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
  L_x is non empty finite & x is uSurreal implies card (L_x)=1
proof
  assume A1: L_x is non empty finite & x is uSurreal;
  then consider Min,Max be Surreal such that
     A2:Min in L_x & Max in L_x and
    A3: for y st y in L_x holds Min <= y <= Max by Th12;
  reconsider c = card (L_x) as Nat by A1;
  assume A4:card (L_x)<>1;
  set B=born_eq x;
  x = Unique_No x by Def11,A1;
  then A5:x in (unique_No_op B).B by Def10;
  A6: B in succ B by ORDINAL1:6;
  A7: born_eq x = born x by A5,Th38;
  then not x in union rng ((unique_No_op B)|B) by A5,Th38;
  then consider Y be non empty surreal-membered set such that
  A8:Y = born_eq_set x/\made_of union rng ((unique_No_op B)|B)
  & x = the Y -smallest Surreal by A5,A6,Def9;
  A9:for y st y in L_x holds y <= Max by A3;
  then reconsider Mx=[{Max},R_x] as Surreal by A2,Th23;
  A10: Mx==x & born Mx c= born x by A9,A2,Th23;
  x in Y by A8,Def7;
  then x in made_of union rng ((unique_No_op B)|B) by A8,XBOOLE_0:def 4;
  then A11: L_x\/R_x c= union rng ((unique_No_op B)|B) by Def8;
  L_Mx \/ R_Mx c= L_x\/R_x by XBOOLE_1:9,A2,ZFMISC_1:31;
  then L_Mx \/ R_Mx c= union rng ((unique_No_op B)|B) by A11,XBOOLE_1:1;
  then A12: Mx in made_of union rng ((unique_No_op B)|B) by Def8;
  Mx in Day born Mx c= Day born x by A9,A2,Th23,SURREAL0:def 18,35;
  then Mx in born_eq_set x by A7,A10,Def6;
  then Mx in Y by A8,A12,XBOOLE_0:def 4;
  then A13:card (L_x) (+) card (R_x) c= card (L_Mx) (+) card (R_Mx)
  by A10,A8,Def7;
  A14:card (L_Mx) = 1 by CARD_1:30;
  1 in Segm c by A4,A1,NAT_1:25,NAT_1:44;
  then card (L_Mx) (+) card (R_Mx) in card (L_x) (+) card (R_x)
  by A14,ORDINAL7:94;
  hence thesis by A13,ORDINAL1:12;
end;
