let x, y, z be set ; :: thesis: for s being State of (GFA1CarryCirc 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*>,and2c ] = a1 '&' ('not' a2) & (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
let s be State of (GFA1CarryCirc 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*>,and2c ] = a1 '&' ('not' a2) & (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
set xy = [<*x,y*>,and2c ];
set yz = [<*y,z*>,and2a ];
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*>,and2c ] = a1 '&' ('not' a2) & (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 ) )
assume that
A1: a1 = s . x and
A2: a2 = s . y and
A3: a3 = s . z ; :: thesis: ( (Following s) . [<*x,y*>,and2c ] = a1 '&' ('not' a2) & (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
set S = GFA1CarryStr x,y,z;
A4: InnerVertices (GFA1CarryStr x,y,z) = the carrier' of (GFA1CarryStr x,y,z) by FACIRC_1:37;
A5: y in the carrier of (GFA1CarryStr x,y,z) by Th55;
A6: x in the carrier of (GFA1CarryStr x,y,z) by Th55;
A7: dom s = the carrier of (GFA1CarryStr x,y,z) by CIRCUIT1:4;
[<*x,y*>,and2c ] in InnerVertices (GFA1CarryStr x,y,z) by Th56;
hence (Following s) . [<*x,y*>,and2c ] = and2c . (s * <*x,y*>) by A4, FACIRC_1:35
.= and2c . <*a1,a2*> by A1, A2, A7, A6, A5, FINSEQ_2:145
.= a1 '&' ('not' a2) by Def3 ;
:: thesis: ( (Following s) . [<*y,z*>,and2a ] = ('not' a2) '&' a3 & (Following s) . [<*z,x*>,and2 ] = a3 '&' a1 )
A8: z in the carrier of (GFA1CarryStr x,y,z) by Th55;
[<*y,z*>,and2a ] in InnerVertices (GFA1CarryStr x,y,z) by Th56;
hence (Following s) . [<*y,z*>,and2a ] = and2a . (s * <*y,z*>) by A4, FACIRC_1:35
.= and2a . <*a2,a3*> by A2, A3, A7, A5, A8, FINSEQ_2:145
.= ('not' a2) '&' a3 by TWOSCOMP:def 2 ;
:: thesis: (Following s) . [<*z,x*>,and2 ] = a3 '&' a1
[<*z,x*>,and2 ] in InnerVertices (GFA1CarryStr x,y,z) by Th56;
hence (Following s) . [<*z,x*>,and2 ] = and2 . (s * <*z,x*>) by A4, FACIRC_1:35
.= and2 . <*a3,a1*> by A1, A3, A7, A6, A8, FINSEQ_2:145
.= a3 '&' a1 by TWOSCOMP:def 1 ;
:: thesis: verum