
theorem Lemma154ii:
  for I being BinOp of [.0,1.],    :: for Fuzzy_Implication it is trivial
      N being Fuzzy_Negation st
    I is satisfying_(I2) N-satisfying_CP holds
      I is satisfying_(I1)
  proof
    let I be BinOp of [.0,1.],
        N be Fuzzy_Negation;
    assume that
A0: I is satisfying_(I2) and
AA: I is N-satisfying_CP;
    for x,y,z being Element of [.0,1.] st x <= y holds
      I.(x,z) >= I.(y,z)
    proof
      let x,y,z be Element of [.0,1.];
      assume x <= y; then
F0:   N.y <= N.x by FUZIMPL3:7;
A3:   I.(N.z,N.x) = I.(x,z) by AA;
      I.(N.z,N.x) >= I.(N.z,N.y) by F0,A0;
      hence thesis by A3,AA;
    end;
    hence thesis;
  end;
