set f1 = and2b ;
set f2 = and2b ;
set f3 = and2b ;
let x, y, z be set ; ( x <> [<*y,z*>,and2b ] & y <> [<*z,x*>,and2b ] & z <> [<*x,y*>,and2b ] implies 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,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) ) )
assume A1:
( x <> [<*y,z*>,and2b ] & y <> [<*z,x*>,and2b ] & z <> [<*x,y*>,and2b ] )
; 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,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
set S = GFA3CarryStr x,y,z;
reconsider x9 = x, y9 = y, z9 = z as Vertex of (GFA3CarryStr x,y,z) by Th129;
let s be 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,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
y in InputVertices (GFA3CarryStr x,y,z)
by A1, Th131;
then A2:
(Following s) . y9 = s . y
by CIRCUIT2:def 5;
z in InputVertices (GFA3CarryStr x,y,z)
by A1, Th131;
then A3:
(Following s) . z9 = s . z
by CIRCUIT2:def 5;
set xy = [<*x,y*>,and2b ];
set yz = [<*y,z*>,and2b ];
set zx = [<*z,x*>,and2b ];
A4:
Following s,2 = Following (Following s)
by FACIRC_1:15;
let a1, a2, a3 be Element of BOOLEAN ; ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following s,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) ) )
assume A5:
( a1 = s . x & a2 = s . y & a3 = s . z )
; ( (Following s,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) & (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
A6:
(Following s) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1)
by A5, Th133;
( (Following s) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) )
by A5, Th133;
hence
(Following s,2) . (GFA3CarryOutput x,y,z) = 'not' (((('not' a1) '&' ('not' a2)) 'or' (('not' a2) '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1)))
by A4, A6, Th134; ( (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
x in InputVertices (GFA3CarryStr x,y,z)
by A1, Th131;
then
(Following s) . x9 = s . x
by CIRCUIT2:def 5;
hence
( (Following s,2) . [<*x,y*>,and2b ] = ('not' a1) '&' ('not' a2) & (Following s,2) . [<*y,z*>,and2b ] = ('not' a2) '&' ('not' a3) & (Following s,2) . [<*z,x*>,and2b ] = ('not' a3) '&' ('not' a1) )
by A5, A2, A3, A4, Th133; verum