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 Th4:
  (for Sub holds ex S st S`1 = p & S`2 = Sub) implies for Sub holds
  ex S st S`1 = 'not' p & S`2 = Sub
proof
  assume
A1: for Sub holds ex S st S`1 = p & S`2 = Sub;
  let Sub;
  consider S such that
A2: S`1 = p & S`2 = Sub by A1;
  S = [p,Sub] by A2,SUBSTUT1:10;
  then [p,Sub] in QC-Sub-WFF(Al);
  then [@p,Sub] in QC-Sub-WFF(Al) by QC_LANG1:def 13;
  then [<*[1, 0]*>^@p,Sub] in QC-Sub-WFF(Al) by SUBSTUT1:def 16;
  then reconsider S = ['not' p,Sub] as Element of QC-Sub-WFF(Al)
   by QC_LANG1:def 15;
    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 A3: 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 = 'not' p;
  then reconsider S as Element of CQC-Sub-WFF(Al) by A3;
  S`2 = Sub;
  hence thesis;
end;
