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
  (p '&' q).(x,y) = (p.(x,y)) '&' (q.(x,y)) & ( QuantNbr(p) = QuantNbr(p
.(x,y)) & QuantNbr(q) = QuantNbr(q.(x,y)) implies QuantNbr(p '&'q) = QuantNbr((
  p '&' q).(x,y)))
proof
  set S = [p '&' q,Sbst(x,y)];
  set S1 = [p,Sbst(x,y)];
  set S2 = [q,Sbst(x,y)];
A1: S1`2 = Sbst(x,y) & S2`2 = Sbst(x,y);
  S = CQCSub_&(S1,S2) by Th19;
  then
A2: S = Sub_&(S1,S2) by A1,SUBLEMMA:def 3;
  then
A3: (p '&' q).(x,y) = (CQC_Sub(S1)) '&' (CQC_Sub(S2)) by A1,SUBSTUT1:31;
  QuantNbr(p) = QuantNbr(p.(x,y)) & QuantNbr(q) = QuantNbr(q.(x,y))
  implies QuantNbr(p '&' q) = QuantNbr((p '&' q).(x,y))
  proof
    assume
A4: QuantNbr(p) = QuantNbr(p.(x,y)) & QuantNbr(q) = QuantNbr(q.(x,y));
    QuantNbr((p '&' q).(x,y)) = QuantNbr(p.(x,y)) + QuantNbr(q.(x,y)) by A3,
CQC_SIM1:17;
    hence thesis by A4,CQC_SIM1:17;
  end;
  hence thesis by A1,A2,SUBSTUT1:31;
end;
