
theorem Th32:
  for X being RealNormSpace-Sequence,
      Y be RealNormSpace
  for f being Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
  for g be Lipschitzian MultilinearOperator of X,Y st g = f
  holds
    for t be VECTOR of product X holds ||. g.t .|| <= ||.f.|| * NrProduct t
  proof
    let X be RealNormSpace-Sequence,
        Y be RealNormSpace;
    let f being Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y);
    let g be Lipschitzian MultilinearOperator of X,Y such that
    A1: g = f;
    A2: PreNorms(g) is non empty bounded_above by Th27;
    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
          A7: t.i = 0.(X.i);
          F.i = ||.0.(X.i).|| by A3,A7
          .= 0; then
          A9: Product F = 0 by A3,RVSUM_1:103;
          g.t = 0.Y by A7,LM32;
          hence ||.g.t.|| <= ||.f.|| * NrProduct t by A3,A9;
        end;
        suppose
          A10: for i be Element of dom X holds t.i <> 0.(X.i);
          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;
            A12: F.j = ||.t.j.|| by A3;
            t.j <> 0.(X.j) by A10; then
            F.j <> 0 by A12,NORMSP_0:def 5;
            hence thesis by A12;
          end; then
          A13: 0 < Product F by LM31;
          consider d be FinSequence of REAL such that
          A14: 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
          A15: for i be Element of dom X holds t1.i= ( d/.i) * t.i
            by LM33;
          A16: for i be Element of dom X holds ||.t1.i.|| <= 1
          proof
            let i be Element of dom X;
            A17: d.i= ||.t.i.||" by A14;
            A18:t1.i= ( d/.i) * t.i by A15;
            t.i <> 0.(X.i) by A10; then
            A19: ||.t.i.|| <> 0 by NORMSP_0:def 5;
            ||.t1.i.|| = |.d/.i.| * ||.t.i.|| by A18,NORMSP_1:def 1
            .= |. ||.t.i.||" .| * ||.t.i.|| by A14,A17,PARTFUN1:def 6
            .= ||.t.i.||" * ||.t.i.|| by ABSVALUE:def 1
            .= 1 by A19,XCMPLX_0:def 7;
            hence thesis;
          end;
          A20: (Product d)* g.t = g.t1 by A14,A15,LM35;
          A23: 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.i = ||.t.j.||" by A14;
            hence thesis by A3;
          end;
          A24: |. (Product d) .| * ||.g.t.|| = ||.g.t1.||
            by A20,NORMSP_1:def 1;
          A25: |. (Product F)".| = |. 1*(Product F)".|
          .= |. 1/(Product F) .| by XCMPLX_0:def 9
          .= 1 / (Product F) by A13,ABSVALUE:def 1;
          A26: |. (Product d) .| * ||.g.t.||
           = (1/(Product F)) * ||.g.t.|| by A3,A14,A23,A25,LM36
          .= ||.g.t.||/ Product F by XCMPLX_1:99;
          ||.g.t1.|| in {||.g.t .|| where t is VECTOR of product X :
            for i be Element of dom X holds ||.t.i.|| <= 1 } by A16; then
          ||.g.t.||/ Product F <= upper_bound PreNorms(g)
            by A2,A24,A26,SEQ_4:def 1; then
          A28: ||.g.t.||/( Product F) <= ||.f.|| by A1,Th30;
          ||.g.t.|| / (Product F) * (Product F) =||.g.t.|| by A13,XCMPLX_1:87;
          hence ||.g.t.|| <= ||.f.||*NrProduct t by A3,A28,XREAL_1:64;
        end;
      end;
      hence ||.g.t.|| <= ||.f.||*NrProduct t;
    end;
    hence thesis;
  end;
