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 X1,X2,Y be set st X1 << Y & X2 << Y holds
    X1\/X2 << Y
proof
  let X1,X2,Y be set such that A1: X1 << Y & X2 << Y;
  let l,r be Surreal such that A2: l in X1\/X2 & r in Y;
  l in X1 or l in X2 by A2,XBOOLE_0:def 3;
  hence thesis by A1,A2;
end;
