set f1 = and2b ;
set f2 = and2b ;
set f3 = and2b ;
let x, y, z be set ; :: thesis: for s being State of (GFA3CarryCirc x,y,z)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
let s be State of (GFA3CarryCirc x,y,z); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
set xy = [<*x,y*>,and2b ];
set yz = [<*y,z*>,and2b ];
set zx = [<*z,x*>,and2b ];
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following s) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) ) )
assume A1: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: ( (Following s) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
set S = GFA3CarryStr x,y,z;
A2: InnerVertices (GFA3CarryStr x,y,z) = the carrier' of (GFA3CarryStr x,y,z) by FACIRC_1:37;
A3: dom s = the carrier of (GFA3CarryStr x,y,z) by CIRCUIT1:4;
A4: ( x in the carrier of (GFA3CarryStr x,y,z) & y in the carrier of (GFA3CarryStr x,y,z) & z in the carrier of (GFA3CarryStr x,y,z) ) by Th129;
[<*x,y*>,and2b ] in InnerVertices (GFA3CarryStr x,y,z) by Th130;
hence (Following s) . [<*x,y*>,and2b ] = and2b . (s * <*x,y*>) by A2, FACIRC_1:35
.= and2b . <*a1,a2*> by A1, A3, A4, FINSEQ_2:145
.= ('not' a1) '&' ('not' a2) by TWOSCOMP:def 3 ;
:: thesis: ( (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
[<*y,z*>,and2b ] in InnerVertices (GFA3CarryStr x,y,z) by Th130;
hence (Following s) . [<*y,z*>,and2b ] = and2b . (s * <*y,z*>) by A2, FACIRC_1:35
.= and2b . <*a2,a3*> by A1, A3, A4, FINSEQ_2:145
.= ('not' a2) '&' ('not' a3) by TWOSCOMP:def 3 ;
:: thesis: (Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1)
[<*z,x*>,and2b ] in InnerVertices (GFA3CarryStr x,y,z) by Th130;
hence (Following s) . [<*z,x*>,and2b ] = and2b . (s * <*z,x*>) by A2, FACIRC_1:35
.= and2b . <*a3,a1*> by A1, A3, A4, FINSEQ_2:145
.= ('not' a3) '&' ('not' a1) by TWOSCOMP:def 3 ;
:: thesis: verum