theorem :: GFACIRC1:39

for x, y, z being set st z <> [<*x,y*>,xor2] & x <> [<*y,z*>,and2] & y <> [<*z,x*>,and2] & z <> [<*x,y*>,and2] holds
for s being State of (BitGFA0Circ (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)) . (GFA0AdderOutput (x,y,z)) = (a1 'xor' a2) 'xor' a3 & (Following (s,2)) . (GFA0CarryOutput (x,y,z)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) )