theorem Th1:
  r<=s implies r <= (r+s)/2 & (r+s)/2 <= s
proof
  assume
A1: r<=s;
  per cases by A1,XXREAL_0:1;
  suppose r<s;
    hence thesis by XREAL_1:226;
  end;
  suppose r=s;
    hence thesis;
  end;
end;
