
theorem Lemma154i:
  for I being BinOp of [.0,1.],    :: for Fuzzy_Implication it is trivial
      N being Fuzzy_Negation st
    I is satisfying_(I1) N-satisfying_CP holds
      I is satisfying_(I2)
  proof
    let I be BinOp of [.0,1.],
        N be Fuzzy_Negation;
    assume that
A0: I is satisfying_(I1) and
AA: I is N-satisfying_CP;
    for x,y,z being Element of [.0,1.] st y <= z holds
      I.(x,y) <= I.(x,z)
    proof
      let x,y,z be Element of [.0,1.];
      assume y <= z; then
      N.z <= N.y by FUZIMPL3:7; then
A1:   I.(N.y,N.x) <= I.(N.z,N.x) by A0;
      I.(x,y) = I.(N.y,N.x) by AA;
      hence thesis by A1,AA;
    end;
    hence thesis;
  end;
