let X be Subset of CQC-WFF ; X c= { 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 ) }
let a be set ; TARSKI:def 3 ( not a in X or a 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 ) } )
assume A1:
a in X
; a 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 ) }
reconsider p = a as Element of CQC-WFF by A1;
reconsider pp = [p,0 ] as Element of [:CQC-WFF ,Proof_Step_Kinds :] by Th43, ZFMISC_1:106;
set f = <*pp*>;
A2:
len <*pp*> = 1
by FINSEQ_1:57;
A3:
<*pp*> . 1 = pp
by FINSEQ_1:57;
A4:
(<*pp*> . (len <*pp*>)) `1 = p
by A2, A3, MCART_1:7;
A5:
Effect <*pp*> = p
by A4, Def6;
A6:
for n being Element of NAT st 1 <= n & n <= len <*pp*> holds
<*pp*>,n is_a_correct_step_wrt X
proof
let n be
Element of
NAT ;
( 1 <= n & n <= len <*pp*> implies <*pp*>,n is_a_correct_step_wrt X )
assume A7:
( 1
<= n &
n <= len <*pp*> )
;
<*pp*>,n is_a_correct_step_wrt X
A8:
n = 1
by A2, A7, XXREAL_0:1;
A9:
(<*pp*> . 1) `2 = 0
by A3, MCART_1:7;
A10:
(<*pp*> . n) `1 in X
by A1, A3, A8, MCART_1:7;
thus
<*pp*>,
n is_a_correct_step_wrt X
by A8, A9, A10, Def4;
verum
end;
A11:
<*pp*> is_a_proof_wrt X
by A6, Def5;
thus
a 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 A5, A11; verum