theorem Th40:
  (P * Partial_Union ASeq).0 = Partial_Sums(P * ASeq).0
proof
A1: dom (P * ASeq) = NAT by SEQ_1:1;
  dom (P * Partial_Union ASeq) = NAT by SEQ_1:1;
  then
A2: (P * Partial_Union ASeq).0 = P.((Partial_Union ASeq).0) by FUNCT_1:12
    .= P.(ASeq.0) by Def2;
  Partial_Sums(P * ASeq).0 = (P * ASeq).0 by SERIES_1:def 1
    .= P.(ASeq.0) by A1,FUNCT_1:12;
  hence thesis by A2;
end;
