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
  No_uOrdinal_op succ A = [{No_uOrdinal_op A},{}]
proof
   set OA=No_uOrdinal_op A,x = [{OA},{}];
A1: {OA}<<{};
A2: born OA = A by Th73;
   then
A3: OA in Day A by SURREAL0:def 18;
   o in {OA}\/{} implies ex O st O in succ A & o in Day O
   proof
     assume o in {OA}\/{};
     then o = OA by TARSKI:def 1;
     hence thesis by A3,ORDINAL1:6;
   end;
   then
A4:x in Day succ A by A1,SURREAL0:46;
   then reconsider x as Surreal;
A5: No_Ordinal_op succ A = [{No_Ordinal_op A},{}] by Th65;
   OA == No_Ordinal_op A by Th73;
   then {OA} <==> {No_Ordinal_op A} by SURREALO:32;
   then
A6: x == No_Ordinal_op succ A by A5,SURREALO:29;
A7: born x c= succ A by A4,SURREAL0:def 18;
   OA in L_x \/ R_x by TARSKI:def 1;
   then
A8: succ A c= born x by A2,SURREALO:1,ORDINAL1:21;
   for y be Surreal st y == x holds succ A c= born y
   proof
     let y be Surreal such that
A9:  y == x & not succ A c= born y;
     No_uOrdinal_op born y < No_uOrdinal_op succ A == No_Ordinal_op succ A
     by Th75,Th73,A9,ORDINAL1:16;
     then No_uOrdinal_op born y < No_Ordinal_op succ A
     by SURREALO:4;
     then
A10: No_uOrdinal_op born y < x by A6,SURREALO:4;
     born y c= A by A9,ORDINAL1:16,ORDINAL1:22;
     then y in Day born y c= Day A by SURREAL0:35,SURREAL0:def 18;
     then y <= No_uOrdinal_op born y by Th76;
     hence thesis by A9,A10,SURREALO:4;
   end;
   then
A11: born_eq x = succ A by A8,SURREALO:def 5,A7,XBOOLE_0:def 10;
   then
A12:x in born_eq_set x by A4,SURREALO:def 6;
A13: L_x\/R_x is uniq-surreal-membered;
   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)
   proof
     let z such that
A14: z in born_eq_set x & L_z\/R_z is uniq-surreal-membered & x<>z
     and
A15: not card (L_x) (+) card (R_x) in card (L_z) (+) card (R_z);
A16: z == x & z in Day succ A by A11,A14,SURREALO:def 6;
     then L_z << {x} & L_x << {z} by SURREAL0:43;
     then (ex xR be Surreal st xR in R_OA & OA < xR <= z) or
     (ex yL be Surreal st yL in L_z & OA <= yL < z) by SURREALO:13,21;
     then consider zL be Surreal such that
A17: zL in L_z & OA <= zL < z by Def10;
     zL in L_z\/R_z by A17,XBOOLE_0:def 3;
     then
A18: zL is uSurreal by A14;
     zL in L_z \/ R_z by A17,XBOOLE_0:def 3;
     then born zL in born z c= succ A
     by SURREALO:1,A16,SURREAL0:def 18;
     then zL in Day born zL c= Day A
     by SURREAL0:35,SURREAL0:def 18,ORDINAL1:22;
     then zL == OA by Th76,A17;
     then
A19: zL = OA by A18,SURREALO:50;
     card (L_x) = 1 & card (R_x)=0 by CARD_1:30;
     then card (L_x) (+) card (R_x) = 1 by ORDINAL7:69;
     then card (L_z) (+) card (R_z) = 1 by
     CARD_1:49,A15,ORDINAL1:16,A17,ZFMISC_1:33;
     then consider w be Surreal such that
A20: z= [{},{w}] or z = [{w},{}] by SURREALO:47;
     thus thesis by A20,A17,A14,A19,TARSKI:def 1;
   end;
   then
A21:x is uSurreal by A12,A13,SURREALO:49;
   No_Ordinal_op succ A == No_uOrdinal_op succ A by Th73;
   then No_uOrdinal_op succ A == x by A6,SURREALO:4;
   hence thesis by A21,SURREALO:50;
end;
