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

theorem Th89:
  for r be non-zero Sequence of REAL,
    y be strictly_decreasing Surreal-Sequence
  holds Sum(r,y) in_meets_terms Partial_Sums(r,y),y,r,dom r/\dom y
proof
  let r be non-zero Sequence of REAL,
    y be strictly_decreasing Surreal-Sequence;
    dom r/\dom y in succ(dom r/\dom y) by ORDINAL1:6;
    then
A1: dom r/\dom y in dom Partial_Sums(r,y) by Def17;
    Partial_Sums(r,y),y,r simplest_on_position dom r/\dom y by A1,Def17;
    hence thesis;
end;
