let X, Y be Subset of MC-wff; :: thesis: ( X c= Y implies CnS4 X c= CnS4 Y )
assume A1: X c= Y ; :: thesis: CnS4 X c= CnS4 Y
thus CnS4 X c= CnS4 Y :: thesis: verum

proof

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in CnS4 X or a in CnS4 Y )
assume A2: a in CnS4 X ; :: thesis: a in CnS4 Y
then reconsider t = a as Element of MC-wff ;
for T being Subset of MC-wff st T is S4_theory & Y c= T holds
t in T

proof

let T be Subset of MC-wff; :: thesis: ( T is S4_theory & Y c= T implies t in T )
assume that
A3: T is S4_theory and
A4: Y c= T ; :: thesis: t in T
X c= T by A1, A4;
hence t in T by A2, A3, Def23; :: thesis: verum

end;

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

end;