let X be Subset of CQC-WFF ; :: thesis: for p being Element of CQC-WFF holds
( p in Cn X iff ex f being FinSequence of [:CQC-WFF ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = p ) )
let p be Element of CQC-WFF ; :: thesis: ( p in Cn X iff ex f being FinSequence of [:CQC-WFF ,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 ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = p ) ) :: thesis: ( ex f being FinSequence of [:CQC-WFF ,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 ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = p )
then p in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = F ) } by Th69;
then ex F being Element of CQC-WFF st
( F = p & ex f being FinSequence of [:CQC-WFF ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = F ) ) ;
hence ex f being FinSequence of [:CQC-WFF ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = p ) ; :: thesis: verum

end;

thus ( ex f being FinSequence of [:CQC-WFF ,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 ,Proof_Step_Kinds :] such that A1: f is_a_proof_wrt X and
A2: Effect f = p ; :: thesis: p in Cn X
p in { F where F is Element of CQC-WFF : ex f being FinSequence of [:CQC-WFF ,Proof_Step_Kinds :] st
( f is_a_proof_wrt X & Effect f = F ) } by A1, A2;
hence p in Cn X by Th69; :: thesis: verum

end;