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 Th5:
  (for Sub holds ex S st S`1 = p & S`2 = Sub) & (for Sub holds ex S
st S`1 = q & S`2 = Sub) implies for Sub holds ex S st S`1 = p '&' q & S`2 = Sub
proof
  assume that
A1: for Sub holds ex S st S`1 = p & S`2 = Sub and
A2: for Sub holds ex S st S`1 = q & S`2 = Sub;
  let Sub;
  consider S1 such that
A3: S1`1 = p & S1`2 = Sub by A1;
  consider S2 such that
A4: S2`1 = q & S2`2 = Sub by A2;
  S2 = [q,Sub] by A4,SUBSTUT1:10;
  then [q,Sub] in QC-Sub-WFF(Al);
  then
A5: [@q,Sub] in QC-Sub-WFF(Al) by QC_LANG1:def 13;
  S1 = [p,Sub] by A3,SUBSTUT1:10;
  then [p,Sub] in QC-Sub-WFF(Al);
  then [@p,Sub] in QC-Sub-WFF(Al) by QC_LANG1:def 13;
  then [<*[2, 0]*>^@p^@q,Sub] in QC-Sub-WFF(Al) by A5,SUBSTUT1:def 16;
  then reconsider S = [p '&' q,Sub] as Element of QC-Sub-WFF(Al)
     by QC_LANG1:def 16;
    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 A6: 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 '&' q;
  then reconsider S as Element of CQC-Sub-WFF(Al) by A6;
  S`2 = Sub;
  hence thesis;
end;
