theorem :: GFACIRC1:103

for x, y, z being set st z <> [<*x,y*>,xor2c] & x <> [<*y,z*>,and2c] & y <> [<*z,x*>,nor2] & z <> [<*x,y*>,and2a] holds
for s being State of (BitGFA2Circ (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)) . (GFA2AdderOutput (x,y,z)) = (('not' a1) 'xor' a2) 'xor' ('not' a3) & (Following (s,2)) . (GFA2CarryOutput (x,y,z)) = 'not' (((('not' a1) '&' a2) 'or' (a2 '&' ('not' a3))) 'or' (('not' a3) '&' ('not' a1))) )