
theorem Th27:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for g be Lipschitzian MultilinearOperator of X,Y
  holds PreNorms(g) is bounded_above
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let g be Lipschitzian MultilinearOperator of X,Y;
    consider K be Real such that
    A1: 0 <= K and
    A2: for x being Point of product X
        holds ||. g.x .|| <= K * NrProduct x by Def8;
    take K;
    let r be ExtReal;
    assume r in PreNorms(g); then
    consider t be VECTOR of product X such that
    A3: r = ||. g.t .|| and
    A4: for i be Element of dom X holds ||. t.i .|| <= 1;
    A5: ||.g.t.|| <= K*NrProduct t by A2;
    0 <= NrProduct t & NrProduct t <= 1 by A4,LM28; then
    K * NrProduct t <= K * 1 by A1,XREAL_1:64;
    hence r <=K by A3,A5,XXREAL_0:2;
  end;
