
theorem Th60:
  for x,y,z being set st z <> [<*x,y*>, xor2c] for s being State
of GFA1AdderCirc(x,y,z) for a1a2,a1,a2,a3 being Element of BOOLEAN st a1a2 = s.
  [<*x,y*>,xor2c] & a3 = s.z holds (Following s).GFA1AdderOutput(x,y,z) = a1a2
  'xor' 'not' a3
proof
  set f = xor2c;
  let x,y,z be set such that
A1: z <> [<*x,y*>,f];
  set A = GFA1AdderCirc(x,y,z);
  set xy = [<*x,y*>,f];
  let s be State of A;
  let a1a2,a1,a2,a3 be Element of BOOLEAN such that
A2: a1a2 = s.xy & a3 = s.z;
  thus (Following s).GFA1AdderOutput(x,y,z) = f.<*s.xy, s.z*> by A1,Lm3
    .= a1a2 'xor' 'not' a3 by A2,Def4;
end;
