reserve Y for non empty set;

theorem
  for a,b,c being Function of Y,BOOLEAN holds ((a 'or' b) 'or' c)
  'imp' (a 'or' (b 'or' c))=I_el(Y)
proof
  let a,b,c be Function of Y,BOOLEAN;
  for x being Element of Y holds (((a 'or' b) 'or' c) 'imp' (a 'or' (b
  'or' c))).x=TRUE
  proof
    let x be Element of Y;
    (((a 'or' b) 'or' c) 'imp' (a 'or' (b 'or' c))).x ='not' ((a 'or' b)
    'or' c).x 'or' (a 'or' (b 'or' c)).x by BVFUNC_1:def 8
      .='not'( (a 'or' b).x 'or' c.x) 'or' (a 'or' (b 'or' c)).x by
BVFUNC_1:def 4
      .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a 'or' (b 'or' c)).x by
BVFUNC_1:def 4
      .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b 'or' c).x)
    by BVFUNC_1:def 4
      .='not'( (a.x 'or' b.x) 'or' c.x) 'or' (a.x 'or' (b.x 'or' (
    c).x)) by BVFUNC_1:def 4
      .=TRUE by XBOOLEAN:102;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
