
theorem LMXFIN1:
  for x be XFinSequence of REAL,
      y be Real_Sequence
  holds x(#)y is finite Sequence of REAL
  & dom (x(#)y) = dom x
  & for i be object st i in dom x
  holds (x(#)y).i = (x.i) * (y.i)
  proof
    let x be XFinSequence of REAL, y be Real_Sequence;
    P1:dom y = NAT by FUNCT_2:def 1;
    P2: dom (x(#)y) = (dom x) /\ (dom y) by VALUED_1:def 4
    .= (dom x) by XBOOLE_1:28,P1; then
    x(#)y is Sequence of rng (x(#)y) by ORDINAL1:31;
    hence thesis by P2,VALUED_1:def 4;
  end;
