reserve A for QC-alphabet;
reserve a,b,b1,b2,c,d for object,
  i,j,k,n for Nat,
  x,y,x1,x2 for bound_QC-variable of A,
  P for QC-pred_symbol of k,A,
  ll for CQC-variable_list of k,A,
  l1 ,l2 for FinSequence of QC-variables(A),
  p for QC-formula of A,
  s,t for QC-symbol of A;
reserve Sub for CQC_Substitution of A;
reserve finSub for finite CQC_Substitution of A;
reserve e for Element of vSUB(A);
reserve S,S9,S1,S2,S19,S29,T1,T2 for Element of QC-Sub-WFF(A);
reserve B for Element of [:QC-Sub-WFF(A),bound_QC-variables(A):];
reserve SQ for second_Q_comp of B;

theorem
  S1`2 = S2`2 & (S19)`2 = (S29)`2 & Sub_&(S1,S2) = Sub_&(S19,S29)
  implies S1 = S19 & S2 = S29
proof
  assume that
A1: S1`2 = S2`2 and
A2: (S19)`2 = (S29)`2 and
A3: Sub_&(S1,S2) = Sub_&(S19,S29);
  Sub_&(S1,S2) = [(S1`1) '&' (S2`1),S1`2] by A1,Def21;
  then
  [(S1`1) '&' (S2`1),S1`2] = [((S19)`1) '&' ((S29)`1),(S19)`2] by A2,A3,Def21;
  then
A4: (S1`1) '&' (S2`1) = ((S19)`1) '&' ((S29)`1) & S1`2 = (S19)`2 by XTUPLE_0:1;
A5: S2 = [S2`1,S2`2] & S29 = [(S29)`1,(S29)`2] by Th10;
  S1 = [S1`1,S1`2] & S19 = [(S19)`1,(S19)`2] by Th10;
  hence thesis by A1,A2,A4,A5,QC_LANG2:2;
end;
