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

theorem Th20:
  0_No <= x infinitely< y implies x*uReal.r < y
proof
  assume
A1:0_No <= x infinitely< y;
  per cases;
  suppose r is positive;
    hence thesis by A1;
  end;
  suppose r <= 0;
    then uReal.r <= 0_No by SURREALN:51,47;
    then uReal.r*x <= 0_No*x by A1,SURREALR:75;
    then uReal.r*x <= x < y by A1,SURREALO:4,Th9;
    hence thesis by SURREALO:4;
  end;
end;
