let ap, bp, cp, dp be non pair set ; :: thesis: 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 ; :: thesis: 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)); :: thesis: 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 ; :: thesis: ( 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 ) ; :: thesis: (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 ;
:: thesis: verum