reserve X,Y,Z for non trivial RealBanachSpace;

theorem LM0:
  for X be RealBanachSpace, s be sequence of X st s is norm_summable
  holds ||. Sum s .|| <= Sum ||.s.||
  proof
    let X be RealBanachSpace, s be sequence of X;
    assume
    A1: s is norm_summable; then
    A3: ||.s.|| is summable;
    A5: now let k be Nat;
        ||. Partial_Sums s .|| . k = ||.((Partial_Sums s) . k).||
        by NORMSP_0:def 4;
        hence ||. Partial_Sums s .|| . k <= (Partial_Sums ||.s.||) . k
          by LOPBAN410;
    end;
    A6: s is summable by A1; then
    A8: ||. Partial_Sums s .|| is convergent by NORMSP_1:23;
    lim ||. Partial_Sums s .|| = ||. Sum s .|| by A6,LOPBAN_1:20;
    hence ||. Sum s .|| <= Sum ||.s.|| by A3,A5,A8,SEQ_2:18;
  end;
