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

theorem
  for r be Real holds uReal.r infinitely< No_omega
proof
  let r be Real;
  let s be positive Real;
A1: r*s+0 < r*s+1 by XREAL_1:6;
  uReal.r * uReal.s == uReal.(r*s) by SURREALN:57;
  then
A2: uReal.r * uReal.s < uReal.(r*s+1) by A1,SURREALN:51,SURREALO:4;
  uReal.(r*s+1) <= No_omega by Th1,SURREALN:76;
  hence thesis by A2,SURREALO:4;
end;
