
theorem LukasiDual:
  for a,b being Element of [.0,1.] holds
    min (a+b,1) = 1 - max (0,1-a+(1-b)-1)
  proof
    let a,b be Element of [.0,1.];
    per cases;
    suppose
A1:   a + b <= 1; then
A2:   a + b - (a + b) <= 1 - (a + b) by XREAL_1:9;
      min (a + b, 1) = 1 - (1 - a - b) by A1,XXREAL_0:def 9
         .= 1 - max (0,1-a+(1-b)-1) by A2,XXREAL_0:def 10;
      hence thesis;
    end;
    suppose
A1:   a + b > 1; then
A2:   a + b - (a + b) > 1 - (a + b) by XREAL_1:9;
      min (a+b,1) = 1 - 0 by A1,XXREAL_0:def 9
         .= 1 - max (0,1-a+(1-b)-1) by A2,XXREAL_0:def 10;
      hence thesis;
    end;
  end;
