let ap, bp, cp, dp be non pair set ; for cin being set
for s being State of (BitFTA0Circ (ap,bp,cp,dp,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] & a4 = s . dp holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) = a1235 'xor' a4
let cin be set ; for s being State of (BitFTA0Circ (ap,bp,cp,dp,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] & a4 = s . dp holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) = a1235 'xor' a4
set S = BitFTA0Str (ap,bp,cp,dp,cin);
set C = BitFTA0Circ (ap,bp,cp,dp,cin);
set A1 = GFA0AdderOutput (ap,bp,cp);
set A2 = GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp);
set A1cin = [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2];
let s be State of (BitFTA0Circ (ap,bp,cp,dp,cin)); for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] & a4 = s . dp holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) = a1235 'xor' a4
let a1235, a4 be Element of BOOLEAN ; ( a1235 = s . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] & a4 = s . dp implies (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) = a1235 'xor' a4 )
assume A1:
( a1235 = s . [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] & a4 = s . dp )
; (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) = a1235 'xor' a4
A2:
dom s = the carrier of (BitFTA0Str (ap,bp,cp,dp,cin))
by CIRCUIT1:3;
A3:
( [<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2] in the carrier of (BitFTA0Str (ap,bp,cp,dp,cin)) & dp in the carrier of (BitFTA0Str (ap,bp,cp,dp,cin)) )
by Th4;
InnerVertices (BitFTA0Str (ap,bp,cp,dp,cin)) = the carrier' of (BitFTA0Str (ap,bp,cp,dp,cin))
by FACIRC_1:37;
then
GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp) in the carrier' of (BitFTA0Str (ap,bp,cp,dp,cin))
by Th5;
hence (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (ap,bp,cp)),cin,dp)) =
xor2 . (s * <*[<*(GFA0AdderOutput (ap,bp,cp)),cin*>,xor2],dp*>)
by FACIRC_1:35
.=
xor2 . <*a1235,a4*>
by A1, A3, A2, FINSEQ_2:125
.=
a1235 'xor' a4
by FACIRC_1:def 4
;
verum