
theorem DrasticDef:
  for a,b being Element of [.0,1.] holds
    (a = 1 implies drastic_norm.(a,b) = b) &
    (b = 1 implies drastic_norm.(a,b) = a) &
    (a <> 1 & b <> 1 implies drastic_norm.(a,b) = 0)
  proof
    let a,b be Element of [.0,1.];
A1: b <= 1 & a <= 1 by XXREAL_1:1;
    thus a = 1 implies drastic_norm.(a,b) = b
    proof
      assume
B1:   a = 1; then
      max (a,b) = 1 by A1,XXREAL_0:def 10; then
      drastic_norm.(a,b) = min (a,b) by Drastic2Def
                        .= b by XXREAL_0:def 9,A1,B1;
      hence thesis;
    end;
    thus b = 1 implies drastic_norm.(a,b) = a
    proof
      assume
B1:   b = 1; then
      max (a,b) = 1 by A1,XXREAL_0:def 10; then
      drastic_norm.(a,b) = min (a,b) by Drastic2Def
                        .= a by XXREAL_0:def 9,A1,B1;
      hence thesis;
    end;
    assume a <> 1 & b <> 1; then
    max (a,b) <> 1 by XXREAL_0:16;
    hence thesis by Drastic2Def;
  end;
