let am, bp, cm, dp be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA2Circ (am,bp,cm,dp,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] & a4 = s . dp holds
(Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = a1235 'xor' ('not' a4)
let cin be set ; :: thesis: for s being State of (BitFTA2Circ (am,bp,cm,dp,cin))
for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] & a4 = s . dp holds
(Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = a1235 'xor' ('not' a4)
set S = BitFTA2Str (am,bp,cm,dp,cin);
set C = BitFTA2Circ (am,bp,cm,dp,cin);
set A1 = GFA2AdderOutput (am,bp,cm);
set A2 = GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp);
set A1cin = [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c];
let s be State of (BitFTA2Circ (am,bp,cm,dp,cin)); :: thesis: for a1235, a4 being Element of BOOLEAN st a1235 = s . [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] & a4 = s . dp holds
(Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = a1235 'xor' ('not' a4)
let a1235, a4 be Element of BOOLEAN ; :: thesis: ( a1235 = s . [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] & a4 = s . dp implies (Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = a1235 'xor' ('not' a4) )
assume A1: ( a1235 = s . [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] & a4 = s . dp ) ; :: thesis: (Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = a1235 'xor' ('not' a4)
A2: dom s = the carrier of (BitFTA2Str (am,bp,cm,dp,cin)) by CIRCUIT1:3;
A3: ( [<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c] in the carrier of (BitFTA2Str (am,bp,cm,dp,cin)) & dp in the carrier of (BitFTA2Str (am,bp,cm,dp,cin)) ) by Th24;
InnerVertices (BitFTA2Str (am,bp,cm,dp,cin)) = the carrier' of (BitFTA2Str (am,bp,cm,dp,cin)) by FACIRC_1:37;
then GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp) in the carrier' of (BitFTA2Str (am,bp,cm,dp,cin)) by Th25;
hence (Following s) . (GFA1AdderOutput ((GFA2AdderOutput (am,bp,cm)),cin,dp)) = xor2c . (s * <*[<*(GFA2AdderOutput (am,bp,cm)),cin*>,xor2c],dp*>) by FACIRC_1:35
.= xor2c . <*a1235,a4*> by A1, A3, A2, FINSEQ_2:125
.= a1235 'xor' ('not' a4) by GFACIRC1:def 4 ;
:: thesis: verum