let ap, bm, cp, dm be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] & a4 = s . dm holds
(Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = a1235 'xor' ('not' a4)
let cin be set ; :: thesis: for s being State of (BitFTA1Circ (ap,bm,cp,dm,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] & a4 = s . dm holds
(Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = a1235 'xor' ('not' a4)
set S = BitFTA1Str (ap,bm,cp,dm,cin);
set C = BitFTA1Circ (ap,bm,cp,dm,cin);
set A1 = GFA1AdderOutput (ap,bm,cp);
set A2 = GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm);
set A1cin = [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c];
let s be State of (BitFTA1Circ (ap,bm,cp,dm,cin)); :: thesis: for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] & a4 = s . dm holds
(Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = a1235 'xor' ('not' a4)
let a1235, a4 be Element of BOOLEAN ; :: thesis: ( a1235 = s . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] & a4 = s . dm implies (Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = a1235 'xor' ('not' a4) )
assume A1: ( a1235 = s . [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] & a4 = s . dm ) ; :: thesis: (Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = a1235 'xor' ('not' a4)
A2: dom s = the carrier of (BitFTA1Str (ap,bm,cp,dm,cin)) by CIRCUIT1:3;
A3: ( [<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c] in the carrier of (BitFTA1Str (ap,bm,cp,dm,cin)) & dm in the carrier of (BitFTA1Str (ap,bm,cp,dm,cin)) ) by Th14;
InnerVertices (BitFTA1Str (ap,bm,cp,dm,cin)) = the carrier' of (BitFTA1Str (ap,bm,cp,dm,cin)) by FACIRC_1:37;
then GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm) in the carrier' of (BitFTA1Str (ap,bm,cp,dm,cin)) by Th15;
hence (Following s) . (GFA2AdderOutput ((GFA1AdderOutput (ap,bm,cp)),cin,dm)) = xor2c . (s * <*[<*(GFA1AdderOutput (ap,bm,cp)),cin*>,xor2c],dm*>) by FACIRC_1:35
.= xor2c . <*a1235,a4*> by A1, A3, A2, FINSEQ_2:125
.= a1235 'xor' ('not' a4) by GFACIRC1:def 4 ;
:: thesis: verum