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: ( [<*(GFA0AdderOutput ap,bp,cp),cin*>,xor2 ] in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) & GFA0AdderOutput ap,bp,cp in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) & cin in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) & dp in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) ) by ThFTA0S6;
A3: dom s = the carrier of (BitFTA0Str ap,bp,cp,dp,cin) by CIRCUIT1:4;
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 ThFTA0S7;
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, A2, A3, FINSEQ_2:145
.= a1235 'xor' a4 by TWOSCOMP:def 13 ;
:: thesis: verum