reserve x,y,w,z for ExtReal,
  a for Real;

theorem
  min(x,y) + max(x,y)= x + y
proof
  per cases;
  suppose
A1: x <= y;
    then min(x,y) = x by XXREAL_0:def 9;
    hence thesis by A1,XXREAL_0:def 10;
  end;
  suppose
A2: not x <= y;
    then min(x,y) = y by XXREAL_0:def 9;
    hence thesis by A2,XXREAL_0:def 10;
  end;
end;
