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

theorem
  for r be non-zero Sequence of REAL,
      y be strictly_decreasing Surreal-Sequence
    for A be Ordinal st A c= dom r /\ dom y
    holds Partial_Sums(r,y).A = Sum(r|A,y|A)
proof
  let r be non-zero Sequence of REAL,
      y be strictly_decreasing Surreal-Sequence;
  let A be Ordinal such that
A1: A c= dom r /\ dom y;
  dom (r|A) = dom r /\ A & dom (y|A) = dom y /\ A by RELAT_1:61;
  then
A2: dom (r|A)/\dom (y|A) = (A/\dom r) /\ dom y /\A by XBOOLE_1:16
  .= A/\(dom r /\ dom y) /\A by XBOOLE_1:16
  .= (dom r /\ dom y) /\(A/\A) by XBOOLE_1:16
  .= A by A1,XBOOLE_1:28;
  thus Partial_Sums(r,y).A = (Partial_Sums(r,y)|succ A).A
  by ORDINAL1:6,FUNCT_1:49
  .= Sum(r|A,y|A) by A2,Th85;
end;
