let Al be QC-alphabet ; :: thesis: for X being Subset of (CQC-WFF Al)
for p being Element of CQC-WFF Al holds
( p in Cn X iff ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) )
let X be Subset of (CQC-WFF Al); :: thesis: for p being Element of CQC-WFF Al holds
( p in Cn X iff ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) )
let p be Element of CQC-WFF Al; :: thesis: ( p in Cn X iff ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) )
thus ( p in Cn X implies ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) ) :: thesis: ( ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) implies p in Cn X )

proof

assume p in Cn X ; :: thesis: ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p )
then p in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } by Th31;
then ex F being Element of CQC-WFF Al st
( F = p & ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) ) ;
hence ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) ; :: thesis: verum

end;

thus ( ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = p ) implies p in Cn X ) :: thesis: verum

proof

given f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] such that A1: ( f is_a_proof_wrt X & Effect f = p ) ; :: thesis: p in Cn X
p in { F where F is Element of CQC-WFF Al : ex f being FinSequence of [:(CQC-WFF Al),Proof_Step_Kinds:] st
( f is_a_proof_wrt X & Effect f = F ) } by A1;
hence p in Cn X by Th31; :: thesis: verum

end;