let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al) holds X c= { F where F is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt X & Effect f = F )
}

let X be Subset of (CQC-WFF Al); :: thesis: X c= { F where F is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt X & Effect f = F )
}

let a be object ; :: according to TARSKI:def 3 :: thesis: ( not a in X or a in { F where F is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt X & Effect f = F )
}
)

assume A1: a in X ; :: thesis: a in { F where F is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt X & Effect f = F )
}

then reconsider p = a as Element of CQC-WFF Al ;
reconsider pp = [p,0] as Element of by ;
set f = <*pp*>;
A2: len <*pp*> = 1 by FINSEQ_1:40;
A3: <*pp*> . 1 = pp by FINSEQ_1:40;
then (<*pp*> . (len <*pp*>)) `1 = p by A2;
then A4: Effect <*pp*> = p by Def6;
for n being Nat st 1 <= n & n <= len <*pp*> holds
<*pp*>,n is_a_correct_step_wrt X
proof
let n be Nat; :: thesis: ( 1 <= n & n <= len <*pp*> implies <*pp*>,n is_a_correct_step_wrt X )
assume ( 1 <= n & n <= len <*pp*> ) ; :: thesis:
then A5: n = 1 by ;
A6: (<*pp*> . 1) `2 = 0 by A3;
(<*pp*> . n) `1 in X by A1, A3, A5;
hence <*pp*>,n is_a_correct_step_wrt X by A5, A6, Def4; :: thesis: verum
end;
then <*pp*> is_a_proof_wrt X ;
hence a in { F where F is Element of CQC-WFF Al : ex f being FinSequence of st
( f is_a_proof_wrt X & Effect f = F )
}
by A4; :: thesis: verum