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
  (for z st z in born_eq_set x & L_z\/R_z is uniq-surreal-membered & x<>z
         holds card (L_x) (+) card (R_x) in card (L_z) (+) card (R_z)) &
    x in born_eq_set x &
    L_x\/R_x is uniq-surreal-membered
  implies x is uSurreal
proof
  assume that A1: for z st z in born_eq_set x & L_z\/R_z
  is uniq-surreal-membered
  & x<>z holds card (L_x) (+) card (R_x) in card (L_z) (+) card (R_z) and
  A2:  x in born_eq_set x &
  L_x\/R_x is uniq-surreal-membered;
  set c = Unique_No x;
  set B=born_eq x;
  A3: c in (unique_No_op B).B by Def10;
  A4: B in succ B by ORDINAL1:6;
  x in Day born_eq x by A2,Def6;
  then born x c= born_eq x c= born x by SURREAL0:def 18,Def5;
  then A5: born x = B by XBOOLE_0:def 10;
  A6: x == c by Def10;
  A7:born_eq c = B & born_eq_set c = born_eq_set x by Def10,Th33,Th34;
  born_eq c = born c by A3,Th38;
  then not c in union rng ((unique_No_op B)|B) by A3,Th38,A7;
  then consider Y be non empty surreal-membered set such that
  A8:Y = born_eq_set c/\made_of union rng ((unique_No_op B)|B)
  & c = the Y -smallest Surreal by A3,A4,Def9;
  c in Y by A8,Def7;
  then A9: c in born_eq_set x & L_c\/R_c is uniq-surreal-membered
  by A7,A8,XBOOLE_0:def 4;
  A10: x in born_eq_set c by A2,Def10,Th34;
  L_x\/ R_x c= union rng ((unique_No_op B)|B)
  proof
    let o;
    assume A11: o in L_x\/ R_x;
    then reconsider y=o as uSurreal by A2;
    set C = born_eq y;
    A12: born y = C by Th48;
    y = Unique_No y by Def11;
    then A13:y in (unique_No_op C).C by Def10;
    A14: C in B by A5,A12,A11,Th1;
    C c= B by A5,A12,A11,Th1,ORDINAL1:def 2;
    then A15:(unique_No_op C).C = (unique_No_op B).C by Th39;
    A16: ((unique_No_op B)|B).C = (unique_No_op B).C
      by A5,A12,A11,Th1,FUNCT_1:49;
    dom (unique_No_op B)=succ B by Def9;
    then dom ((unique_No_op B)|B) = B by A4,RELAT_1:62,ORDINAL1:def 2;
    then (unique_No_op C).C in rng ((unique_No_op B)|B)
      by A15,A14,A16,FUNCT_1:def 3;
    hence thesis by A13,TARSKI:def 4;
  end;
  then
  x in made_of union rng ((unique_No_op B)|B) by Def8;
  then x in Y by A8,A10,XBOOLE_0:def 4;
  then card (L_c) (+) card (R_c) c= card (L_x) (+) card (R_x) by A8,A6,Def7;
  hence thesis by A9,A1,ORDINAL1:5;
end;
