reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;
reserve n for Nat;

theorem Th38:
  x in (unique_No_op A).B implies born_eq x = born x c= B &
     x in (unique_No_op A).born x &
     not x in union rng ((unique_No_op A)|born x)
proof
  set M=unique_No_op A;
  assume A1:x in M.B;
  then B in dom M by FUNCT_1:def 2;
  then A2:B in succ A by Def9;
  defpred M[Ordinal] means x in M.$1 & $1 in succ A;
  A3: ex C be Ordinal st M[C] by A1,A2;
  consider D be Ordinal such that
  A4: M[D] & for E be Ordinal st M[E] holds D c= E from ORDINAL1:sch 1(A3);
  A5:not x in union rng (M|D)
  proof
    assume x in union rng (M|D);
    then consider Z be set such that
    A6: x in Z & Z in rng (M|D) by TARSKI:def 4;
    consider E be object such that
    A7:E in dom (M|D) & (M|D).E = Z by A6,FUNCT_1:def 3;
    reconsider E as Ordinal by A7;
    M[E] by A4,A7,FUNCT_1:47,A6,ORDINAL1:10;
    hence thesis by A4,A7,ORDINAL1:5;
  end;
  then A8: D = born_eq x by A4,Def9;
  consider Y be non empty surreal-membered set such that
  A9:Y=born_eq_set x/\made_of union rng (M|D)
  & x= the Y -smallest Surreal by A5,A4,Def9;
  x in born_eq_set x/\made_of union rng (M|D) by A9,Def7;
  then x in born_eq_set x by XBOOLE_0:def 4;
  then x in Day born_eq x by Def6;
  then born x c= born_eq x c= born x by Def5,SURREAL0:def 18;
  then born x = born_eq x by XBOOLE_0:def 10;
  hence thesis by A1,A2,A4,A5,A8;
end;
