let X be Subset of MC-wff; :: thesis: X c= { F where F is Element of MC-wff : ex f being FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:] st
( f is_a_proof_wrt_IPC X & Effect_IPC 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 MC-wff : ex f being FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:] st
( f is_a_proof_wrt_IPC X & Effect_IPC f = F ) } )
assume A1: a in X ; :: thesis: a in { F where F is Element of MC-wff : ex f being FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:] st
( f is_a_proof_wrt_IPC X & Effect_IPC f = F ) }
then reconsider p = a as Element of MC-wff ;
reconsider pp = [p,0] as Element of [:MC-wff,Proof_Step_Kinds_IPC:] by Th1, ZFMISC_1:87;
set f = <*pp*>;
A2: len <*pp*> = 1 by FINSEQ_1:40;
(<*pp*> . (len <*pp*>)) `1 = p by A2;
then A4: Effect_IPC <*pp*> = p by Def5;
for n being Nat st 1 <= n & n <= len <*pp*> holds
<*pp*>,n is_a_correct_step_wrt_IPC X

proof

let n be Nat; :: thesis: ( 1 <= n & n <= len <*pp*> implies <*pp*>,n is_a_correct_step_wrt_IPC X )
assume ( 1 <= n & n <= len <*pp*> ) ; :: thesis: <*pp*>,n is_a_correct_step_wrt_IPC X
then A5: n = 1 by A2, XXREAL_0:1;
A6: (<*pp*> . 1) `2 = 0 ;
(<*pp*> . n) `1 in X by A1, A5;
hence <*pp*>,n is_a_correct_step_wrt_IPC X by A5, A6, Def3; :: thesis: verum

end;

then <*pp*> is_a_proof_wrt_IPC X ;
hence a in { F where F is Element of MC-wff : ex f being FinSequence of [:MC-wff,Proof_Step_Kinds_IPC:] st
( f is_a_proof_wrt_IPC X & Effect_IPC f = F ) } by A4; :: thesis: verum