let X be Subset of MC-wff; :: thesis: CnCPC X c= CnS4 X
let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnCPC X or a in CnS4 X )
assume A1: a in CnCPC X ; :: thesis: a in CnS4 X
then reconsider r = a as Element of MC-wff ;
for T being Subset of MC-wff st T is S4_theory & X c= T holds
r in T

proof

let T be Subset of MC-wff; :: thesis: ( T is S4_theory & X c= T implies r in T )
assume that
A2: T is S4_theory and
A3: X c= T ; :: thesis: r in T
T is CPC_theory by A2;
hence r in T by A1, A3, Def20; :: thesis: verum

end;

hence a in CnS4 X by Def23; :: thesis: verum