reserve D for non empty set,
  i,j,k for Nat,
  n,m for Nat,
  r for Real,
  e for real-valued FinSequence;

theorem Th34:
  for i for e1,e2 being Element of i-tuples_on REAL for f1,f2
  being Element of i-tuples_on the carrier of F_Real st e1 = f1 & e2 = f2 holds
  mlt(e1,e2) = mlt(f1,f2)
proof
  let i;
  let e1,e2 be Element of i-tuples_on REAL;
  let f1,f2 be Element of i-tuples_on the carrier of F_Real such that
A1: e1 = f1 & e2 = f2;
A2: dom (mlt(e1,e2)) = Seg len (mlt(e1,e2)) by FINSEQ_1:def 3
    .= Seg i by CARD_1:def 7
    .= Seg len mlt(f1,f2) by CARD_1:def 7
    .= dom mlt(f1,f2) by FINSEQ_1:def 3;
  for i be Nat st i in dom(mlt(e1,e2)) holds (mlt(e1,e2)).i = (mlt(f1,f2)) . i
  proof
    let i be Nat such that
A3: i in dom mlt(e1,e2);
    e1.i in REAL & e2.i in REAL by XREAL_0:def 1;
    then reconsider a1 = e1.i, a2 = e2.i as Element of F_Real
         by VECTSP_1:def 5;
    thus (mlt(e1,e2)).i = a1 * a2 by RVSUM_1:59
      .= (mlt(f1,f2)).i by A1,A2,A3,FVSUM_1:60;
  end;
  hence thesis by A2,FINSEQ_1:13;
end;
