
theorem Prop136d: :: Proposition 1.3.6 d) p. 13
  for I being Fuzzy_Implication,
      f being bijective increasing UnOp of [.0,1.] st
   I is satisfying_(OP) holds
   ConjImpl (I,f) is satisfying_(OP)
  proof
    let I be Fuzzy_Implication,
        f be bijective increasing UnOp of [.0,1.];
    assume
B0: I is satisfying_(OP);
    set g = ConjImpl (I,f);
    let x,y be Element of [.0,1.];
b3: rng f = [.0,1.] by FUNCT_2:def 3;
b4: dom f = [.0,1.] by FUNCT_2:def 1;
    thus g.(x,y) = 1 implies x <= y
    proof
      assume g.(x,y) = 1; then
      f".(I.(f.x,f.y)) = 1 by BIDef; then
      f.(f".(I.(f.x,f.y))) = 1 by LemmaBound; then
      I.(f.x,f.y) = 1 by FUNCT_1:35,b3; then
      f.x <= f.y by FUZIMPL2:def 4,B0; then
      f".(f.x) <= f".(f.y) by Rosnie; then
      x <= f".(f.y) by b4,FUNCT_1:34;
      hence thesis by b4,FUNCT_1:34;
    end;
    assume x <= y; then
    f.x <= f.y by Rosnie; then
    I.(f.x, f.y) = 1 by FUZIMPL2:def 4,B0; then
    f".(I.(f.x,f.y)) = 1 by LemmaBound;
    hence thesis by BIDef;
  end;
