let X, Y be Subset of HP-WFF; :: thesis: ( X c= Y implies CnPos X c= CnPos Y )
assume A1: X c= Y ; :: thesis: CnPos X c= CnPos Y
thus CnPos X c= CnPos Y :: thesis: verum

proof

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

proof

let T be Subset of HP-WFF; :: thesis: ( T is Hilbert_theory & Y c= T implies t in T )
assume that
A3: T is Hilbert_theory and
A4: Y c= T ; :: thesis: t in T
X c= T by A1, A4;
hence t in T by A2, A3, Def11; :: thesis: verum

end;

hence a in CnPos Y by Def11; :: thesis: verum

end;