
theorem Lemma156i:
  for I being BinOp of [.0,1.],
      N being negation-strong Fuzzy_Negation st
    N = FNegation I holds
     I is N-satisfying_CP implies
       I is satisfying_(NP)
  proof
    let I be BinOp of [.0,1.],
        N being negation-strong Fuzzy_Negation;
    assume
A0: N = FNegation I;
    assume
A1: I is N-satisfying_CP;
A2: N is involutive by FUZIMPL3:def 13,def 11;
    for y being Element of [.0,1.] holds
      I.(1,y) = y
    proof
      let y be Element of [.0,1.];
      1 in [.0,1.] by XXREAL_1:1; then
      I.(1,y) = I.(N.y,N.1) by A1
             .= I.(N.y,0) by FUZIMPL3:def 4
             .= N.(N.y) by A0,FUZIMPL3:def 16
             .= y by A2,FUZIMPL3:def 8;
      hence thesis;
    end;
    hence thesis by FUZIMPL2:def 1;
  end;
