let X, Y be Subset of MC-wff; :: thesis: ( X c= Y implies CnCPC X c= CnCPC Y )
assume A1: X c= Y ; :: thesis: CnCPC X c= CnCPC Y
thus CnCPC X c= CnCPC Y :: thesis: verum

proof

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnCPC X or a in CnCPC Y )
assume A2: a in CnCPC X ; :: thesis: a in CnCPC Y
then reconsider t = a as Element of MC-wff ;
for T being Subset of MC-wff st T is CPC_theory & Y c= T holds
t in T

proof

let T be Subset of MC-wff; :: thesis: ( T is CPC_theory & Y c= T implies t in T )
assume that
A3: T is CPC_theory and
A4: Y c= T ; :: thesis: t in T
X c= T by A1, A4;
hence t in T by A2, A3, Def20; :: thesis: verum

end;

hence a in CnCPC Y by Def20; :: thesis: verum

end;