reserve Al for QC-alphabet;
reserve a,b,b1 for object,
  i,j,k,n for Nat,
  p,q,r,s for Element of CQC-WFF(Al),
  x,y,y1 for bound_QC-variable of Al,
  P for QC-pred_symbol of k,Al,
  l,ll for CQC-variable_list of k,Al,
  Sub,Sub1 for CQC_Substitution of Al,
  S,S1,S2 for Element of CQC-Sub-WFF(Al),
  P1,P2 for Element of QC-pred_symbols(Al);

theorem Th2:
  for Sub holds ex S st S`1 = P!ll & S`2 = Sub
proof
  let Sub;
  P is QC-pred_symbol of the_arity_of P,Al by QC_LANG3:1;
  then k = the_arity_of P by Lm1;
  then [<*P*>^ll,Sub] in QC-Sub-WFF(Al) & len ll = the_arity_of P
    by CARD_1:def 7,SUBSTUT1:def 16;
  then reconsider S = [P!ll,Sub] as Element of QC-Sub-WFF(Al)
    by QC_LANG1:def 12;
    set X = { G where G is Element of QC-Sub-WFF(Al) :
              G`1 is Element of CQC-WFF(Al) };
    X = CQC-Sub-WFF(Al) by SUBSTUT1:def 39;
    then A1: for G being Element of QC-Sub-WFF(Al) holds
        G`1 is Element of CQC-WFF(Al) implies G in CQC-Sub-WFF(Al);
  take S;
  S`1 = P!ll;
  then reconsider S as Element of CQC-Sub-WFF(Al) by A1;
  S`2 = Sub;
  hence thesis;
end;
