
theorem
  for I being Fuzzy_Negation,
      f being bijective increasing UnOp of [.0,1.] holds
    ConjNeg (I,f) = f" * I * f
  proof
    let I be Fuzzy_Negation,
        f be bijective increasing UnOp of [.0,1.];
    set g = ConjNeg (I,f);
AA: dom f = [.0,1.] by FUNCT_2:def 1;
A0: dom g = [.0,1.] by FUNCT_2:def 1;
A1: dom g = dom (f" * I * f) by A0,FUNCT_2:def 1;
    for x being object st x in dom g holds
      g.x = (f" * I * f).x
    proof
      let x be object;
      assume x in dom g; then
      reconsider x as Element of [.0,1.] by FUNCT_2:def 1;
      f.x in [.0,1.]; then
AB:   f.x in dom I by FUNCT_2:def 1;
      g.x = f".(I.(f.x)) by CNDef
         .= (f" * I).(f.x) by FUNCT_1:13,AB
         .= (f" * I * f).x by FUNCT_1:13,AA;
      hence thesis;
    end;
    hence thesis by A1,FUNCT_1:2;
  end;
