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 a123, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a5 = s . cin holds
(Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = a123 'xor' ('not' a5)
let cin be set ; :: thesis: for s being State of (BitFTA1Circ ap,bm,cp,dm,cin)
for a123, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a5 = s . cin holds
(Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = a123 'xor' ('not' a5)
set S = BitFTA1Str ap,bm,cp,dm,cin;
set C = BitFTA1Circ ap,bm,cp,dm,cin;
set A1 = GFA1AdderOutput ap,bm,cp;
set A1cin = [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ];
let s be State of (BitFTA1Circ ap,bm,cp,dm,cin); :: thesis: for a123, a5 being Element of BOOLEAN st a123 = s . (GFA1AdderOutput ap,bm,cp) & a5 = s . cin holds
(Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = a123 'xor' ('not' a5)
let a123, a5 be Element of BOOLEAN ; :: thesis: ( a123 = s . (GFA1AdderOutput ap,bm,cp) & a5 = s . cin implies (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = a123 'xor' ('not' a5) )
assume A1: ( a123 = s . (GFA1AdderOutput ap,bm,cp) & a5 = s . cin ) ; :: thesis: (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = a123 'xor' ('not' a5)
A2: ( GFA1AdderOutput ap,bm,cp in the carrier of (BitFTA1Str ap,bm,cp,dm,cin) & cin in the carrier of (BitFTA1Str ap,bm,cp,dm,cin) ) by ThFTA1S6;
A3: dom s = the carrier of (BitFTA1Str ap,bm,cp,dm,cin) by CIRCUIT1:4;
InnerVertices (BitFTA1Str ap,bm,cp,dm,cin) = the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) by FACIRC_1:37;
then [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] in the carrier' of (BitFTA1Str ap,bm,cp,dm,cin) by ThFTA1S7;
hence (Following s) . [<*(GFA1AdderOutput ap,bm,cp),cin*>,xor2c ] = xor2c . (s * <*(GFA1AdderOutput ap,bm,cp),cin*>) by FACIRC_1:35
.= xor2c . <*a123,a5*> by A1, A2, A3, FINSEQ_2:145
.= a123 'xor' ('not' a5) by GFACIRC1:def 4 ;
:: thesis: verum