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
  born x = born_eq x & not born x is limit_ordinal implies
    ex y,z be Surreal st x == z & (z = [L_y\/{y},R_y] or z = [L_y,R_y\/{y}])
proof
  assume A1: born x = born_eq x & not born x is limit_ordinal;
  then consider B be Ordinal such that
    A2: born x = succ B by ORDINAL1:29;
  defpred L[object] means for z st z = $1 holds born z in B & z < x;
  consider L be set such that
    A3: o in L iff o in Day B & L[o] from XBOOLE_0:sch 1;
  defpred R[object] means for z st z = $1 holds born z in B & x < z;
  consider R be set such that
    A4: o in R iff o in Day B & R[o] from XBOOLE_0:sch 1;
A5:  L << R
proof
  let l,r such that A6:l in L & r in R;
  l < x <= r by A3,A4,A6;
  hence thesis by Th4;
end;
A7: for o be object st o in L \/ R
    ex O st O in B & o in Day O
proof
  let o be object such that A8: o in L \/ R;
  A9:o in L or o in R by A8,XBOOLE_0:def 3;
  then o in Day B by A3,A4;
  then reconsider o as Surreal;
  born o in B & o in Day born o by A9,A3,A4,SURREAL0:def 18;
  hence thesis;
end;
then A10:[L,R] in Day B by A5,SURREAL0:46;
then reconsider LR=[L,R] as Surreal;
A11: not LR == x
proof
  assume LR == x;
  then born x = born_eq LR c= born LR c= B
    by A1,A10,SURREAL0:def 18,Th33,Def5;
  then succ B c= B c= succ B by A2,XBOOLE_1:1,7;
  then succ B = B by XBOOLE_0:def 10;
then B in B by ORDINAL1:6;
hence thesis;
end;
per cases by A11;
  suppose A12: LR < x;
   A13: L\/{LR} << R
proof
  let l,r such that A14:l in L\/{LR} & r in R;
l in L or l = LR by A14,ZFMISC_1:136;
then l < x <= r by A12,A3,A4,A14;
  hence thesis by Th4;
end;
 for o be object st o in L\/{LR} \/ R
    ex O st O in succ B & o in Day O
proof
  let o be object such that A15: o in L\/{LR} \/ R;
  o in L\/{LR} or o in R by A15,XBOOLE_0:def 3;
  then A16:o in L or o=LR or o in R by ZFMISC_1:136;
  then o = LR or o in Day B by A3,A4;
  then reconsider o as Surreal;
born o in B  or born o c= B by A10,A16,A3,A4,SURREAL0:def 18;
then born o c= B by ORDINAL1:def 2;
then born o in succ B & o in Day born o by ORDINAL1:22,SURREAL0:def 18;
  hence thesis;
end;
then A17: [L\/{LR},R] in Day succ B by A13,SURREAL0:46;
then reconsider L1R=[L\/{LR},R] as Surreal;
take LR,L1R;

not born L1R c= B
proof
  assume A18: born L1R c= B;
LR in L\/{LR} by ZFMISC_1:136;
then LR in L_L1R\/R_L1R by XBOOLE_0:def 3;
then born LR in born L1R by Th1;
then L[LR] by A12,A18;
then LR in L = L_LR << {LR} & LR in {LR}
by A3,A7,A5,SURREAL0:46,Th11,TARSKI:def 1;
hence thesis by Th3;
end;
then A19:succ B c= born L1R c= succ B by A17,ORDINAL1:16,21,SURREAL0:def 18;
for y be Surreal st y == L1R holds succ B c= born y
proof
  let y be Surreal such that A20: y == L1R;
  assume A21:not succ B c= born y;
  then A22:born y c= B by ORDINAL1:16,22;
A23: L_L1R << {y} & {y} << R_L1R by A20,SURREAL0:43;
LR in L\/{LR} & y in {y} by TARSKI:def 1,ZFMISC_1:136;
then per cases by Th13,A23;
  suppose ex xLR be Surreal st xLR in R_LR & LR < xLR <= y;
