reserve A,B for Ordinal,
        o for object,
        x,y,z for Surreal,
        n for Nat,
        r,r1,r2 for Real;

theorem Th21:
  for S be c=-monotone Function-yielding Sequence st
    for B st B in dom S
      ex SB be ManySortedSet of Day B st S.B = SB &
        for x be object st x in Day B holds
          SB.x = [{0_No}\/{(union rng (S|B).xL) *' (uReal.r)
                           where xL is Element of L_x, r is Element of REAL:
                           xL in L_x & r is positive},
                  {(union rng (S|B).xR) *' (uReal.r)
                           where xR is Element of R_x, r is Element of REAL:
                           xR in R_x & r is positive}]
    holds
  for A st A in dom S holds No_omega_op A = S.A
proof
  deffunc D(Ordinal) = Day $1;
  deffunc H(object,c=-monotone Function-yielding Sequence) = [{0_No}\/
  {(union rng $2.xL) *' (uReal.r) where xL is Element of L_$1, r is
  Element of REAL:xL in L_$1 & r is positive},
  {(union rng $2.xR) *' (uReal.r) where xR is Element of R_$1, r is
  Element of REAL:xR in R_$1 & r is positive}];
  let S1 be c=-monotone Function-yielding Sequence such that
A1:for B be Ordinal st B in dom S1
       ex SB be ManySortedSet of D(B) st S1.B = SB &
         for x be object st x in D(B) holds SB.x = H(x,S1|B);
  let A be Ordinal such that
A2: A in dom S1;
A3:succ A c= dom S1 by A2,ORDINAL1:21;
  consider S2 be c=-monotone Function-yielding Sequence such that
A4:dom S2 = succ A & S2.A = No_omega_op A &
      for B be Ordinal st B in succ A
         ex SB be ManySortedSet of D(B) st S2.B = SB &
           for x be object st x in D(B) holds
             SB.x = H(x,S2|B) by Def4;
A5: for B be Ordinal st B in succ A
    ex SB be ManySortedSet of D(B) st S1.B = SB &
        for x be object st x in D(B) holds SB.x = H(x,S1|B) by A1,A3;
A6: for B be Ordinal st B in succ A
    ex SB be ManySortedSet of D(B) st S2.B = SB &
      for x be object st x in D(B) holds SB.x = H(x,S2|B) by A4;
A7: succ A c= dom S1 & succ A c= dom S2 by A2,ORDINAL1:21,A4;
A8: S1|succ A = S2|succ A from SURREALR:sch 2(A7,A5,A6);
  A in succ A by ORDINAL1:8;
  hence thesis by A4,A8,FUNCT_1:49;
end;
