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 a123, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a5 = s . cin holds
(Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = a123 'xor' a5
let cin be set ; :: thesis: for s being State of (BitFTA3Circ (am,bm,cm,dm,cin))
for a123, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a5 = s . cin holds
(Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = a123 'xor' a5
set S = BitFTA3Str (am,bm,cm,dm,cin);
set C = BitFTA3Circ (am,bm,cm,dm,cin);
set A1 = GFA3AdderOutput (am,bm,cm);
set A1cin = [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2];
let s be State of (BitFTA3Circ (am,bm,cm,dm,cin)); :: thesis: for a123, a5 being Element of BOOLEAN st a123 = s . (GFA3AdderOutput (am,bm,cm)) & a5 = s . cin holds
(Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = a123 'xor' a5
let a123, a5 be Element of BOOLEAN ; :: thesis: ( a123 = s . (GFA3AdderOutput (am,bm,cm)) & a5 = s . cin implies (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = a123 'xor' a5 )
assume A1: ( a123 = s . (GFA3AdderOutput (am,bm,cm)) & a5 = s . cin ) ; :: thesis: (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = a123 'xor' a5
A2: dom s = the carrier of (BitFTA3Str (am,bm,cm,dm,cin)) by CIRCUIT1:3;
A3: ( 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)) ) by Th34;
InnerVertices (BitFTA3Str (am,bm,cm,dm,cin)) = the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by FACIRC_1:37;
then [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] in the carrier' of (BitFTA3Str (am,bm,cm,dm,cin)) by Th35;
hence (Following s) . [<*(GFA3AdderOutput (am,bm,cm)),cin*>,xor2] = xor2 . (s * <*(GFA3AdderOutput (am,bm,cm)),cin*>) by FACIRC_1:35
.= xor2 . <*a123,a5*> by A1, A3, A2, FINSEQ_2:125
.= a123 'xor' a5 by FACIRC_1:def 4 ;
:: thesis: verum