
theorem Lemma154va:
  for I being BinOp of [.0,1.],    :: for Fuzzy_Implication it is trivial
      N being Fuzzy_Negation st
    I is satisfying_(NP) N-satisfying_CP holds
      N = FNegation I & FNegation I is negation-strong
  proof
    let I be BinOp of [.0,1.],
        N be Fuzzy_Negation;
    assume that
A1: I is satisfying_(NP) and
A2: I is N-satisfying_CP;
    set NI = FNegation I;
A3: N.0 = 1 by FUZIMPL3:def 4;
B1: 0 in [.0,1.] by XXREAL_1:1;
B2: 1 in [.0,1.] by XXREAL_1:1;
    for x being Element of [.0,1.] holds
      NI.x = N.x
    proof
      let x be Element of [.0,1.];
      NI.x = I.(x,0) by FUZIMPL3:def 16
          .= I.(N.0,N.x) by B1,A2
          .= N.x by A1,A3,FUZIMPL2:def 1;
      hence thesis;
    end;
    hence
T1: N = NI by FUNCT_2:63; then
C1: NI.1 = 0 by FUZIMPL3:def 4;
    for y being Element of [.0,1.] holds
      y = NI.(NI.y)
    proof
      let y be Element of [.0,1.];
      y = I.(1,y) by A1,FUZIMPL2:def 1
       .= I.(NI.y,NI.1) by B2,A2,T1
       .= NI.(NI.y) by C1,FUZIMPL3:def 16;
      hence thesis;
    end; then
    NI is involutive by FUZIMPL3:def 8;
    hence thesis by FUZIMPL3:def 13,def 11;
  end;
