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

theorem Th29:
   for xL,x be Surreal
      st xL <= x & xL,No_omega^ y are_commensurate &
          not x,No_omega^ y are_commensurate holds
        No_omega^ y infinitely< x
proof
  set O=No_omega^ y;
  let xL,x be Surreal such that
A1: xL <= x & xL,No_omega^ y are_commensurate  and
A2: not x,O are_commensurate and
A3: not O infinitely< x;
  consider r be positive Real such that
A4:not O * uReal.r < x by A3;
  consider n be Nat such that
A5:r < n by SEQ_4:3;
  reconsider n as positive Nat by A5;
A6:O is positive;
  uReal.r < uReal.n = uDyadic.n = uInt.n by A5,SURREALN:51,46,def 5;
  then O * uReal.r < O*uInt.n by A6,SURREALR:70;
  then
A7: x < O*uInt.n by A4,SURREALO:4;
  consider k be positive Nat such that
A8: O < uInt.k * xL by A1;
  uInt.k is positive;
  then 0_No <= uInt.k;
  then uInt.k * xL <= uInt.k * x by A1,SURREALR:75;
  then O < uInt.k * x by A8,SURREALO:4;
  hence thesis by A2,A7;
end;
