reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for x,y be Real holds min(x,y)*max(x,y) = x*y &
    min(x,y) + max(x,y) = x+y
  proof
    let x,y be Real;
    per cases;
    suppose x >= y; then
      min (x,y)= y & max(x,y) = x by XXREAL_0:def 9, def 10;
      hence thesis;
    end;
    suppose x < y; then
      min (x,y) = x & max (x,y) = y by XXREAL_0:def 9, def 10;
      hence thesis;
    end;
  end;
