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