reserve Al for QC-alphabet;
reserve a,b,c,d for object,
  i,k,n for Nat,
  p,q for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  A for non empty set,
  J for interpretation of Al,A,
  v,w for Element of Valuations_in(Al,A),
  f,g for Function,
  P,P9 for QC-pred_symbol of k,Al,
  ll,ll9 for CQC-variable_list of k,Al,
  l1 for FinSequence of QC-variables(Al),
  Sub,Sub9,Sub1 for CQC_Substitution of Al,
  S,S9,S1,S2 for Element of CQC-Sub-WFF(Al),
  s for QC-symbol of Al;
reserve vS,vS1,vS2 for Val_Sub of A,Al;

theorem Th20:
  S1`2 = S2`2 implies CQCSub_&(S1,S2)`1 = (S1`1) '&' (S2`1) &
  CQCSub_&(S1,S2)`2 = S1`2
proof
  assume
A1: S1`2 = S2`2;
  then Sub_&(S1,S2) = [(S1`1) '&' (S2`1),S1`2] by SUBSTUT1:def 21;
  then CQCSub_&(S1,S2) = [(S1`1) '&' (S2`1),S1`2] by A1,Def3;
  hence thesis;
end;
