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

theorem Th30:
   for x,xR be Surreal
      st 0_No < x <= xR & xR,No_omega^ y are_commensurate &
          not x,No_omega^ y are_commensurate holds
        x infinitely< No_omega^ y
proof
  set O=No_omega^ y;
  let x,xR be Surreal such that
A1: 0_No < x <= xR & xR,No_omega^ y are_commensurate and
A2: not x,O are_commensurate and
A3: not x infinitely< O;
  consider r be positive Real such that
A4:not x * uReal.r < O by A3;
  consider n be Nat such that
A5:r < n by SEQ_4:3;
  reconsider n as positive Nat by A5;
  uReal.r < uReal.n = uDyadic.n = uInt.n by A5,SURREALN:51,46,def 5;
  then x * uReal.r < x*uInt.n by A1,SURREALR:70;
  then
A6: O < x*uInt.n by A4,SURREALO:4;
  consider k be positive Nat such that
A7: xR < uInt.k * O by A1;
  x < uInt.k * O by A7,A1,SURREALO:4;
  hence thesis by A2,A6;
end;
