theorem :: GFACIRC1:71

for x, y, z being set st z <> [<*x,y*>,xor2c] & x <> [<*y,z*>,and2a] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2c] holds
for s being State of (BitGFA1Circ (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)) . (GFA1AdderOutput (x,y,z)) = 'not' ((a1 'xor' ('not' a2)) 'xor' a3) & (Following (s,2)) . (GFA1CarryOutput (x,y,z)) = ((a1 '&' ('not' a2)) 'or' (('not' a2) '&' a3)) 'or' (a3 '&' a1) )