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

proof

let T be Subset of CQC-WFF; :: thesis: ( T is being_a_theory & X c= T implies t in T )
assume that
A3: T is being_a_theory and
A4: X c= T ; :: thesis: t in T
Cn X c= T by A3, A4, Th37;
hence t in T by A1, A3, Def2; :: thesis: verum

end;

hence a in Cn X by Def2; :: thesis: verum