reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem
  for X,Y1,Y2 be set st X << Y1 & X << Y2 holds
    X << Y1\/Y2
proof
  let X,Y1,Y2 be set such that A1: X << Y1 & X << Y2;
  let l,r be Surreal such that A2: l in X & r in Y1\/Y2;
  r in Y1 or r in Y2 by A2,XBOOLE_0:def 3;
  hence thesis by A1,A2;
end;
