set f1 = nor2 ;
set f2 = nor2 ;
set f3 = nor2 ;
let x, y, z be set ; ( x <> [<*y,z*>,nor2] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,nor2] 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*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) ) )
assume A1:
( x <> [<*y,z*>,nor2] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,nor2] )
; 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*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('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 Th111;
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*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
y in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then A2:
(Following s) . y9 = s . y
by CIRCUIT2:def 5;
z in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then A3:
(Following s) . z9 = s . z
by CIRCUIT2:def 5;
set xy = [<*x,y*>,nor2];
set yz = [<*y,z*>,nor2];
set zx = [<*z,x*>,nor2];
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*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('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*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
A6:
(Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1)
by A5, Th115;
( (Following s) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) )
by A5, Th115;
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, Th116; ( (Following (s,2)) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
x in InputVertices (GFA3CarryStr (x,y,z))
by A1, Th113;
then
(Following s) . x9 = s . x
by CIRCUIT2:def 5;
hence
( (Following (s,2)) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following (s,2)) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following (s,2)) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
by A5, A2, A3, A4, Th115; verum