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: dom s = the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by CIRCUIT1:3;
A3: ( [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] in the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) & dm in the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) ) by Th34;
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 Th35;
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, A3, A2, FINSEQ_2:125
.= a1235 'xor' a4 by FACIRC_1:def 4 ;
:: thesis: verum