reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds ((a '&' b) '&' c)
  'imp' (a '&' (b '&' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  for x being Element of Y holds (((a '&' b) '&' c) 'imp' (a '&' (b '&' c)
  )).x=TRUE
  proof
    let x be Element of Y;
    (((a '&' b) '&' c) 'imp' (a '&' (b '&' c))).x ='not' ((a '&' b) '&' c)
    .x 'or' (a '&' (b '&' c)).x by BVFUNC_1:def 8
      .='not'( (a '&' b).x '&' c.x) 'or' (a '&' (b '&' c)).x by
MARGREL1:def 20
      .='not'( (a.x '&' b.x) '&' c.x) 'or' (a '&' (b '&' c)).x by
MARGREL1:def 20
      .='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b '&' c).x) by
MARGREL1:def 20
      .='not'( (a.x '&' b.x) '&' c.x) 'or' (a.x '&' (b.x '&' c.x
    )) by MARGREL1:def 20
      .=TRUE by XBOOLEAN:102;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
