 reserve x,y for Element of [.0,1.];

theorem Lemma01:
  (x <= y implies I_LK.(x,y) = 1) &
  (x > y implies I_LK.(x,y) = 1 - x + y)
  proof
    thus x <= y implies I_LK.(x,y) = 1
    proof
      assume x <= y; then
      1 - x >= 1 - y by XREAL_1:10; then
a1:   1 - x + y >= 1 - y + y by XREAL_1:6;
      I_LK.(x,y) = min (1, 1 - x + y) by FUZIMPL1:def 14
                .= 1 by a1,XXREAL_0:def 9;
      hence thesis;
    end;
    assume x > y; then
    1 - x <= 1 - y by XREAL_1:10; then
a1: 1 - x + y <= 1 - y + y by XREAL_1:6;
    I_LK.(x,y) = min (1, 1 - x + y) by FUZIMPL1:def 14
              .= 1 - x + y by a1,XXREAL_0:def 9;
    hence thesis;
  end;
