
theorem Lemma156ii:
  for I being BinOp of [.0,1.],
      N being negation-strong Fuzzy_Negation st
      N = FNegation I holds
     I is satisfying_(EP) implies
       I is satisfying_(I3) satisfying_(I4) satisfying_(I5)
         satisfying_(NP) N-satisfying_CP
  proof
    let I be BinOp of [.0,1.],
        N being negation-strong Fuzzy_Negation;
    assume
A0: N = FNegation I;
    assume
A1: I is satisfying_(EP);
A2: N is involutive by FUZIMPL3:def 13,def 11;
    for x,y being Element of [.0,1.] holds
      I.(N.y,N.x) = I.(x,y)
    proof
      let x,y be Element of [.0,1.];
ZZ:   N.y in [.0,1.] & 0 in [.0,1.] by XXREAL_1:1;
      I.(N.y,N.x) = I.(N.y,I.(x,0)) by A0,FUZIMPL3:def 16
                 .= I.(x,I.(N.y,0)) by A1,FUZIMPL2:def 2,ZZ
                 .= I.(x,N.(N.y)) by FUZIMPL3:def 16,A0
                 .= I.(x,y) by A2,FUZIMPL3:def 8;
      hence thesis;
    end; then
T1: I is N-satisfying_CP; then
    I is satisfying_(NP) by A0,Lemma156i;
    hence thesis by T1,Lemma154vb;
  end;
