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

proof

let T be Subset of HP-WFF; :: thesis: ( T is Hilbert_theory & X c= T implies t in T )
assume that
A2: T is Hilbert_theory and
A3: X c= T ; :: thesis: t in T
CnPos X c= T by A2, A3, Th8;
hence t in T by A1, A2, Def11; :: thesis: verum

end;

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