let am, bm, cm, dm be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] & a4 = s . dm holds
(Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = a1235 'xor' a4
let cin be set ; :: thesis: for s being State of (BitFTA3Circ am,bm,cm,dm,cin)
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] & a4 = s . dm holds
(Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = a1235 'xor' a4
set S = BitFTA3Str am,bm,cm,dm,cin;
set C = BitFTA3Circ am,bm,cm,dm,cin;
set A1 = GFA3AdderOutput am,bm,cm;
set A2 = GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm;
set A1cin = [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ];
let s be State of (BitFTA3Circ am,bm,cm,dm,cin); :: thesis: for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] & a4 = s . dm holds
(Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = a1235 'xor' a4
let a1235, a4 be Element of BOOLEAN ; :: thesis: ( a1235 = s . [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] & a4 = s . dm implies (Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = a1235 'xor' a4 )
assume A1: ( a1235 = s . [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] & a4 = s . dm ) ; :: thesis: (Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = a1235 'xor' a4
A2: ( [<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ] in the carrier of (BitFTA3Str am,bm,cm,dm,cin) & GFA3AdderOutput am,bm,cm in the carrier of (BitFTA3Str am,bm,cm,dm,cin) & cin in the carrier of (BitFTA3Str am,bm,cm,dm,cin) & dm in the carrier of (BitFTA3Str am,bm,cm,dm,cin) ) by ThFTA3S6;
A3: dom s = the carrier of (BitFTA3Str am,bm,cm,dm,cin) by CIRCUIT1:4;
InnerVertices (BitFTA3Str am,bm,cm,dm,cin) = the carrier' of (BitFTA3Str am,bm,cm,dm,cin) by FACIRC_1:37;
then GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm in the carrier' of (BitFTA3Str am,bm,cm,dm,cin) by ThFTA3S7;
hence (Following s) . (GFA3AdderOutput (GFA3AdderOutput am,bm,cm),cin,dm) = xor2 . (s * <*[<*(GFA3AdderOutput am,bm,cm),cin*>,xor2 ],dm*>) by FACIRC_1:35
.= xor2 . <*a1235,a4*> by A1, A2, A3, FINSEQ_2:145
.= a1235 'xor' a4 by TWOSCOMP:def 13 ;
:: thesis: verum