reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem
  for R be real-valued FinSequence, r,s st r <> 0 holds R"{s/r} = (r*R)"{ s }
proof
  let R be real-valued FinSequence, r,s;
A1: Seg len R = dom R & dom(r*R) = Seg len(r*R) by FINSEQ_1:def 3;
  assume
A2: r <> 0;
  reconsider R1 = R as FinSequence of REAL by RVSUM_1:145;
  thus R"{s/r} c= (r*R)"{s}
  proof
    let x be object;
    assume
A3: x in R"{s/r};
    then x in R1"{s/r};
    then reconsider i = x as Element of NAT;
    R.i in {s/r} by A3,FUNCT_1:def 7;
    then R.i = s/r by TARSKI:def 1;
    then r*(R.i) = s by A2,XCMPLX_1:87;
    then (r*R).i = s by RVSUM_1:44;
    then
A4: (r*R).i in {s} by TARSKI:def 1;
    i in dom R by A3,FUNCT_1:def 7;
    then i in dom (r*R1) by A1,FINSEQ_2:33;
    hence thesis by A4,FUNCT_1:def 7;
  end;
  let x be object;
  assume
A5: x in (r*R)"{s};
  then reconsider i = x as Element of NAT;
  (r*R).i in {s} by A5,FUNCT_1:def 7;
  then (r*R).i = s by TARSKI:def 1;
  then r*R.i = s by RVSUM_1:44;
  then R.i = s/r by A2,XCMPLX_1:89;
  then
A6: R.i in {s/r} by TARSKI:def 1;
  i in dom(r*R) by A5,FUNCT_1:def 7;
  then i in dom R1 by A1,FINSEQ_2:33;
  hence thesis by A6,FUNCT_1:def 7;
end;
