
theorem Th37:
  for X being RealNormSpace-Sequence,Y be RealNormSpace
  for f, g being Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y)
  for a be Real holds
  ( ||.f.|| = 0 iff f = 0.R_NormSpace_of_BoundedMultilinearOperators(X,Y) )
  & ||.a*f.|| = |.a.| * ||.f.|| & ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    let X be RealNormSpace-Sequence,Y be RealNormSpace;
    let f, g be Point of R_NormSpace_of_BoundedMultilinearOperators(X,Y);
    let a be Real;
    A2: now
      assume
      A3: f = 0.R_NormSpace_of_BoundedMultilinearOperators(X,Y);
      thus ||.f.|| = 0
      proof
        reconsider g=f as Lipschitzian MultilinearOperator of X,Y by Def9;
        set z = (the carrier of product X) --> 0.Y;
        reconsider z as Function of the carrier of product X,the carrier of Y;
        consider r0 be object such that
        A4: r0 in PreNorms(g) by XBOOLE_0:def 1;
        reconsider r0 as Real by A4;
        A5: (for s be Real st s in PreNorms(g) holds s <= 0)
            implies upper_bound PreNorms(g) <= 0 by SEQ_4:45;
        A6: PreNorms(g) is non empty bounded_above by Th27;
        A7: z = g by A3,Th31;
        A8: now
          let r be Real;
          assume r in PreNorms(g); then
          consider t be VECTOR of product X such that
          A9: r = ||.g.t .|| and
          for i be Element of dom X holds ||.t.i.|| <= 1;
          ||.g.t.|| = ||.0.Y.|| by A7,FUNCOP_1:7
          .= 0;
          hence 0 <= r & r <=0 by A9;
        end; then
        0 <= r0 by A4; then
        upper_bound PreNorms(g) = 0 by A4,A5,A6,A8,SEQ_4:def 1;
        hence thesis by Th30;
      end;
    end;
    A11: ||.f+g.|| <= ||.f.|| + ||.g.||
    proof
      reconsider f1=f, g1=g, h1=f+g as Lipschitzian MultilinearOperator of X,Y
        by Def9;
      A12: ( for s be Real st s in PreNorms(h1) holds s <= ||.f.|| + ||.g.||)
        implies upper_bound PreNorms(h1) <= ||.f.|| + ||.g.|| by SEQ_4:45;
      A13: now
        let t be VECTOR of product X;
        assume
        A14: for i be Element of dom X holds ||.t.i.|| <= 1;
        A15: 0 <= NrProduct t & NrProduct t <= 1 by A14,LM28;
        0 <= ||.g.|| by Th33; then
        A16: ||.g.|| * (NrProduct t) <= ||.g.|| * 1 by A15,XREAL_1:64;
        0 <= ||.f.|| by Th33; then
        ||.f.|| * ( NrProduct t) <= ||.f.|| * 1 by A15,XREAL_1:64; then
        A17: ||.f.|| * NrProduct t + ||.g.|| * NrProduct t
          <= ||.f.|| * 1 + ||.g.|| * 1 by A16,XREAL_1:7;
        A18: ||.f1.t+g1.t.|| <= ||.f1.t.|| + ||.g1.t.|| by NORMSP_1:def 1;
        A19: ||.g1.t.|| <= ||.g.|| * NrProduct t by Th32;
        ||.f1.t.|| <= ||.f.|| * NrProduct t by Th32; then
        ||.f1.t.|| + ||.g1.t.||
          <= ||.f.|| * NrProduct t + ||.g.|| * NrProduct t
          by A19,XREAL_1:7; then
        A20: ||.f1.t.|| + ||.g1.t.|| <= ||.f.|| + ||.g.|| by A17,XXREAL_0:2;
        ||.h1.t.||= ||.f1.t+g1.t.|| by Th35;
        hence ||.h1.t.|| <= ||.f.|| + ||.g.|| by A18,A20,XXREAL_0:2;
      end;
      now
        let r be Real;
        assume r in PreNorms(h1); then
        consider t be VECTOR of product X such that
        A22: r = ||. h1.t .|| and
        A23: for i be Element of dom X holds ||.t.i.|| <= 1;
        thus r <= ||.f.|| + ||.g.|| by A13,A22,A23;
      end;
      hence thesis by A12,Th30;
    end;
    reconsider f1=f, h1=a*f as Lipschitzian MultilinearOperator of X,Y
      by Def9;
    A25: (for s be Real st s in PreNorms(h1) holds s <= |.a.|*||.f.||)
        implies upper_bound PreNorms(h1) <= |.a.|*||.f.|| by SEQ_4:45;
    A24: ||.a*f.|| = |.a.| * ||.f.||
    proof
      A26: now
        A27: 0 <= ||.f.|| by Th33;
        let t be VECTOR of product X;
        assume
        A28: for i be Element of dom X holds ||.t.i.|| <= 1;
        NrProduct t <= 1 by A28,LM28; then
        A29: ||.f.|| * (NrProduct t) <= ||.f.|| * 1 by A27,XREAL_1:64;
