reserve x,X,Y for set;
reserve g,r,r1,r2,p,p1,p2 for Real;
reserve R for Subset of REAL;
reserve seq,seq1,seq2,seq3 for Real_Sequence;
reserve Ns for increasing sequence of NAT;
reserve n for Nat;
reserve W for non empty set;
reserve h,h1,h2 for PartFunc of W,REAL;

theorem Th9:
  for h being PartFunc of W,REAL, seq being sequence of W holds
  for r being Real holds rng seq c= dom h implies
  (r(#)h)/*seq = r(#) (h/*seq)
proof
  let h be PartFunc of W,REAL, seq be sequence of W;
  let r be Real;
  assume
A1: rng seq c= dom h;
  then
A2: rng seq c= dom (r(#)h) by VALUED_1:def 5;
  now
    let n;
A3: n in NAT by ORDINAL1:def 12;
A4: seq.n in rng seq by VALUED_0:28;
    thus ((r(#)h)/*seq).n = (r(#)h).(seq.n) by A2,FUNCT_2:108,A3
      .= r * (h.(seq.n)) by A2,A4,VALUED_1:def 5
      .= r * (h/*seq).n by A1,FUNCT_2:108,A3;
  end;
  hence thesis by SEQ_1:9;
end;
