
theorem Th4:
  for x,y being Element of BOOLEAN holds xor2c.<*x,y*> = x 'xor'
'not' y & xor2c.<*x,y*> = xor2a.<*x,y*> & xor2c.<*x,y*> = or2.<*nor2.<*x,y*>,
and2.<*x,y*> *> & xor2c.<*0,0*>=1 & xor2c.<*0,1*>=0 & xor2c.<*1,0*>=0 & xor2c.
  <*1,1*>=1
proof
  let x,y be Element of BOOLEAN;
  thus xor2c.<*x,y*> = x 'xor' 'not' y by Def4;
  thus xor2c.<*x,y*> = x 'xor' 'not' y by Def4
    .= 'not' x 'xor' y
    .= xor2a.<*x,y*> by TWOSCOMP:def 14;
  thus xor2c.<*x,y*> = x 'xor' 'not' y by Def4
    .= ('not' x '&' 'not' y) 'or' (x '&' 'not' 'not' y)
    .= or2.<*('not' x '&' 'not' y), (x '&' y) *> by FACIRC_1:def 5
    .= or2.<*nor2.<*x,y*>, (x '&' y) *> by TWOSCOMP:54
    .= or2.<*nor2.<*x,y*>,and2.<*x,y*> *> by FACIRC_1:def 6;
  thus xor2c.<*0,0*> = FALSE 'xor' 'not' FALSE by Def4
    .= 1 by XBOOLEAN:102;
  thus xor2c.<*0,1*> = FALSE 'xor' 'not' TRUE by Def4
    .= 0;
  thus xor2c.<*1,0*> = TRUE 'xor' 'not' FALSE by Def4
    .= 0;
  thus xor2c.<*1,1*> = TRUE 'xor' 'not' TRUE by Def4
    .= 1 by BINARITH:7;
end;