then consider xLR be Surreal such that
   A24: xLR in R_LR & LR < xLR <= y;
xLR <= L1R by A20,A24,Th4;
then L_xLR << {L1R} & {xLR} << R_L1R by SURREAL0:43;
then xLR in {xLR} << R by TARSKI:def 1;
hence thesis by Th3,A24;
end;
  suppose ex yL be Surreal st yL in L_y & LR <= yL < y;
then consider yL be Surreal such that
    A25: yL in L_y & LR <= yL < y;
yL in L_y\/R_y by A25,XBOOLE_0:def 3;
then A26:born yL in born y by Th1;
 then A27: yL in Day born yL c= Day B
   by A21,ORDINAL1:22,SURREAL0:def 18,SURREAL0:35;
per cases;
  suppose x==yL;
then B in succ B in B by A2,A26,A22,A1,ORDINAL1:6,12,Def5;
hence thesis;
end;
suppose yL < x;
then L[yL] by A26,A22;
then yL in L =L_LR << {LR} & LR in {LR} by A3,A27, TARSKI:def 1,Th11;
hence thesis by A25;
end;
suppose x < yL;
then R[yL] by A26,A22;
then yL in R & L1R in {L1R} << R_L1R by A4,A27, TARSKI:def 1,Th11;
then L1R <= yL < L1R by A25,A20,Th4;
hence thesis;
end;
end;
end;
then A28: born_eq L1R = succ B by A19,XBOOLE_0:def 10,Def5;
A29:L_L1R << {x}
proof
   let l,r such that A30:l in L_L1R & r in {x};
l in L or l = LR by A30,ZFMISC_1:136;
then l < x by A12,A3;
  hence thesis by A30,TARSKI:def 1;
end;
A31: {L1R} << R_x
proof
   let l,r such that A32:l in {L1R} & r in R_x;
A33:r in L_x\/R_x by A32,XBOOLE_0:def 3;
then A34: born r c= B by Th1,A2,ORDINAL1:22;
A35: x in {x} << R_x by Th11,TARSKI:def 1;
A36:l=L1R by A32,TARSKI:def 1;
assume A37:r <= l;
not l <= r
proof
  assume l<=r;
  then L1R == r by A37,A32,TARSKI:def 1;
  then born_eq L1R = born_eq r c= born r in succ B by A2,A33,Th1,Th33,Def5;
hence thesis by A28,ORDINAL1:12;
end;
then per cases by A36,Th13;
suppose ex xR be Surreal st xR in R_r & r < xR <= L1R;
then consider xR be Surreal such that
     A38: xR in R_r & r < xR <= L1R;
xR in L_r\/R_r by A38,XBOOLE_0:def 3;
then A39:born xR in born r by Th1;
then A40:xR in Day born xR c= Day B by A34,SURREAL0:35,def 18,ORDINAL1:def 2;
r <= xR by A38;
then R[xR] by A35,A32,Th4,A39,A34;
then xR in R & L1R in {L1R} << R_L1R by TARSKI:def 1,Th11,A40,A4;
hence thesis by A38;
end;
suppose ex yL be Surreal st yL in L_L1R & r <= yL < L1R;
then consider yL be Surreal such that
  A41: yL in L_L1R & r <= yL < L1R;
per cases by A41,ZFMISC_1:136;
   suppose yL in L;
then yL <=x by A3;
hence thesis by A35,A32,Th4,A41;
   end;
   suppose yL = LR;
then LR <= x < LR by A12,A35,A32,A41,Th4;
hence thesis;
   end;
end;
end;
A42:L_x << {L1R}
proof
   let r,l such that A43:r in L_x & l in {L1R};
A44:r in L_x\/R_x by A43,XBOOLE_0:def 3;
then A45: born r c= B by Th1,A2,ORDINAL1:22;
A46:x in {x} & L_x << {x} by Th11,TARSKI:def 1;
A47:l=L1R by A43,TARSKI:def 1;
assume A48:l <= r;
not r <= l
proof
  assume r<=l;
  then L1R == r by A48,A43,TARSKI:def 1;
  then born_eq L1R = born_eq r c= born r in succ B by A2,A44,Th1,Th33,Def5;
hence thesis by A28,ORDINAL1:12;
end;
then per cases by A47,Th13;
suppose ex xR be Surreal st xR in L_r & L1R <= xR < r;
then consider xR be Surreal such that
     A49: xR in L_r & L1R <= xR < r;
xR in L_r\/R_r by A49,XBOOLE_0:def 3;
