let X be Subset of MC-wff; :: thesis: for p being Element of MC-wff st p in CnS4 X holds
Nes p in CnS4 X
let p be Element of MC-wff ; :: thesis: ( p in CnS4 X implies Nes p in CnS4 X )
assume A1: p in CnS4 X ; :: thesis: Nes p in CnS4 X
for T being Subset of MC-wff st T is S4_theory & X c= T holds
Nes p in T

proof

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

end;

hence Nes p in CnS4 X by Def23; :: thesis: verum