let X be Subset of MC-wff; :: thesis: CnCPC (CnCPC X) c= CnCPC X
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnCPC (CnCPC X) or a in CnCPC X )
assume A1: a in CnCPC (CnCPC X) ; :: thesis: a in CnCPC X
then reconsider t = a as Element of MC-wff ;
for T being Subset of MC-wff st T is CPC_theory & X c= T holds
t in T

proof

let T be Subset of MC-wff; :: thesis: ( T is CPC_theory & X c= T implies t in T )
assume that
A2: T is CPC_theory and
A3: X c= T ; :: thesis: t in T
CnCPC X c= T by A2, A3, Th71;
hence t in T by A1, A2, Def20; :: thesis: verum

end;

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