[R1] and
A3: NT[R1] and
A4: T[R2] and
A5: NT[R2];
    defpred P[set] means
    for x being object, s being Element of S holds [$1,x] in
    R1.s iff [$1,x] in R2.s;
A6: for nt being Symbol of D, ts being FinSequence of TS(D) st nt ==>
roots ts & for t being DecoratedTree of the carrier of D st t in rng ts holds P
    [t] holds P[nt-tree ts]
    proof
      let nt be Symbol of D, ts be FinSequence of TS(D) such that
A7:   nt ==> roots ts and
A8:   for t being DecoratedTree of the carrier of D st t in rng ts holds P[t];
      nt in { s where s is Symbol of D: ex n being FinSequence st s ==> n
      } by A7;
      then reconsider nt1 = nt as NonTerminal of D by LANG1:def 3;
      reconsider tss = ts as SubtreeSeq of nt1 by A7,DTCONSTR:def 6;
      let x be object, s be Element of S;
A9:   rng ts c= TS D by FINSEQ_1:def 4;
      [nt,roots ts] in the Rules of D by A7,LANG1:def 1;
      then reconsider rt = roots ts as Element of OU* by ZFMISC_1:87;
      reconsider sy = nt as Element of OU;
      [sy,rt] in OSREL(X) by A7,LANG1:def 1;
      then sy in [:the carrier' of S,{the carrier of S}:] by Def4;
      then consider
      o being Element of the carrier' of S, x2 being Element of {the
      carrier of S} such that
A10:  sy = [o,x2] by DOMAIN_1:1;
A11:  x2 = the carrier of S by TARSKI:def 1;
      then
A12:  (nt-tree tss).{} = [o,the carrier of S] by A10,TREES_4:def 4;
      then consider ts2 being SubtreeSeq of OSSym(o,X) such that
A13:  nt1-tree tss = OSSym(o,X)-tree ts2 and
      OSSym(o,X) ==> roots ts2 and
A14:  ts2 in Args(o,PTA) and
A15:  nt1-tree tss = Den(o,PTA).ts2 by Th11;
A16:  (the Sorts of PTA).s = ParsedTerms(X,s) by Def8
        .= {a1 where a1 is Element of TS(DTConOSA(X)): (ex s1 being Element
      of S, x be object st s1 <= s & x in X.s1 & a1 = root-tree [x,s1])
    or ex o be
OperSymbol of S st [o,the carrier of S] = a1.{} & the_result_sort_of o <= s};
      hereby
        assume
A17:    [nt-tree ts,x] in R1.s;
        then x in (the Sorts of PTA).s by ZFMISC_1:87;
        then consider a1 being Element of TS(DTConOSA(X)) such that
A18:    x = a1 and
A19:    (ex s1 being Element of S, y be object st s1 <= s & y in X.s1 &
a1 = root-tree [y,s1]) or ex o be OperSymbol of S st [o,the carrier of S] = a1.
        {} & the_result_sort_of o <= s by A16;
        not ex s1 being Element of S, y be set st s1 <= s & y in X.s1 &
        a1 = root-tree [y,s1]
        proof
          given s1 being Element of S, y be set such that
          s1 <= s and
A20:      y in X.s1 and
A21:      a1 = root-tree [y,s1];
          nt-tree ts = root-tree [y,s1] by A2,A17,A18,A20,A21;
          then [y,s1] = nt by TREES_4:17;
          then the carrier of S = s1 by A10,A11,XTUPLE_0:1;
          then s1 in s1;
          hence contradiction;
        end;
        then consider o1 be OperSymbol of S such that
A22:    [o1,the carrier of S] = a1.{} and
        the_result_sort_of o1 <= s by A19;
        consider ts1 being SubtreeSeq of OSSym(o1,X) such that
        a1 = OSSym(o1,X)-tree ts1 and
        OSSym(o1,X) ==> roots ts1 and
A23:    ts1 in Args(o1,PTA) and
A24:    a1 = Den(o1,PTA).ts1 by A22,Th11;
        consider ts2 being SubtreeSeq of OSSym(o,X) such that
