let X, Y be Subset of CQC-WFF ; :: thesis: ( X c= Y implies Cn X c= Cn Y )
assume A1: X c= Y ; :: thesis: Cn X c= Cn Y
thus Cn X c= Cn Y :: thesis: verum

proof

let a be set ; :: according to TARSKI:def 3 :: thesis: ( not a in Cn X or a in Cn Y )
assume A2: a in Cn X ; :: thesis: a in Cn Y
then reconsider t = a as Element of CQC-WFF ;
for T being Subset of CQC-WFF st T is being_a_theory & Y c= T holds
t in T

proof

let T be Subset of CQC-WFF ; :: thesis: ( T is being_a_theory & Y c= T implies t in T )
assume that
A4: T is being_a_theory and
A5: Y c= T ; :: thesis: t in T
X c= T by A1, A5, XBOOLE_1:1;
hence t in T by A2, A4, Def2; :: thesis: verum

end;

hence a in Cn Y by Def2; :: thesis: verum

end;