reserve X,G for RealNormSpace-Sequence,
          Y for RealNormSpace;
reserve f for MultilinearOperator of X,Y;

theorem NDIFF823:
  for n be Nat, F be Element of REAL n, s be Real
  st for i be Nat st i in dom F holds 0 <= F.i & F.i <= s
  holds |.F.| <= sqrt((s*s) * len F)
  proof
    let n be Nat,
        F be Element of REAL n,
        s be Real;
    assume
    A1: for i be Nat st i in dom F holds 0 <= F.i & F.i <= s;
    A2: 0 <= Sum sqr F by RVSUM_1:86;
    set G = len F |-> s;
    A3: sqr G = len F |-> s^2 by RVSUM_1:56;
    len F is natural Number & s is Element of REAL by XREAL_0:def 1; then
    reconsider G as Element of len F -tuples_on REAL by FINSEQ_2:112;
    reconsider F0 = F as Element of len F -tuples_on REAL by FINSEQ_2:92;
    for j being Nat st j in Seg len F0 holds
    (sqr F0).j <= (sqr G).j
    proof
      let j be Nat;
      assume
      A4: j in Seg len F0; then
      A5: j in dom F by FINSEQ_1:def 3;
      A6: (sqr F0).j = (F.j)^2 by VALUED_1:11;
      A7: (sqr G).j = (G.j)^2 by VALUED_1:11;
      A8: 0 <= F0.j by A1,A5;
      F0.j <= s by A1,A5; then
      F0.j <= G.j by A4,FINSEQ_2:57;
      hence thesis by A6,A7,A8,SQUARE_1:15;
    end; then
    Sum(sqr F0) <= Sum(sqr G) by RVSUM_1:82; then
    Sum sqr F <= (s*s) * len F by A3,RVSUM_1:80;
    hence thesis by A2,SQUARE_1:26;
  end;
