
theorem LemmaOnto2:
  for N1,N2 being Fuzzy_Negation st N1~ = N2 holds
    N1 is onto
  proof
    let N1,N2 be Fuzzy_Negation;
    assume N1~ = N2; then
AB: N1 = N2~; then
FF: N2 is one-to-one by LemmaOne;
    for y being object st y in [.0,1.]
     ex x being object st x in [.0,1.] & y = N1.x
    proof
      let y be object;
      assume
A2:   y in [.0,1.];
      take x = N2.y;
      N1 = N2" by AB,LemmaOne,FUNCT_1:def 5;
      hence thesis by A2,FF,FUNCT_2:26,FUNCT_2:5;
    end; then
    rng N1 = [.0,1.] by FUNCT_2:10;
    hence thesis by FUNCT_2:def 3;
  end;
