
theorem
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for g be MultilinearOperator of X,Y
  holds g is Lipschitzian iff PreNorms(g) is bounded_above
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let g be MultilinearOperator of X,Y;
    now
      reconsider K = upper_bound PreNorms(g) as Real;
      assume
      A1: PreNorms(g) is bounded_above;
      A2: now
        let t be VECTOR of product X;
        consider F be FinSequence of REAL such that
        A3: dom F = dom X
           & ( for i be Element of dom X holds F.i = ||.t.i.|| )
           & NrProduct t = Product F by DefNrPro;
        now
          per cases;
          suppose
            ex i be Element of dom X st t.i = 0.(X.i); then
            consider i be Element of dom X such that
            A6: t.i = 0.(X.i);
            A7: F.i = ||.t.i.|| by A3;
            F.i = 0 by A6,A7; then
            A10: Product F = 0 by A3,RVSUM_1:103;
            g.t = 0.Y by A6,LM32;
            hence ||.g.t.|| <= K * NrProduct t by A3,A10;
          end;
          suppose
            A11: not ex i be Element of dom X st t.i = 0.(X.i);
            A12: for i be Element of dom F holds F.i > 0
            proof
              let i be Element of dom F;
              reconsider j = i as Element of dom X by A3;
              A13: F.j = ||. t.j .|| by A3;
              t.j <> 0.(X.j) by A11; then
              F.j <> 0 by A13,NORMSP_0:def 5;
              hence thesis by A13;
            end;
            A15: 0 < Product F by A12,LM31;
            consider d be FinSequence of REAL such that
            A16: dom d = dom X
                & for i be Element of dom X holds d.i= ||.t.i.||" by LM34;
            consider t1 be Element of product X such that
            A17: for i be Element of dom X holds t1.i= ( d/.i) * t.i
                by LM33;
            A18: (Product d) * g.t = g.t1 by A16,A17,LM35;
            A22: for i be Element of dom F holds d.i = (F.i)"
            proof
              let i be Element of dom F;
              reconsider j = i as Element of dom X by A3;
              d.j = ||.t.j.||" by A16;
              hence thesis by A3;
            end;
       A23: |. (Product d) .| * ||.g.t.|| = ||.g.t1.|| by A18,NORMSP_1:def 1;
            |. (Product F)" .| = |. 1*(Product F)" .|
            .= |. 1/(Product F) .| by XCMPLX_0:def 9
            .= 1 / (Product F) by A15,ABSVALUE:def 1; then
            A25: |. (Product d) .| * ||.g.t.||
             = (1/(Product F)) * ||.g.t.|| by A3,A16,A22,LM36
            .= ||.g.t.||/ Product F by XCMPLX_1:99;
            A26: for i be Element of dom X holds ||.t1.i.|| <= 1
            proof
              let i be Element of dom X;
              A27: d.i = ||.t.i.||" by A16;
              A28: t1.i = (d/.i) * t.i by A17;
              t.i <> 0.(X.i) by A11; then
              A29: ||.t.i.|| <> 0 by NORMSP_0:def 5;
              ||.t1.i.|| = |.d/.i.| * ||.t.i.|| by A28,NORMSP_1:def 1
              .= |. ||.t.i.||" .| * ||.t.i.|| by A16,A27,PARTFUN1:def 6
              .= ||.t.i.||" * ||.t.i.|| by ABSVALUE:def 1
              .= 1 by A29,XCMPLX_0:def 7;
              hence thesis;
            end;
            ||.g.t1.|| in {||.g.t .|| where t is VECTOR of product X :
              for i be Element of dom X holds ||.t.i.|| <= 1 } by A26; then
            ||.g.t.|| / Product F <= K by A1,A23,A25,SEQ_4:def 1; then
            ||.g.t.|| / Product F * Product F <= K * Product F
              by A15,XREAL_1:64;
            hence ||.g.t.|| <= K * NrProduct t by A3,A15,XCMPLX_1:87;
          end;
        end;
        hence ||.g.t .|| <= K *NrProduct t;
      end;
      take K;
      0 <= K
      proof
        consider r0 be object such that
        A30: r0 in PreNorms(g) by XBOOLE_0:def 1;
        reconsider r0 as Real by A30;
        now
          let r be Real;
          assume r in PreNorms(g); then
          ex t be VECTOR of product X st r = ||. g.t .||
           & for i be Element of dom X holds ||.t.i.|| <= 1;
          hence 0 <= r;
        end; then
        0 <= r0 by A30;
        hence thesis by A1,A30,SEQ_4:def 1;
      end;
      hence g is Lipschitzian by A2;
    end;
    hence thesis by Th27;
  end;
