set f1 = and2b ;
set f2 = and2b ;
set f3 = and2b ;
let x, y, z be set ; :: thesis: ( 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 ] ) ; :: 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,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 xy = [<*x,y*>,and2b ];
set yz = [<*y,z*>,and2b ];
set zx = [<*z,x*>,and2b ];
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,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) )
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( 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 A2: ( a1 = s . x & a2 = s . y & a3 = s . z ) ; :: thesis: ( (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 x' = x, y' = y, z' = z as Vertex of (GFA3CarryStr x,y,z) by Th129;
( x in InputVertices (GFA3CarryStr x,y,z) & y in InputVertices (GFA3CarryStr x,y,z) & z in InputVertices (GFA3CarryStr x,y,z) ) by A1, Th131;
then A3: ( (Following s) . x' = s . x & (Following s) . y' = s . y & (Following s) . z' = s . z ) by CIRCUIT2:def 5;
A4: Following s,2 = Following (Following s) by FACIRC_1:15;
( (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) ) by A2, 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, Th134; :: thesis: ( (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) )
thus ( (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 A2, A3, A4, Th133; :: thesis: verum