
theorem :: Theorem 1.4.21
  for I being Fuzzy_Implication,
      h being bijective increasing UnOp of [.0,1.] holds
    ConjNeg (FNegation I,h) = FNegation ConjImpl (I,h)
  proof
    let I be Fuzzy_Implication,
        h be bijective increasing UnOp of [.0,1.];
    set f = ConjNeg (FNegation I,h);
    set g = FNegation ConjImpl (I,h);
AA: 0 in [.0,1.] by XXREAL_1:1;
    for x being Element of [.0,1.] holds f.x = g.x
    proof
      let x be Element of [.0,1.];
      f.x = h".((FNegation I).(h.x)) by CNDef
         .= h".(I.(h.x,0)) by FNeg
         .= h".(I.(h.x,h.0)) by LemmaBound
         .= (ConjImpl (I,h)).(x,0) by AA,BIDef
         .= g.x by FNeg;
      hence thesis;
    end;
    hence thesis by FUNCT_2:63;
  end;
