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 Th73:
   No_uOrdinal_op A == No_Ordinal_op A & born No_uOrdinal_op A = A
proof
A1:No_uOrdinal_op A == No_Ordinal_op A by SURREALO:def 10;
  born_eq No_uOrdinal_op A = born_eq No_Ordinal_op A c= born No_Ordinal_op A
  = A by SURREALO:33,def 5,def 10,Th70;
  then
A2:born No_uOrdinal_op A c= A by SURREALO:48;
  A c= born No_uOrdinal_op A
  proof
    set B=born No_uOrdinal_op A;
    assume not A c= born No_uOrdinal_op A;
    then
A3: No_Ordinal_op B < No_Ordinal_op A by Th68,ORDINAL1:16;
    No_uOrdinal_op A in Day B by SURREAL0:def 18;
    then No_uOrdinal_op A <= No_Ordinal_op B by Th69;
    hence thesis by A1,A3,SURREALO:4;
  end;
  hence thesis by SURREALO:def 10,A2,XBOOLE_0:def 10;
end;
