
theorem Th37:
  for X,Y,Z be RealNormSpace
  for f, g being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z)
  for a be Real
  holds
  ( ||.f.|| = 0 iff f = 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z) )
  & ||.a*f.|| = |.a.| * ||.f.||
  & ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    let X,Y,Z be RealNormSpace;
    let f, g being Point of R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
    let a be Real;
    A1: now
      assume
      A2: f = 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
      thus ||.f.|| = 0
      proof
        reconsider g=f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
        set z = (the carrier of [:X,Y:]) --> 0.Z;
        reconsider z as Function of the carrier of [:X,Y:], the carrier of Z;
        consider r0 be object such that
        A3: r0 in PreNorms(g) by XBOOLE_0:def 1;
        reconsider r0 as Real by A3;
        A4: (for s be Real st s in PreNorms(g) holds s <= 0)
            implies upper_bound PreNorms(g) <= 0 by SEQ_4:45;
        A6: z=g by A2,Th31;
        A7: now
          let r be Real;
          assume r in PreNorms(g); then
          consider t be VECTOR of X, s be VECTOR of Y such that
          A8: r = ||.g.(t,s).|| and
             ||.t.|| <= 1 & ||.s.|| <= 1;
          [t,s] is Point of [:X,Y:]; then
          g.(t,s) = 0.Z by A6,FUNCOP_1:7;
          hence 0 <= r & r <=0 by A8;
        end;
        then 0<=r0 by A3;
        then upper_bound PreNorms(g) = 0 by A3,A4,A7,SEQ_4:def 1;
        hence thesis by Th30;
      end;
    end;
    A9: ||.f+g.|| <= ||.f.|| + ||.g.||
    proof
      reconsider f1=f, g1=g, h1=f+g
        as Lipschitzian BilinearOperator of X,Y,Z by Def9;
      A10: ( for s be Real st s in PreNorms(h1)
             holds s <= ||.f.|| + ||.g.||)
           implies upper_bound PreNorms(h1) <= ||.f.|| + ||.g.|| by SEQ_4:45;
      A11: now
        let t be VECTOR of X,s be VECTOR of Y such that
        A12: ||.t.|| <= 1 & ||.s.|| <= 1;
        A13: ||.t.|| * ||.s.|| <= 1 * 1 by A12,XREAL_1:66;
        0 <= ||.g.|| by Th33; then
        A14: ||.g.|| * ( ||.t.|| * ||.s.|| ) <= ||.g.|| * 1 by A13,XREAL_1:64;
        0 <= ||.f.|| by Th33; then
        ||.f.|| * ( ||.t.|| * ||.s.|| ) <= ||.f.|| * 1 by A13,XREAL_1:64; then
        A15: ||.f.|| * ||.t.|| * ||.s.|| + ||.g.|| * ||.t.|| * ||.s.||
          <= ||.f.|| * 1 + ||.g.|| * 1 by A14,XREAL_1:7;
        A16: ||.f1.(t,s)+g1.(t,s).|| <= ||.f1.(t,s).|| + ||.g1.(t,s).||
          by NORMSP_1:def 1;
        A17: ||.g1.(t,s).|| <= ||.g.|| * ||.t.|| * ||.s.|| by Th32;
        ||.f1.(t,s).|| <= ||.f.||*||.t.||*||.s.|| by Th32; then
        ||.f1.(t,s).||+||.g1.(t,s).||
          <= ||.f.|| * ||.t.|| * ||.s.|| + ||.g.|| * ||.t.|| * ||.s.||
          by A17,XREAL_1:7; then
        A18: ||.f1.(t,s).|| + ||.g1.(t,s).||
          <= ||.f.|| + ||.g.|| by A15,XXREAL_0:2;
        ||.h1.(t,s).|| = ||.f1.(t,s) + g1.(t,s).|| by Th35;
        hence ||.h1.(t,s).|| <= ||.f.|| + ||.g.|| by A16,A18,XXREAL_0:2;
      end;
      now
        let r be Real;
        assume r in PreNorms(h1); then
        ex t be VECTOR of X,s be VECTOR of Y
        st r = ||.h1.(t,s).|| & ||.t.|| <= 1 & ||.s.|| <= 1;
        hence r <= ||.f.|| + ||.g.|| by A11;
      end;
      hence thesis by A10,Th30;
    end;
    A20: ||.a*f.|| = |.a.| * ||.f.||
    proof
      reconsider f1=f, h1=a*f
        as Lipschitzian BilinearOperator of X,Y,Z by Def9;
      A21: (for s be Real st s in PreNorms(h1) holds s <= |.a.| * ||.f.|| )
           implies upper_bound PreNorms(h1) <= |.a.| * ||.f.||
