reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN holds a '&' b = a '&' ('not'
  a 'or' b)
proof
  let a,b be Function of Y,BOOLEAN;
    let x be Element of Y;
    (a '&' ('not' a 'or' b)).x =a.x '&' ('not' a 'or' b).x by MARGREL1:def 20
      .=a.x '&' (('not' a).x 'or' b.x) by BVFUNC_1:def 4
      .=a.x '&' ('not' a).x 'or' a.x '&' b.x by XBOOLEAN:8
      .=a.x '&' 'not' a.x 'or' a.x '&' b.x by MARGREL1:def 19
      .=FALSE 'or' a.x '&' b.x by XBOOLEAN:138
      .=a.x '&' b.x 
      .=(a '&' b).x by MARGREL1:def 20;
    hence thesis;
end;
