reserve Y for non empty set;

theorem
  for a,b being Function of Y,BOOLEAN holds a 'imp' 'not' b=I_el(Y
  ) implies b 'imp' 'not' a=I_el(Y)
proof
  let a,b be Function of Y,BOOLEAN;
  assume
A1: a 'imp' 'not' b=I_el(Y);
  for x being Element of Y holds (b 'imp' 'not' a).x=TRUE
  proof
    let x be Element of Y;
    (a 'imp' 'not' b).x=TRUE by A1,BVFUNC_1:def 11;
    then ('not' a.x) 'or' ('not' b).x=TRUE by BVFUNC_1:def 8;
    then
A2: 'not' a.x 'or' 'not' b.x=TRUE by MARGREL1:def 19;
    (b 'imp' 'not' a).x =('not' b.x) 'or' ('not' a).x by BVFUNC_1:def 8
      .=TRUE by A2,MARGREL1:def 19;
    hence thesis;
  end;
  hence thesis by BVFUNC_1:def 11;
end;
