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 CIRCUIT1:4;
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_2:145
.= a1235 'xor' ('not' a4) by GFACIRC1:def 4 ;
:: thesis: verum