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.x 'or' (b '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 'or' b).x '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;
