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;
reserve Z for Element of [:QC-WFF(A),vSUB(A):];

theorem Th37:
  S1`2 = S2`2 & CQC_Sub(S1) is Element of CQC-WFF(A) & CQC_Sub(S2) is
  Element of CQC-WFF(A) implies CQC_Sub(Sub_&(S1,S2)) is Element of CQC-WFF(A)
proof
  assume
A1: S1`2 = S2`2 & CQC_Sub(S1) is Element of CQC-WFF(A) & CQC_Sub(S2) is
  Element of CQC-WFF(A);
  S1`2 = S2`2 implies CQC_Sub(Sub_&(S1,S2)) = (CQC_Sub(S1)) '&' (CQC_Sub(
  S2)) by Th31;
  hence thesis by A1,CQC_LANG:9;
end;
