
theorem
  for x,y being Element of BOOLEAN holds inv1.<* xor2.<*x,y*> *> = xor2a
.<*x,y*> & inv1.<* xor2.<*x,y*> *> = xor2c.<*x,y*> & xor2.<* inv1.<*x*>, inv1.
  <*y*> *> = xor2.<*x,y*>
proof
  let x,y be Element of BOOLEAN;
  thus inv1.<* xor2.<*x,y*> *> = inv1.<* x 'xor' y *> by FACIRC_1:def 4
    .= 'not' (x 'xor' y) by Def1
    .= 'not' x 'xor' y by XBOOLEAN:74
    .= xor2a.<*x,y*> by TWOSCOMP:def 14;
  hence inv1.<* xor2.<*x,y*> *> = xor2c.<*x,y*> by Th4;
  thus xor2.<* inv1.<*x*>, inv1.<*y*> *> = xor2.<* 'not' x, inv1.<*y*> *> by
Th1
    .= xor2.<* 'not' x, 'not' y *> by Th1
    .= 'not' x 'xor' 'not' y by FACIRC_1:def 4
    .= x 'xor' y
    .= xor2.<*x,y*> by FACIRC_1:def 4;
end;