A25:    nt1-tree tss = OSSym(o,X)-tree ts2 and
        OSSym(o,X) ==> roots ts2 and
A26:    ts2 in Args(o,PTA) and
A27:    nt1-tree tss = Den(o,PTA).ts2 by A12,Th11;
A28:    len the_arity_of o = len the_arity_of o1 by A3,A17,A18,A23,A24,A26,A27;
        reconsider tsb = ts2 as Element of Args(o,PTA) by A26;
        reconsider tsa = ts1 as Element of Args(o1,PTA) by A23;
        consider w3 being Element of (the carrier of S)* such that
A29:    dom w3 = dom tsb and
A30:    for y being Nat st y in dom w3 holds [tsb.y,tsa.y] in R1.(w3
        /.y) by A3,A17,A18,A24,A27;
A31:    ts2 = tss by A25,TREES_4:15;
A32:    for y being Nat st y in dom w3 holds [tsb.y,tsa.y] in R2.(w3/.y)
        proof
          let y be Nat such that
A33:      y in dom w3;
A34:      tsb.y in rng ts by A31,A29,A33,FUNCT_1:3;
          then reconsider t = tsb.y as Element of TS(D) by A9;
          [t,tsa.y] in R1.(w3/.y) by A30,A33;
          hence thesis by A8,A34;
        end;
A35:    the_result_sort_of o1 <= s by A3,A17,A18,A23,A24,A26,A27;
A36:    the_result_sort_of o <= s by A3,A17,A18,A23,A24,A26,A27;
        o ~= o1 by A3,A17,A18,A23,A24,A26,A27;
        hence [nt-tree ts,x] in R2.s by A5,A18,A24,A27,A28,A36,A35,A29,A32;
      end;
      reconsider tsb = ts2 as Element of Args(o,PTA) by A14;
      assume
A37:  [nt-tree ts,x] in R2.s;
      then x in (the Sorts of PTA).s by ZFMISC_1:87;
      then consider a1 being Element of TS(DTConOSA(X)) such that
A38:  x = a1 and
A39:  (ex s1 being Element of S, y be object st s1 <= s & y in X.s1 & a1
= root-tree [y,s1]) or ex o be OperSymbol of S st [o,the carrier of S] = a1.{}
      & the_result_sort_of o <= s by A16;
      not ex s1 being Element of S, y be set st s1 <= s & y in X.s1 & a1
      = root-tree [y,s1]
      proof
        given s1 being Element of S, y be set such that
        s1 <= s and
A40:    y in X.s1 and
A41:    a1 = root-tree [y,s1];
        nt-tree ts = root-tree [y,s1] by A4,A37,A38,A40,A41;
        then [y,s1] = nt by TREES_4:17;
        then the carrier of S = s1 by A10,A11,XTUPLE_0:1;
        then s1 in s1;
        hence contradiction;
      end;
      then consider o1 be OperSymbol of S such that
A42:  [o1,the carrier of S] = a1.{} and
      the_result_sort_of o1 <= s by A39;
      consider ts1 being SubtreeSeq of OSSym(o1,X) such that
      a1 = OSSym(o1,X)-tree ts1 and
      OSSym(o1,X) ==> roots ts1 and
A43:  ts1 in Args(o1,PTA) and
A44:  a1 = Den(o1,PTA).ts1 by A42,Th11;
A45:  len the_arity_of o = len the_arity_of o1 by A5,A37,A38,A43,A44,A14,A15;
      reconsider tsa = ts1 as Element of Args(o1,PTA) by A43;
      consider w3 being Element of (the carrier of S)* such that
A46:  dom w3 = dom tsb and
A47:  for y being Nat st y in dom w3 holds [tsb.y,tsa.y] in R2.(w3/.y
      ) by A5,A37,A38,A44,A15;
A48:  ts2 = tss by A13,TREES_4:15;
A49:  for y being Nat st y in dom w3 holds [tsb.y,tsa.y] in R1.(w3/.y)
      proof
        let y be Nat such that
A50:    y in dom w3;
A51:    tsb.y in rng ts by A48,A46,A50,FUNCT_1:3;
        then reconsider t = tsb.y as Element of TS(D) by A9;
        [t,tsa.y] in R2.(w3/.y) by A47,A50;
        hence thesis by A8,A51;
      end;
A52:  the_result_sort_of o1 <= s by A5,A37,A38,A43,A44,A14,A15;
