reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem
  for X1,X2,Y1,Y2 be set st X1 <=_ X2 & Y1 <=_ Y2 holds
     X1 \/ Y1 <=_ X2 \/ Y2
proof
  let X1,X2,Y1,Y2 be set such that
  A1:  X1 <=_ X2 & Y1 <=_ Y2;
  let x such that A2: x in X1\/Y1;
  per cases by A2,XBOOLE_0:def 3;
  suppose x in X1;
    then consider x2,x3 be Surreal such that
    A3:x2 in X2 & x3 in X2 & x2 <= x <= x3 by A1;
    take x2,x3;
    thus thesis by A3,XBOOLE_0:def 3;
  end;
  suppose x in Y1;
    then consider y2,y3 be Surreal such that
    A4:  y2 in Y2 & y3 in Y2 & y2 <= x <= y3 by A1;
    take y2,y3;
    thus thesis by A4,XBOOLE_0:def 3;
  end;
end;
