reserve X for RealNormSpace;

theorem
  for X be RealNormSpace,
      R1, R2, R3 be FinSequence of X
        st len R1 = len R2 & R3 = R1 - R2 holds Sum(R3) = Sum(R1) - Sum(R2)
proof
  let X be RealNormSpace,
      R1, R2, R3 be FinSequence of X;
  assume A1: len R1 = len R2 & R3 = R1 - R2; then
A2: dom R1 = dom R2 by FINSEQ_3:29;
A3: dom R3 = dom R1 /\ dom R2 by A1,VFUNCT_1:def 2
          .= dom R1 by A2; then
A4: len R3 = len R1 by FINSEQ_3:29;
A5: for k be Nat st k in dom R1 holds R3.k = R1/.k - R2/.k
  proof
    let k be Nat;
    assume A6: k in dom R1;
    thus R3.k = R3/.k by A6,A3,PARTFUN1:def 6
             .= R1/.k - R2/.k by A1,A6,A3,VFUNCT_1:def 2;
  end;
  thus thesis by A1,A4,A5,RLVECT_2:5;
end;
