
theorem
  for N being Fuzzy_Negation holds
    NegationD1 <= N <= NegationD2
  proof
    set f = NegationD1;
    set g = NegationD2;
    let N be Fuzzy_Negation;
    thus f <= N
    proof
      let x be Element of [.0,1.];
      per cases;
      suppose
A1:     x = 0; then
        f.x = 1 by D1Def;
        hence thesis by A1,N1Def;
      end;
      suppose x <> 0; then
        f.x = 0 by D1Def;
        hence thesis by XXREAL_1:1;
      end;
    end;
    let x be Element of [.0,1.];
    per cases;
    suppose
A1:   x = 1; then
      N.x = 0 by N1Def;
      hence thesis by A1,D2Def;
    end;
    suppose x <> 1; then
      g.x = 1 by D2Def;
      hence thesis by XXREAL_1:1;
    end;
  end;
