
theorem IRSIsLCP:
  I_RS is (N_CC)-satisfying_L-CP
  proof
    set N = N_CC;  set I = I_RS;
    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.];
      per cases;
      suppose A1: N.x <= y; then
        1 - x <= y by FUZIMPL3:def 6; then
        1 - y <= x by XREAL_1:12; then
A2:     N.y <= x by FUZIMPL3:def 6;
        I.(N.x,y) = 1 by A1,FUZIMPL1:def 20
                 .= I.(N.y,x) by A2,FUZIMPL1:def 20;
        hence thesis;
      end;
      suppose A1: N.x > y; then
        1 - x > y by FUZIMPL3:def 6; then
        1 - y > x by XREAL_1:12; then
A2:     N.y > x by FUZIMPL3:def 6;
        I.(N.x,y) = 0 by A1,FUZIMPL1:def 20
                 .= I.(N.y,x) by A2,FUZIMPL1:def 20;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
