
theorem Th59:
  for x,y,z being set st z <> [<*x,y*>, xor2c] for s being State
of GFA1AdderCirc(x,y,z) for a1,a2,a3 being Element of BOOLEAN st a1 = s.x & a2
  = s.y & a3 = s.z holds (Following s).[<*x,y*>,xor2c] = a1 'xor' 'not' a2 & (
  Following s).x = a1 & (Following s).y = a2 & (Following s).z = 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 a1,a2,a3 be Element of BOOLEAN such that
A2: a1 = s.x & a2 = s.y and
A3: a3 = s.z;
  (Following s).xy = f.<*a1, a2*> by A1,A2,Lm3;
  hence (Following s).xy = a1 'xor' 'not' a2 by Def4;
  thus thesis by A1,A2,A3,Lm3;
end;
