
theorem Th126:
  for x,y,z being set st z <> [<*x,y*>, xor2] for s being State
of GFA3AdderCirc(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).GFA3AdderOutput(x,y,z) = 'not' ('not' a1
  'xor' 'not' a2 'xor' 'not' a3)
proof
  set f = xor2;
  let x,y,z be set such that
A1: z <> [<*x,y*>,f];
  set A = GFA3AdderCirc(x,y,z);
  let s be State of A;
  let a1,a2,a3 be Element of BOOLEAN;
  assume a1 = s.x & a2 = s.y & a3 = s.z;
  hence (Following(s,2)).GFA3AdderOutput(x,y,z) = a1 'xor' a2 'xor' a3 by A1
,Th125
    .= 'not' ('not' a1 'xor' 'not' a2 'xor' 'not' a3) by XBOOLEAN:74;
end;
