reserve Al for QC-alphabet;
reserve i,j,n,k,l for Nat;
reserve a for set;
reserve T,S,X,Y for Subset of CQC-WFF(Al);
reserve p,q,r,t,F,H,G for Element of CQC-WFF(Al);
reserve s for QC-formula of Al;
reserve x,y for bound_QC-variable of Al;

theorem Th14:
  X c= Y implies Cn(X) c= Cn(Y)
proof
  assume
A1: X c= Y;
  thus Cn(X) c= Cn(Y)
  proof
    let a be object;
    assume
A2: a in Cn(X);
    then reconsider t=a as Element of CQC-WFF(Al);
 for T st T is being_a_theory & Y c= T holds t in T
    proof
      let T such that
A3:   T is being_a_theory and
A4:   Y c= T;
   X c= T by A1,A4;
      hence thesis by A2,A3,Def2;
    end;
    hence thesis by Def2;
  end;
end;
