set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
let x, y, z be set ; :: thesis: for s being State of (GFA0CarryCirc 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*>,and2 ] = a1 '&' a2 & (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
let s be State of (GFA0CarryCirc 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*>,and2 ] = a1 '&' a2 & (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
set xy = [<*x,y*>,and2 ];
set yz = [<*y,z*>,and2 ];
set zx = [<*z,x*>,and2 ];
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following s) . [<*x,y*>,and2 ] = a1 '&' a2 & (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 ) )
assume A1: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: ( (Following s) . [<*x,y*>,and2 ] = a1 '&' a2 & (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
set S = GFA0CarryStr x,y,z;
A2: InnerVertices (GFA0CarryStr x,y,z) = the carrier' of (GFA0CarryStr x,y,z) by FACIRC_1:37;
A3: dom s = the carrier of (GFA0CarryStr x,y,z) by CIRCUIT1:4;
A4: ( x in the carrier of (GFA0CarryStr x,y,z) & y in the carrier of (GFA0CarryStr x,y,z) & z in the carrier of (GFA0CarryStr x,y,z) ) by Th19;
[<*x,y*>,and2 ] in InnerVertices (GFA0CarryStr x,y,z) by Th20;
hence (Following s) . [<*x,y*>,and2 ] = and2 . (s * <*x,y*>) by A2, FACIRC_1:35
.= and2 . <*a1,a2*> by A1, A3, A4, FINSEQ_2:145
.= a1 '&' a2 by TWOSCOMP:def 1 ;
:: thesis: ( (Following s) . [<*y,z*>,and2 ] = a2 '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
[<*y,z*>,and2 ] in InnerVertices (GFA0CarryStr x,y,z) by Th20;
hence (Following s) . [<*y,z*>,and2 ] = and2 . (s * <*y,z*>) by A2, FACIRC_1:35
.= and2 . <*a2,a3*> by A1, A3, A4, FINSEQ_2:145
.= a2 '&' a3 by TWOSCOMP:def 1 ;
:: thesis: (Following s) . [<*z,x*>,and2 ] = a3 '&' a1
[<*z,x*>,and2 ] in InnerVertices (GFA0CarryStr x,y,z) by Th20;
hence (Following s) . [<*z,x*>,and2 ] = and2 . (s * <*z,x*>) by A2, FACIRC_1:35
.= and2 . <*a3,a1*> by A1, A3, A4, FINSEQ_2:145
.= a3 '&' a1 by TWOSCOMP:def 1 ;
:: thesis: verum