reserve A,B,O for Ordinal,
        o for object,
        x,y,z for Surreal,
        n,m for Nat;
reserve d,d1,d2 for Dyadic;
reserve i,j for Integer,
        n,m,p for Nat;
reserve r,r1,r2 for Real;

theorem
  born uReal.r = omega iff not r is Dyadic
proof
  uReal.r = Unique_No sReal.r by Def7;
  then
A1: born_eq (sReal.r) = born_eq (uReal.r) by SURREALO:def 10,33;
A2: born_eq uReal.r = born uReal.r by SURREALO:48;
  thus born (uReal.r) = omega implies not r is Dyadic
  proof
    assume
A3: born(uReal.r) = omega & r is Dyadic;
    then reconsider r as Dyadic;
    born  uDyadic.r is finite by Th37;
    hence thesis by Th46,A3;
  end;
  assume that
A4:not r is Dyadic and
A5:born (uReal.r) <> omega;
  born_eq (sReal.r) c= born (sReal.r) c= omega by SURREALO:def 5,Th49;
  then born_eq (uReal.r) c< omega
  by A1,A2,A5,XBOOLE_0:def 8,XBOOLE_1:1;
  then born_eq (uReal.r) in omega by ORDINAL1:11;
  then reconsider B=born_eq (uReal.r) as Nat;
  consider x be Surreal such that
A6: born x = B & uReal.r ==x by SURREALO:def 5;
  x in Day B by A6,SURREAL0:def 18;
  then consider d be Dyadic such that
A7:x == uDyadic.d & uDyadic.d in Day B by Th35;
  uReal.r == uDyadic.d == uReal.d by A7,A6,SURREALO:4,Th46;
  then r <= d <= r by SURREALO:4,Th51;
  hence thesis by XXREAL_0:1,A4;
end;
