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 Th63:
  for S be Sequence st dom S = succ A &
    (for O st succ O in succ A holds S.succ O = [{S.O},{}]) &
    (for O st O in succ A & O is limit_ordinal holds S.O = [rng (S|O),{}])
  for O st O in succ A holds S.O = No_Ordinal_op O
proof
  let S be Sequence such that
A1:dom S = succ A and
A2:(for O st succ O in succ A holds S.succ O = [{S.O},{}]) &
  (for O st O in succ A & O is limit_ordinal holds
  S.O = [rng (S|O),{}]);
  let O be Ordinal such that
A3: O in succ A;
  consider So be Sequence such that
A4:No_Ordinal_op O = So.O & dom So = succ O and
A5:(for B st succ B in succ O holds So.succ B = [{So.B},{}]) &
  for B st B in succ O & B is limit_ordinal holds
  So.B = [rng (So|B),{}] by Def11;
  defpred P[Ordinal] means $1 c= O implies So.$1=S.$1;
A6: for B be Ordinal st for C be Ordinal st C in B holds P[C] holds P[B]
  proof
    let B be Ordinal such that
A7: for C be Ordinal st C in B holds P[C];
    assume
A8: B c= O;
    then
A9: B in succ O by ORDINAL1:22;
    A10:B in succ A by A3,A8,ORDINAL1:12;
    per cases;
    suppose not B is limit_ordinal;
      then consider C be Ordinal such that
A11:  B = succ C by ORDINAL1:29;
A12:  C in B by A11,ORDINAL1:21;
      S.C = So.C by A8,A12,A7,ORDINAL1:def 2;
      then So.B = [{S.C},{}] by A5,A8,ORDINAL1:22,A11;
      hence thesis by A3,A8,ORDINAL1:12,A2,A11;
    end;
    suppose B is limit_ordinal;
      then
A13:  S.B = [rng (S|B), {}] & So.B = [rng (So|B), {}]
      by A2,A5,A8,ORDINAL1:22,A3,ORDINAL1:12;
A14:  dom (S|B)=B & dom (So|B)=B by A4,A1,A9,ORDINAL1:def 2,RELAT_1:62,A10;
      for x be object st x in B holds (So|B).x = (S|B).x
      proof
        let x be object such that
A15:    x in B;
        reconsider x as Ordinal by A15;
        (So|B).x = So.x by A15,FUNCT_1:49
        .= S.x by A15,A7,A8,ORDINAL1:def 2;
        hence thesis by A15,FUNCT_1:49;
      end;
      hence thesis by A13,A14,FUNCT_1:2;
    end;
  end;
  for B be Ordinal holds P[B] from ORDINAL1:sch 2(A6);
  hence thesis by A4;
end;
