
theorem Th3:
  for X be RealLinearSpace,
      R1, R2 be FinSequence of X,
      a be Element of REAL st R2 = a(#)R1 holds Sum(R2) = a * Sum(R1)
proof
  let X be RealLinearSpace,
      R1, R2 be FinSequence of X,
      a be Element of REAL;
  assume A1: R2 = a(#)R1;
  dom R2 = dom R1 by A1,VFUNCT_1:def 4; then
A2: len R2 = len R1 by FINSEQ_3:29;
  for k be Nat st k in dom R1 holds R2.k = a * R1/.k
  proof
    let k be Nat;
    assume k in dom R1; then
A3: k in dom R2 by A1,VFUNCT_1:def 4;
    hence R2.k = R2/.k by PARTFUN1:def 6
             .= a * R1/.k by A3,A1,VFUNCT_1:def 4;
  end;
  hence thesis by A2,RLVECT_2:3;
end;
