reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th11:
  for x,y,z be Surreal st x <= y infinitely< z
    holds x infinitely< z
proof
  let x,y,z be Surreal such that A1:x <= y infinitely< z;
  let r be positive Real;
  0_No <= uReal.r by SURREALI:def 8;
  then x * uReal.r <= y * uReal.r < z by A1,SURREALR:75;
  hence thesis by SURREALO:4;
end;
