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 Th43:
  x is uSurreal & o in L_x \/ R_x implies o is uSurreal
proof
  set B=born_eq x;
  assume A1: x is uSurreal & o in L_x \/ R_x;
  then x = Unique_No x by Def11;
  then A2:x in (unique_No_op B).B by Def10;
  L_x \/ R_x is surreal-membered;
  then reconsider y=o as Surreal by A1;
  A3: B in succ B by ORDINAL1:6;
  A4: born_eq x = born x by A2,Th38; then
  not x in union rng ((unique_No_op B)|B) by A2,Th38;
  then consider Y be non empty surreal-membered set such that
  A5:Y = born_eq_set x/\made_of union rng ((unique_No_op B)|B)
  & x = the Y -smallest Surreal by A2,A3,Def9;
  x in Y by A5,Def7;
  then x in made_of union rng ((unique_No_op B)|B) by A5,XBOOLE_0:def 4;
  then L_x \/ R_x c= union rng ((unique_No_op B)|B) by Def8;
  then consider Z be set such that
  A6:y in Z & Z in rng ((unique_No_op B)|B) by A1,TARSKI:def 4;
  consider o be object such that
  A7:o in dom ((unique_No_op B)|B) & ((unique_No_op B)|B).o=Z
  by A6,FUNCT_1:def 3;
  reconsider o as Ordinal by A7;
  y in (unique_No_op B).o by A6,A7,FUNCT_1:47;
  then A8:born_eq y = born y c= o &
  y in (unique_No_op B).born y by Th38;
  born y c= B by A4,A1,Th1,ORDINAL1:def 2;
  then y in (unique_No_op born y).born y by A8,Th39;
  then Unique_No y=y by Def10,A8;
  hence thesis by Def11;
end;
