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

proof

let T be Subset of MC-wff; :: thesis: ( T is S4_theory & X c= T implies t in T )
assume that
A2: T is S4_theory and
A3: X c= T ; :: thesis: t in T
CnS4 X c= T by A2, A3, Th88;
hence t in T by A1, A2, Def23; :: thesis: verum

end;

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