reserve i,j for Nat;

theorem Th17:
  for r being Real, fr being Element of F_Real, p being
FinSequence of REAL, fp being FinSequence of F_Real st r=fr & p=fp holds r*p=fr
  *fp
proof
  let r be Real, fr be Element of F_Real, p be FinSequence of REAL,
      fp be FinSequence of F_Real;
  assume that
A1: r=fr and
A2: p=fp;
A3: len (r*p)=len fp by A2,RVSUM_1:117;
  then
A4: len (r*p)=len (fr*fp) by Th16;
  for i be Nat st 1<=i & i<=len (r*p) holds (r*p).i=(fr*fp).i
  proof
    let i be Nat;
    assume 1<=i & i<=len (r*p);
    then
 i in Seg len fp by A3,FINSEQ_1:1;
    then
A5: i in dom (fr*fp) by A3,A4,FINSEQ_1:def 3;
    reconsider s=fp.i as Element of F_Real by XREAL_0:def 1;
    thus (r*p).i=fr*s by A1,A2,RVSUM_1:44
      .= (fr*fp).i by A5,FVSUM_1:50;
  end;
  hence thesis by A4,FINSEQ_1:14;
end;