A53:  the_result_sort_of o <= s by A5,A37,A38,A43,A44,A14,A15;
      o ~= o1 by A5,A37,A38,A43,A44,A14,A15;
      hence thesis by A3,A38,A44,A15,A45,A53,A52,A46,A49;
    end;
A54: for s being Symbol of D st s in Terminals D holds P[root-tree s]
    proof
      let sy be Symbol of D;
      assume sy in Terminals D;
      then consider s being Element of S, x being set such that
A55:  x in X.s and
A56:  sy = [x,s] by Th4;
      let y be object, s1 be Element of S;
      hereby
        assume
A57:    [root-tree sy,y] in R1.s1;
        then
A58:    y = root-tree [x,s] by A2,A55,A56;
        s <= s1 by A2,A55,A56,A57;
        hence [root-tree sy,y] in R2.s1 by A4,A55,A56,A58;
      end;
      assume
A59:  [root-tree sy,y] in R2.s1;
      then
A60:  y = root-tree [x,s] by A4,A55,A56;
      s <= s1 by A4,A55,A56,A59;
      hence thesis by A2,A55,A56,A60;
    end;
A61: for t being DecoratedTree of the carrier of D st t in TS(D) holds P[t
    ] from DTCONSTR:sch 7(A54,A6);
    for i being object st i in the carrier of S holds R1.i = R2.i
    proof
      let i be object;
      assume i in the carrier of S;
      then reconsider s = i as Element of S;
      for a,b being object holds [a,b] in R1.s iff [a,b] in R2.s
      proof
        let a,b be object;
A62:    (the Sorts of PTA).s = ParsedTerms(X,s) by Def8
          .= {a1 where a1 is Element of TS(DTConOSA(X)): (ex s1 being
Element of S, x be object
st s1 <= s & x in X.s1 & a1 = root-tree [x,s1]) or ex o
be OperSymbol of S st [o,the carrier of S] = a1.{} & the_result_sort_of o <= s}
        ;
        hereby
          assume
A63:      [a,b] in R1.s;
          then a in (the Sorts of PTA).s by ZFMISC_1:87;
          then ex a1 being Element of TS(DTConOSA(X)) st a = a1 &( (ex s1
being Element of S, x be object
st s1 <= s & x in X.s1 & a1 = root-tree [x,s1]) or
ex o be OperSymbol of S st [o,the carrier of S] = a1.{} & the_result_sort_of o
          <= s) by A62;
          hence [a,b] in R2.s by A61,A63;
        end;
        assume
A64:    [a,b] in R2.s;
        then a in (the Sorts of PTA).s by ZFMISC_1:87;
        then
        ex a1 being Element of TS(DTConOSA(X)) st a = a1 &( (ex s1 being
Element of S, x be object
st s1 <= s & x in X.s1 & a1 = root-tree [x,s1]) or ex o
be OperSymbol of S st [o,the carrier of S] = a1.{} & the_result_sort_of o <= s)
        by A62;
        hence thesis by A61,A64;
      end;
      hence thesis by RELAT_1:def 2;
    end;
    hence thesis;
  end;
A65: NT[PTCongruence X]
  proof
    let o1,o2 be OperSymbol of S, x1 be Element of Args(o1,ParsedTermsOSA(X)),
    x2 be Element of Args(o2,ParsedTermsOSA(X)), s3 be Element of S;
A66: dom (the_arity_of o2) = dom x2 by MSUALG_3:6;
A67: P.s3 = {[x3,y] where x3,y is Element of TS(DTConOSA(X)): [x3,s3] in
    F.y} by Def22;
    reconsider ts2 = x2 as FinSequence of TS(D) by Th13;
A68: dom (the_arity_of o1) = dom x1 by MSUALG_3:6;
    reconsider ts1 = x1 as FinSequence of TS(D) by Th13;
A69: rng ts1 c= TS D by FINSEQ_1:def 4;
    reconsider x = F * ts1 as FinSequence of C;
    dom F = TS D by FUNCT_2:def 1;
    then
A70: len x = len ts1 by A69,FINSEQ_2:29;
    then
A71: dom x = dom ts1 by FINSEQ_3:29;
A72: OSSym(o1,X) ==> roots x1 by Th13;
    then
