
theorem
  for x,y,k being R_eal st k <= 0 holds k*min(x,y) = max(k*x,k*y) & k*
  max(x,y) = min(k*x,k*y)
proof
  let x,y,k be R_eal;
  assume
A1: k <= 0;
  hereby
    per cases by XXREAL_0:16;
    suppose
      max(x,y) = x;
      then
A2:   y <= x by XXREAL_0:def 10;
      then k*x <= k*y by A1,XXREAL_3:101;
      then max(k*x,k*y) = k*y by XXREAL_0:def 10;
      hence k*min(x,y) = max(k*x,k*y) by A2,XXREAL_0:def 9;
    end;
    suppose
      max(x,y) = y;
      then
A3:   x <= y by XXREAL_0:def 10;
      then k*y <= k*x by A1,XXREAL_3:101;
      then max(k*x,k*y) = k*x by XXREAL_0:def 10;
      hence k*min(x,y) = max(k*x,k*y) by A3,XXREAL_0:def 9;
    end;
  end;
  per cases by XXREAL_0:15;
  suppose
    min(x,y) = x;
    then
A4: x <= y by XXREAL_0:def 9;
    then k*y <= k*x by A1,XXREAL_3:101;
    then min(k*x,k*y) = k*y by XXREAL_0:def 9;
    hence thesis by A4,XXREAL_0:def 10;
  end;
  suppose
    min(x,y) = y;
    then
A5: y <= x by XXREAL_0:def 9;
    then k*x <= k*y by A1,XXREAL_3:101;
    then min(k*y,k*x) = k*x by XXREAL_0:def 9;
    hence thesis by A5,XXREAL_0:def 10;
  end;
end;