then A50:born xR in born r by Th1;
then A51:xR in Day born xR c= Day B by A45,ORDINAL1:def 2,SURREAL0:def 18,35;
 xR <= r by A49;
then L[xR] by A50,A45,A46,A43,Th4;
then xR in L by A51,A3;
then xR in L_L1R & L1R in {L1R} & L_L1R << {L1R}
  by XBOOLE_0:def 3,TARSKI:def 1,Th11;
hence thesis by A49;
end;
suppose ex yL be Surreal st yL in R_L1R & L1R < yL <= r;
then consider yL be Surreal such that
  A52: yL in R_L1R & L1R < yL <= r;
x <= yL by A4,A52;
hence thesis by A52,A46,A43,Th4;
end;
end;
{x} << R_L1R
proof
   let l,r such that A53:l in {x} & r in R_L1R;
   x < r by A4,A53;
   hence thesis by A53,TARSKI:def 1;
end;
hence thesis by A31,SURREAL0:43,A29,A42;
  end;
  suppose A54: x < LR;
   A55: L << R\/{LR}
proof
  let l,r such that A56:l in L & r in R\/{LR};
r in R or r = LR by A56,ZFMISC_1:136;
then l <= x < r by A54,A3,A4,A56;
  hence thesis by Th4;
end;
 for o be object st o in L\/ (R \/{LR})
    ex O st O in succ B & o in Day O
proof
  let o be object such that A57: o in L\/(R\/{LR});
  o in L or o in R\/{LR} by A57,XBOOLE_0:def 3;
  then A58:o in L or o=LR or o in R by ZFMISC_1:136;
  then o = LR or o in Day B by A3,A4;
  then reconsider o as Surreal;
born o in B  or born o c= B by A10,A58,A3,A4,SURREAL0:def 18;
then born o c= B by ORDINAL1:def 2;
then born o in succ B & o in Day born o by ORDINAL1:22,SURREAL0:def 18;
  hence thesis;
end;
then A59: [L,R\/{LR}] in Day succ B by A55,SURREAL0:46;
then reconsider L1R=[L,R\/{LR}] as Surreal;
take LR,L1R;
not born L1R c= B
proof
  assume A60: born L1R c= B;
LR in R\/{LR} by ZFMISC_1:136;
then LR in L_L1R\/R_L1R by XBOOLE_0:def 3;
then born LR in born L1R by Th1;
then R[LR] by A54,A60;
then LR in R = R_LR & {LR} << R_LR & LR in {LR}
by A4,A7,A5,SURREAL0:46,Th11,TARSKI:def 1;
hence thesis by Th3;
end;
then A61: succ B c= born L1R c= succ B by A59,ORDINAL1:16,21,SURREAL0:def 18;
for y be Surreal st y == L1R holds succ B c= born y
proof
  let y be Surreal such that A62: y == L1R;
  assume A63: not succ B c= born y;
  then A64:born y c= B by ORDINAL1:16,22;
A65: L_L1R << {y} & {y} << R_L1R by A62,SURREAL0:43;
LR in R\/{LR} & y in {y} by TARSKI:def 1,ZFMISC_1:136;
then per cases by A65,Th13;
  suppose ex xLR be Surreal st xLR in L_LR & y <= xLR < LR;
then consider xLR be Surreal such that
   A66: xLR in L_LR & y <= xLR < LR;
L1R <= xLR by A62,A66,Th4;
then L_L1R << {xLR} & {L1R} << R_xLR by SURREAL0:43;
then L << {xLR} & xLR in {xLR} by TARSKI:def 1;
hence thesis by Th3,A66;
end;
  suppose ex yL be Surreal st yL in R_y & y < yL <= LR;
then consider yL be Surreal such that
    A67: yL in R_y & y < yL <= LR;
yL in L_y\/R_y by A67,XBOOLE_0:def 3;
then A68:born yL in born y by Th1;
 then A69: yL in Day born yL c= Day B
   by A63,ORDINAL1:22,SURREAL0:def 18,SURREAL0:35;
per cases;
  suppose x==yL;
then B in succ B in B by A2,A68,A64,A1,Def5,ORDINAL1:6,12;
hence thesis;
end;
suppose x < yL;
then R[yL] by A68,A64;
then yL in R =R_LR & {LR} << R_LR & LR in {LR}
by A4,A69,TARSKI:def 1,Th11;