A73: F.(OSSym(o1,X)-tree ts1) = @(OSSym(o1,X),x) by Def21
      .= {[Den(o3,ParsedTermsOSA(X)).x3,s4] where o3 is OperSymbol of S, x3
is Element of Args(o3,ParsedTermsOSA(X)), s4 is Element of S : ( ex o4 being
    OperSymbol of S st OSSym(o1,X) = [o4,the carrier of S] & o4 ~= o3 & len
    the_arity_of o4 = len the_arity_of o3 & the_result_sort_of o4 <= s4 &
the_result_sort_of o3 <= s4 ) & ex w3 being Element of (the carrier of S)* st
    dom w3 = dom x & for y being Nat st y in dom x holds [x3.y,w3/.y] in x.y};
A74: OSSym(o2,X) ==> roots x2 by Th13;
    then reconsider
    tx = OSSym(o1,X)-tree ts1, ty = OSSym(o2,X)-tree ts2 as Element
    of TS D by A72,Th12;
A75: Den(o2,PTA).x2 = ((PTOper(X)).o2).x2 by MSUALG_1:def 6
      .= PTDenOp(o2,X).ts2 by Def10
      .= OSSym(o2,X)-tree ts2 by A74,Def9;
A76: Den(o1,PTA).x1 = ((PTOper(X)).o1).x1 by MSUALG_1:def 6
      .= PTDenOp(o1,X).ts1 by Def10
      .= OSSym(o1,X)-tree ts1 by A72,Def9;
A77: rng ts2 c= TS D by FINSEQ_1:def 4;
    hereby
      assume [Den(o1,PTA).x1,Den(o2,PTA).x2] in P.s3;
      then consider t1,t2 being Element of TS D such that
A78:  [Den(o1,PTA).x1,Den(o2,PTA).x2] = [t1,t2] and
A79:  [t1,s3] in F.t2 by A67;
A80:  Den(o1,PTA).x1 = t1 by A78,XTUPLE_0:1;
A81:  Den(o2,PTA).x2 = t2 by A78,XTUPLE_0:1;
      [t2,s3] in F.t1 by A79,Th19;
      then consider o3 being OperSymbol of S, x3 being Element of Args(o3,
      ParsedTermsOSA(X)), s4 being Element of S such that
A82:  [t2,s3] = [Den(o3,PTA).x3,s4] and
A83:  ex o4 being OperSymbol of S st OSSym(o1,X) = [o4,the carrier
      of S] & o4 ~= o3 & len the_arity_of o4 = len the_arity_of o3 &
      the_result_sort_of o4 <= s4 & the_result_sort_of o3 <= s4 and
A84:  ex w3 being Element of (the carrier of S)* st dom w3 = dom x &
      for y being Nat st y in dom x holds [x3.y,w3/.y] in x.y by A76,A73,A80;
      consider o4 being OperSymbol of S such that
A85:  OSSym(o1,X) = [o4,the carrier of S] and
A86:  o4 ~= o3 and
A87:  len the_arity_of o4 = len the_arity_of o3 and
A88:  the_result_sort_of o4 <= s4 and
A89:  the_result_sort_of o3 <= s4 by A83;
A90:  o1 = o4 by A85,XTUPLE_0:1;
      reconsider ts3 = x3 as FinSequence of TS(D) by Th13;
A91:  OSSym(o3,X) ==> roots x3 by Th13;
A92:  t2 = Den(o3,PTA).x3 by A82,XTUPLE_0:1;
A93:  Den(o3,PTA).x3 = ((PTOper(X)).o3).x3 by MSUALG_1:def 6
        .= PTDenOp(o3,X).ts3 by Def10
        .= OSSym(o3,X)-tree ts3 by A91,Def9;
      then
A94:  OSSym(o3,X) = OSSym(o2,X) by A75,A81,A92,TREES_4:15;
      s3 = s4 by A82,XTUPLE_0:1;
      hence o1 ~= o2 & len the_arity_of o1 = len the_arity_of o2 &
      the_result_sort_of o1 <= s3 & the_result_sort_of o2 <= s3 by A86,A87,A88
,A89,A90,A94,XTUPLE_0:1;
      consider w3 being Element of (the carrier of S)* such that
A95:  dom w3 = dom x and
A96:  for y being Nat st y in dom x holds [x3.y,w3/.y] in x.y by A84;
      take w3;
      thus dom w3 = dom x1 by A70,A95,FINSEQ_3:29;
      let y be Nat such that
A97:  y in dom w3;
A98:  ts1.y in rng ts1 by A71,A95,A97,FUNCT_1:3;
A99:  ts3 = ts2 by A75,A81,A92,A93,TREES_4:15;
      o3 = o2 by A94,XTUPLE_0:1;
      then dom the_arity_of o1 = dom the_arity_of o2 by A87,A90,FINSEQ_3:29;
      then ts2.y in rng ts2 by A71,A68,A66,A95,A97,FUNCT_1:3;
