
theorem Prop136b: :: Proposition 1.3.6 b) p. 13
  for I being Fuzzy_Implication,
      f being bijective increasing UnOp of [.0,1.] st
   I is satisfying_(EP) holds
     ConjImpl (I,f) is satisfying_(EP)
  proof
    let I be Fuzzy_Implication,
        f be bijective increasing UnOp of [.0,1.];
    assume
B0: I is satisfying_(EP);
    set g = ConjImpl (I,f);
    let x,y,z be Element of [.0,1.];
    I.(f.y,f.z) in [.0,1.]; then
BB: I.(f.y,f.z) in rng f by FUNCT_2:def 3;
    I.(f.x,f.z) in [.0,1.]; then
B2: I.(f.x,f.z) in rng f by FUNCT_2:def 3;
    g.(x,g.(y,z)) = g.(x,f".(I.(f.y, f.z))) by BIDef
       .= f".(I.(f.x,f.(f".(I.(f.y, f.z))))) by BIDef
       .= f".(I.(f.x,I.(f.y, f.z))) by FUNCT_1:35,BB
       .= f".(I.(f.y,I.(f.x, f.z))) by B0,FUZIMPL2:def 2
       .= f".(I.(f.y,f.(f".(I.(f.x, f.z))))) by B2,FUNCT_1:35
       .= g.(y,f".(I.(f.x, f.z))) by BIDef
       .= g.(y,g.(x,z)) by BIDef;
    hence thesis;
  end;
