 reserve A,B,O for Ordinal,
      n,m for Nat,
      a,b,o for object,
      x,y,z for Surreal,
      X,Y,Z for set,
      Inv,I1,I2 for Function;

theorem Th25:
  for S be c=-monotone Function-yielding Sequence st
    for B st B in dom S
      ex SB be ManySortedSet of Positives B st S.B = SB &
         for o st o in Positives B holds
           SB.o =
           [ Union divL(||.o.||,(union rng (S|B))),
             Union divR(||.o.||,(union rng (S|B)))]
    holds
  for A st A in dom S holds No_inverse_op A = S.A
proof
  deffunc D(Ordinal) = Positives $1;
  deffunc H(object,c=-monotone Function-yielding Sequence) =
  [Union divL(||.$1.||,(union rng ($2))),
    Union divR(||.$1.||,(union rng ($2)))];
  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_inverse_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 Def11;
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;
