
theorem
  for N being decreasing Fuzzy_Negation holds
    I_RS is N-satisfying_CP
  proof
    let N be decreasing Fuzzy_Negation;
    set I = I_RS;
    for x,y being Element of [.0,1.] holds
      I.(x,y) = I.(N.y,N.x)
    proof
      let x,y be Element of [.0,1.];
      per cases;
      suppose A1: x <= y; then
A2:     N.y <= N.x by FUZIMPL3:7;
        I.(x,y) = 1 by A1,FUZIMPL1:def 20
               .= I.(N.y,N.x) by A2,FUZIMPL1:def 20;
        hence thesis;
      end;
      suppose A1: x > y; then
A2:     N.x < N.y by FUZIMPL3:8;
        I.(x,y) = 0 by A1,FUZIMPL1:def 20
               .= I.(N.y,N.x) by A2,FUZIMPL1:def 20;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
