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

theorem Th17:
  X << {x} & x <= y implies X << {y}
proof
  assume A1: X << {x} & x <= y;
  let l,r such that A2: l in X & r in {y};
  x in {x} by TARSKI:def 1;
  then l < y by A1,Th4,A2;
  hence thesis by A2,TARSKI:def 1;
end;
