
theorem
  for I being BinOp of [.0,1.],
      N being negation-strict Fuzzy_Negation,
      N1 being Fuzzy_Negation st N~ = N1 holds
   I is N-satisfying_L-CP iff
     I is N1-satisfying_R-CP
  proof
    let I be BinOp of [.0,1.],
        N be negation-strict Fuzzy_Negation,
        N1 be Fuzzy_Negation;
    assume
A0: N~ = N1; then
    N is onto by LemmaOnto2; then
A2: rng N = [.0,1.] by FUNCT_2:def 3;
DD: N" = N~ by FUNCT_1:def 5;
    thus I is N-satisfying_L-CP implies I is N1-satisfying_R-CP
    proof
      assume
a1:   I is N-satisfying_L-CP;
      for x,y being Element of [.0,1.] holds
        I.(x,N1.y) = I.(y,N1.x)
      proof
        let x,y be Element of [.0,1.];
        I.(x,N1.y) = I.(N.(N1.x),N1.y) by A0,A2,FUNCT_1:35,DD
             .= I.(N.(N1.y),N1.x) by a1
             .= I.(y,N1.x) by A2,DD,A0,FUNCT_1:35;
        hence thesis;
      end;
      hence I is N1-satisfying_R-CP;
    end;
    assume
a1: I is N1-satisfying_R-CP;
C1: dom N = [.0,1.] by FUNCT_2:def 1;
    for x,y being Element of [.0,1.] holds
      I.(N.x,y) = I.(N.y,x)
    proof
      let x,y be Element of [.0,1.];
B2:   (N1 * N).y = y by A0,C1,FUNCT_1:34,DD;
      (N1 * N).y = N1.(N.y) by C1,FUNCT_1:13; then
      I.(N.x,y) = I.(N.y,N1.(N.x)) by a1,B2
               .= I.(N.y,x) by A0,C1,FUNCT_1:34,DD;
      hence thesis;
    end;
    hence I is N-satisfying_L-CP;
  end;
