let ap, bp, cp, dp be non pair set ; :: thesis: for cin being set
for s being State of (BitFTA0Circ ap,bp,cp,dp,cin)
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA0AdderOutput ap,bp,cp) & a4 = s . dp & a5 = s . cin holds
( (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = a123 '&' a5 & (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 )
let cin be set ; :: thesis: for s being State of (BitFTA0Circ ap,bp,cp,dp,cin)
for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA0AdderOutput ap,bp,cp) & a4 = s . dp & a5 = s . cin holds
( (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = a123 '&' a5 & (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 )
set S = BitFTA0Str ap,bp,cp,dp,cin;
set C = BitFTA0Circ ap,bp,cp,dp,cin;
set A1 = GFA0AdderOutput ap,bp,cp;
set S2 = BitGFA0Str (GFA0AdderOutput ap,bp,cp),cin,dp;
set C2 = BitGFA0Circ (GFA0AdderOutput ap,bp,cp),cin,dp;
set A1cin = [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ];
set cindp = [<*cin,dp*>,and2 ];
set dpA1 = [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ];
let s be State of (BitFTA0Circ ap,bp,cp,dp,cin); :: thesis: for a123, a4, a5 being Element of BOOLEAN st a123 = s . (GFA0AdderOutput ap,bp,cp) & a4 = s . dp & a5 = s . cin holds
( (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = a123 '&' a5 & (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 )
let a123, a4, a5 be Element of BOOLEAN ; :: thesis: ( a123 = s . (GFA0AdderOutput ap,bp,cp) & a4 = s . dp & a5 = s . cin implies ( (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = a123 '&' a5 & (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 ) )
assume A1: ( a123 = s . (GFA0AdderOutput ap,bp,cp) & a4 = s . dp & a5 = s . cin ) ; :: thesis: ( (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = a123 '&' a5 & (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 )
A2: ( GFA0AdderOutput ap,bp,cp in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) & dp in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) & cin in the carrier of (BitFTA0Str ap,bp,cp,dp,cin) ) by ThFTA0S6;
A3: dom s = the carrier of (BitFTA0Str ap,bp,cp,dp,cin) by CIRCUIT1:4;
InnerVertices (BitFTA0Str ap,bp,cp,dp,cin) = the carrier' of (BitFTA0Str ap,bp,cp,dp,cin) by FACIRC_1:37;
then A4: ( [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] in the carrier' of (BitFTA0Str ap,bp,cp,dp,cin) & [<*cin,dp*>,and2 ] in the carrier' of (BitFTA0Str ap,bp,cp,dp,cin) & [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] in the carrier' of (BitFTA0Str ap,bp,cp,dp,cin) ) by ThFTA0S7;
hence (Following s) . [<*(GFA0AdderOutput ap,bp,cp),cin*>,and2 ] = and2 . (s * <*(GFA0AdderOutput ap,bp,cp),cin*>) by FACIRC_1:35
.= and2 . <*a123,a5*> by A1, A2, A3, FINSEQ_2:145
.= a123 '&' a5 by TWOSCOMP:def 1 ;
:: thesis: ( (Following s) . [<*cin,dp*>,and2 ] = a5 '&' a4 & (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123 )
thus (Following s) . [<*cin,dp*>,and2 ] = and2 . (s * <*cin,dp*>) by A4, FACIRC_1:35
.= and2 . <*a5,a4*> by A1, A2, A3, FINSEQ_2:145
.= a5 '&' a4 by TWOSCOMP:def 1 ; :: thesis: (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = a4 '&' a123
thus (Following s) . [<*dp,(GFA0AdderOutput ap,bp,cp)*>,and2 ] = and2 . (s * <*dp,(GFA0AdderOutput ap,bp,cp)*>) by A4, FACIRC_1:35
.= and2 . <*a4,a123*> by A1, A2, A3, FINSEQ_2:145
.= a4 '&' a123 by TWOSCOMP:def 1 ; :: thesis: verum