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