        ||.f1.t.|| <= ||.f.|| * NrProduct t by Th32; then
        A30: ||.f1.t.|| <= ||.f.|| by A29,XXREAL_0:2;
        A31: ||.a*f1.t.|| = |.a.| * ||.f1.t.|| by NORMSP_1:def 1;
        A32: 0 <= |.a.| by COMPLEX1:46;
        ||.h1.t.||= ||.a*f1.t.|| by Th36;
        hence ||.h1.t.|| <= |.a.| * ||.f.|| by A30,A31,A32,XREAL_1:64;
      end;
      A33: now
        let r be Real;
        assume r in PreNorms(h1); then
        consider t be VECTOR of product X such that
        A34: r = ||. h1.t .|| and
        A35: for i be Element of dom X holds ||.t.i.|| <= 1;
        thus r <= |.a.|*||.f.|| by A26,A34,A35;
      end;
      A36: now
        per cases;
        case
          A37: a <> 0;
          A38: now
            A39: 0 <= ||.a*f.|| by Th33;
            let t be VECTOR of product X;
            assume for i be Element of dom X holds ||.t.i.|| <= 1; then
            NrProduct t <= 1 by LM28; then
            A41: ||.a*f.|| * (NrProduct t) <= ||.a*f.|| * 1
              by A39,XREAL_1:64;
            ||.h1.t.||<= ||.a*f.|| * NrProduct t by Th32; then
            A42: ||.h1.t.|| <= ||.a*f.|| by A41,XXREAL_0:2;
            h1.t = a*f1.t by Th36; then
            A43: a" * h1.t = (a"* a) * f1.t by RLVECT_1:def 7
            .= 1 * f1.t by A37,XCMPLX_0:def 7
            .= f1.t by RLVECT_1:def 8;
            A44: |.a".| = |.1*a".|
            .= |. 1/a.| by XCMPLX_0:def 9
            .= 1 / |.a.| by ABSVALUE:7
            .= 1 * |.a.|" by XCMPLX_0:def 9
            .= |.a.|";
            A45: 0 <= |.a".| by COMPLEX1:46;
            ||.a" * h1.t.|| = |.a".| * ||.h1.t.|| by NORMSP_1:def 1;
            hence ||.f1.t.|| <= |.a.|" * ||.a * f.||
              by A42,A43,A44,A45,XREAL_1:64;
          end;
          A46: now
            let r be Real;
            assume r in PreNorms(f1); then
            consider t be VECTOR of product X such that
            A47: r = ||. f1.t .|| and
            A48: for i be Element of dom X holds ||.t.i.|| <= 1;
            thus r <= |.a.|" * ||.a*f.|| by A38,A47,A48;
          end;
          A49: ( for s be Real st s in PreNorms(f1) holds
              s <= |.a.|"* ||.a*f.|| )
                implies upper_bound PreNorms(f1) <= |.a.|"*||.a*f.||
                  by SEQ_4:45;
          A50: 0 <= |.a.| by COMPLEX1:46;
          ||.f.|| <= |.a.|" * ||.a*f.|| by A46,A49,Th30; then
          |.a.| * ||.f.|| <= |.a.| * (|.a.|" * ||.a*f.||)
            by A50,XREAL_1:64; then
          A51: |.a.|*||.f.|| <= (|.a.|*|.a.|") * ||.a*f.||;
          |.a.| <> 0 by A37,COMPLEX1:47; then
          |.a.| * ||.f.|| <= 1 * ||.a*f.|| by A51,XCMPLX_0:def 7;
          hence |.a.| * ||.f.|| <= ||.a*f.||;
        end;
        case
          A52: a=0;
          reconsider fz=f as VECTOR of
            R_VectorSpace_of_BoundedMultilinearOperators(X,Y);
          A53: a * f = a * fz
          .= 0.R_VectorSpace_of_BoundedMultilinearOperators(X,Y)
              by A52,RLVECT_1:10
          .= 0.R_NormSpace_of_BoundedMultilinearOperators(X,Y);
          thus |.a.| * ||.f.|| = 0 * ||.f.|| by A52,ABSVALUE:2
          .= ||.a*f.|| by A53,Th34;
        end;
      end;
      ||.a*f.|| <= |.a.| * ||.f.|| by A25,A33,Th30;
      hence thesis by A36,XXREAL_0:1;
    end;
    now
      reconsider g = f as Lipschitzian MultilinearOperator of X,Y by Def9;
      set z = (the carrier of product X) --> 0.Y;
      reconsider z as Function of the carrier of product X,the carrier of Y;
      assume
      A54: ||.f.|| = 0;
      now
        let t be VECTOR of product X;
        ||.g.t.|| <= ||.f.|| *NrProduct t by Th32; then
        ||.g.t.|| = 0 by A54;
        hence g.t = 0.Y by NORMSP_0:def 5
        .= z.t by FUNCOP_1:7;
      end; then
      g = z by FUNCT_2:63;
      hence f = 0.R_NormSpace_of_BoundedMultilinearOperators(X,Y) by Th31;
    end;
    hence thesis by A2,A11,A24;
  end;
