theorem Th29:
  P * ASeq is Real_Sequence
proof
  rng ASeq c= Sigma by RELAT_1:def 19;
  then rng ASeq c= dom P by FUNCT_2:def 1;
  then
A1: dom (P * ASeq) = dom ASeq by RELAT_1:27
    .= NAT by FUNCT_2:def 1;
  rng (P * ASeq) c= REAL by RELAT_1:def 19;
  hence thesis by A1,FUNCT_2:def 1,RELSET_1:4;
end;
