let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al) holds Cn (Cn X) c= Cn X

let X be Subset of (CQC-WFF Al); :: thesis: Cn (Cn X) c= Cn X

let a be object ; :: 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 Al ;

for T being Subset of (CQC-WFF Al) st T is being_a_theory & X c= T holds

t in T

let X be Subset of (CQC-WFF Al); :: thesis: Cn (Cn X) c= Cn X

let a be object ; :: 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 Al ;

for T being Subset of (CQC-WFF Al) st T is being_a_theory & X c= T holds

t in T

proof

hence
a in Cn X
by Def2; :: thesis: verum
let T be Subset of (CQC-WFF Al); :: thesis: ( T is being_a_theory & X c= T implies t in T )

assume that

A2: T is being_a_theory and

A3: X c= T ; :: thesis: t in T

Cn X c= T by A2, A3, Th12;

hence t in T by A1, A2, Def2; :: thesis: verum

end;assume that

A2: T is being_a_theory and

A3: X c= T ; :: thesis: t in T

Cn X c= T by A2, A3, Th12;

hence t in T by A1, A2, Def2; :: thesis: verum