      then reconsider
      t22 = ts2.y,t11=ts1.y as Element of TS D by A69,A77,A98;
      [x3.y,w3/.y] in x.y by A95,A96,A97;
      then [ts2.y,w3/.y] in F.(ts1.y) by A99,A95,A97,FUNCT_1:12;
      then
A100: [t11,w3/.y] in F.(t22) by Th19;
      P.(w3/.y) = {[x5,y5] where x5,y5 is Element of TS(DTConOSA(X)): [
      x5,w3/.y] in F.y5} by Def22;
      hence [x1.y,x2.y] in P.(w3/.y) by A100;
    end;
    assume that
A101: o1 ~= o2 and
A102: len the_arity_of o1 = len the_arity_of o2 and
A103: the_result_sort_of o1 <= s3 and
A104: the_result_sort_of o2 <= s3 and
A105: ex w3 being Element of (the carrier of S)* st dom w3 = dom x1 &
    for y being Nat st y in dom w3 holds [x1.y,x2.y] in P.(w3/.y);
    consider w3 being Element of (the carrier of S)* such that
A106: dom w3 = dom x1 and
A107: for y being Nat st y in dom w3 holds [x1.y,x2.y] in P.(w3/.y) by A105;
    for y being Nat st y in dom x holds [x2.y,w3/.y] in x.y
    proof
      let y being Nat such that
A108: y in dom x;
A109: P.(w3/.y) = {[x5,y5] where x5,y5 is Element of TS(DTConOSA(X)): [
      x5,w3/.y] in F.y5} by Def22;
      [x1.y,x2.y] in P.(w3/.y) by A71,A106,A107,A108;
      then consider x5,y5 being Element of TS(DTConOSA(X)) such that
A110: [x1.y,x2.y] = [x5,y5] and
A111: [x5,w3/.y] in F.y5 by A109;
A112: x1.y = x5 by A110,XTUPLE_0:1;
A113: x2.y = y5 by A110,XTUPLE_0:1;
      [y5,w3/.y] in F.x5 by A111,Th19;
      hence thesis by A108,A112,A113,FUNCT_1:12;
    end;
    then [Den(o2,PTA).x2,s3] in F.(OSSym(o1,X)-tree ts1) by A71,A73,A101,A102
,A103,A104,A106;
    then [tx,s3] in F.(ty) by A75,Th19;
    hence [Den(o1,PTA).x1,Den(o2,PTA).x2] in P.s3 by A67,A76,A75;
  end;
  T[P]
  proof
    let s1,s2 be Element of S, x be object such that
A114: x in X.s1;
    reconsider sy = [x,s1] as Terminal of D by A114,Th4;
A115: P.s2 = {[x1,y] where x1,y is Element of TS(DTConOSA(X)): [x1,s2] in F
    .y} by Def22;
A116: root-tree [x,s1] in SPTA.s1 by A114,Th10;
    hereby
      assume
A117: s1 <= s2;
      [root-tree sy,s1] in F.(root-tree sy) by A116,Th19;
      then [root-tree sy,s2] in F.(root-tree sy) by A116,A117,Th21;
      hence [root-tree [x,s1],root-tree[x,s1]] in P.s2 by A115;
    end;
    let y be object;
    assume
A118: [root-tree [x,s1],y] in P.s2 or [y,root-tree [x,s1]] in P.s2;
    then
A119: root-tree [x,s1] in SPTA.s2 by ZFMISC_1:87;
    field(P.s2) = SPTA.s2 by ORDERS_1:12;
    then
A120: P.s2 is_symmetric_in SPTA.s2 by RELAT_2:def 11;
A121: F.(root-tree sy) = @(sy) by Def21
      .= {[root-tree sy,s3] where s3 is Element of S: ex s4 be Element of S,
    x be set st x in X.s4 & sy = [x,s4] & s4 <= s3};
    y in SPTA.s2 by A118,ZFMISC_1:87;
    then [y,root-tree sy] in P.s2 by A120,A118,A119,RELAT_2:def 3;
    then consider y1,r1 being Element of TS D such that
A122: [y,root-tree sy] = [y1,r1] and
A123: [y1,s2] in F.(r1) by A115;
A124: y = y1 by A122,XTUPLE_0:1;
    root-tree sy = r1 by A122,XTUPLE_0:1;
    then consider s3 being Element of S such that
A125: [y1,s2] = [root-tree sy,s3] and
A126: ex s4 be Element of S, x be set st x in X.s4 & sy = [x,s4] & s4
    <= s3 by A121,A123;
    s2 = s3 by A125,XTUPLE_0:1;
    hence thesis by A124,A125,A126,XTUPLE_0:1;
  end;
  hence thesis by A1,A65;
end;