hence thesis by A67;
end;
suppose yL < x;
then L[yL] by A68,A64;
then yL in L & L1R in {L1R} & L_L1R << {L1R}
  by A3,A69,TARSKI:def 1,Th11;
then L1R <= yL < L1R by A67,A62,Th4;
hence thesis;
end;
end;
end;
then A70: born_eq L1R = succ B by A61,XBOOLE_0:def 10,Def5;
A71:L_L1R << {x}
proof
   let l,r such that A72:l in L_L1R & r in {x};
l < x by A3,A72;
  hence thesis by A72,TARSKI:def 1;
end;
A73:{L1R} << R_x
proof
   let l,r such that A74:l in {L1R} & r in R_x;
A75:r in L_x\/R_x by A74,XBOOLE_0:def 3;
then A76: born r c= B by A2,Th1,ORDINAL1:22;
A77: x in {x} << R_x by Th11,TARSKI:def 1;
A78:l=L1R by A74,TARSKI:def 1;
assume A79:r <= l;
not l <= r
proof
  assume l<=r;
  then L1R == r by A79,A74,TARSKI:def 1;
  then born_eq L1R = born_eq r c= born r in succ B by A2,A75,Th1,Th33,Def5;
hence thesis by A70,ORDINAL1:12;
end;
then per cases by A78,Th13;
suppose ex xR be Surreal st xR in R_r & r < xR <= L1R;
then consider xR be Surreal such that
     A80: xR in R_r & r < xR <= L1R;
xR in L_r\/R_r by A80,XBOOLE_0:def 3;
then A81:born xR in born r by Th1;
then A82:xR in Day born xR c= Day B by A76,SURREAL0:def 18,35,ORDINAL1:def 2;
r <= xR by A80;
then R[xR] by A81,A76,A77,A74,Th4;
then xR in R by A82,A4;
then xR in R\/{LR} & L1R in {L1R} << R_L1R
by TARSKI:def 1,Th11,XBOOLE_0:def 3;
hence thesis by A80;
end;
suppose ex yL be Surreal st yL in L_L1R & r <= yL < L1R;
then consider yL be Surreal such that
  A83: yL in L_L1R & r <= yL < L1R;
yL <=x by A3,A83;
hence thesis by A83,A77,A74,Th4;
end;
end;
A84:L_x << {L1R}
proof
   let r,l such that A85:r in L_x & l in {L1R};
A86:r in L_x\/R_x by A85,XBOOLE_0:def 3;
then A87: born r c= B by A2,Th1,ORDINAL1:22;
A88: x in {x} & L_x << {x} by Th11,TARSKI:def 1;
A89:l=L1R by A85,TARSKI:def 1;
assume A90:l <= r;
not r <= l
proof
  assume r<=l;
  then L1R == r by A90,A85,TARSKI:def 1;
  then born_eq L1R = born_eq r c= born r in succ B by A2,A86,Th1,Th33,Def5;
hence thesis by A70,ORDINAL1:12;
end;
then per cases by A89,Th13;
suppose ex xR be Surreal st xR in L_r & L1R <= xR < r;
then consider xR be Surreal such that
     A91: xR in L_r & L1R <= xR < r;
xR in L_r\/R_r by A91,XBOOLE_0:def 3;
then A92:born xR in born r by Th1;
then A93:xR in Day born xR c= Day B by A87,ORDINAL1:def 2,SURREAL0:def 18,35;
 xR <= r by A91;
then L[xR] by A92,A87,A88,A85,Th4;
then xR in L_L1R & L1R in {L1R} & L_L1R << {L1R}
  by A93,A3,TARSKI:def 1,Th11;
hence thesis by A91;
end;
suppose ex yL be Surreal st yL in R_L1R & L1R < yL <= r;
then consider yL be Surreal such that
  A94: yL in R_L1R & L1R < yL <= r;
per cases by A94,ZFMISC_1:136;
   suppose yL in R;
then x <= yL by A4;
hence thesis by A94,A88,A85,Th4;
   end;
   suppose yL = LR;
then LR <= x < LR by A54,A88,A85,A94,Th4;
hence thesis;
   end;
end;
end;
{x} << R_L1R
proof
   let l,r such that A95:l in {x} & r in R_L1R;
   r in R or r = LR by A95,ZFMISC_1:136;
   then x < r by A54,A4;
   hence thesis by A95,TARSKI:def 1;
end;
hence thesis by A73,SURREAL0:43,A71,A84;
  end;
end;