           by SEQ_4:45;
      A22: now
        A23: 0 <= ||.f.|| by Th33;
        let t be VECTOR of X,s be VECTOR of Y;
        assume ||.t.|| <= 1 & ||.s.|| <= 1; then
        ||.t.|| * ||.s.|| <= 1 * 1 by XREAL_1:66; then
        A24: ||.f.|| * (||.t.|| * ||.s.|| ) <= ||.f.|| * 1 by A23,XREAL_1:64;
        ||.f1.(t,s).||<= ||.f.|| * ||.t.|| * ||.s.|| by Th32; then
        A25: ||.f1.(t,s).|| <= ||.f.|| by A24,XXREAL_0:2;
        A26: ||.a * f1.(t,s).|| =|.a.| * ||.f1.(t,s).|| by NORMSP_1:def 1;
        A27: 0 <= |.a.| by COMPLEX1:46;
        ||.h1.(t,s).|| = ||.a * f1.(t,s).|| by Th36;
        hence ||.h1.(t,s).|| <= |.a.| * ||.f.|| by A25,A26,A27,XREAL_1:64;
      end;
      A28: now
        let r be Real;
        assume r in PreNorms(h1); then
        ex t be VECTOR of X, s be VECTOR of Y
        st r = ||.h1.(t,s).||
         & ||.t.|| <= 1
         & ||.s.|| <= 1;
        hence r <= |.a.|*||.f.|| by A22;
      end;
      A29: now
        per cases;
        case
          A30: a <> 0;
          A31: now
            A32: 0 <= ||.a*f.|| by Th33;
            let t be VECTOR of X,s be VECTOR of Y;
            assume ||.t.|| <= 1 & ||.s.|| <= 1; then
            ||.t.|| * ||.s.|| <= 1 * 1 by XREAL_1:66; then
            A33: ||.a * f.|| * ( ||.t.|| * ||.s.|| )
              <= ||.a * f.|| * 1 by A32,XREAL_1:64;
            ||.h1.(t,s).||<= ||.a*f.||*||.t.||*||.s.|| by Th32; then
            A34: ||.h1.(t,s).|| <= ||.a*f.|| by A33,XXREAL_0:2;
            h1.(t,s) = a * f1.(t,s) by Th36; then
            A35: a"*h1.(t,s) = ( a"* a) * f1.(t,s) by RLVECT_1:def 7
             .= 1 * f1.(t,s) by A30,XCMPLX_0:def 7
             .= f1.(t,s) by RLVECT_1:def 8;
            A36: |.a".| = |.1 * a".|
             .= |. 1 / a .| by XCMPLX_0:def 9
             .= 1 / |.a.| by ABSVALUE:7
             .= 1 * |.a.|" by XCMPLX_0:def 9
             .= |.a.|";
            A37: 0 <= |.a".| by COMPLEX1:46;
            ||.a" * h1.(t,s).||
              = |.a".| * ||.h1.(t,s).|| by NORMSP_1:def 1;
            hence ||.f1.(t,s).||
              <= |.a.|"*||.a*f.|| by A34,A35,A36,A37,XREAL_1:64;
          end;
          A38: now
            let r be Real;
            assume r in PreNorms(f1); then
            ex t be VECTOR of X, s be VECTOR of Y
            st r = ||.f1.(t,s).||
             & ||.t.|| <= 1 & ||.s.|| <= 1;
            hence r <= |.a.|"*||.a*f.|| by A31;
          end;
          A39: ( 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;
          A40: 0 <= |.a.| by COMPLEX1:46;
          ||.f.|| <=|.a.|" * ||.a*f.|| by A38,A39,Th30; then
          |.a.| * ||.f.|| <= |.a.| * (|.a.|" * ||.a*f.||)
            by A40,XREAL_1:64; then
          A41: |.a.| * ||.f.|| <= (|.a.| * |.a.|") * ||.a*f.||;
          |.a.| <> 0 by A30,COMPLEX1:47; then
          |.a.| * ||.f.|| <= 1 * ||.a*f.|| by A41,XCMPLX_0:def 7;
          hence |.a.|* ||.f.|| <= ||.a*f.||;
        end;
        case
          A42: a=0;
          reconsider fz=f
            as VECTOR of R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z);
          A43: a*f = a*fz
           .= 0.R_VectorSpace_of_BoundedBilinearOperators(X,Y,Z)
               by A42,RLVECT_1:10
           .= 0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z);
          thus |.a.|* ||.f.|| = 0 * ||.f.|| by A42,ABSVALUE:2
           .= ||.a*f.|| by A43,Th34;
        end;
      end;
      ||.a*f.|| <= |.a.| * ||.f.|| by A21,A28,Th30;
      hence thesis by A29,XXREAL_0:1;
    end;
    now
      reconsider g=f as Lipschitzian BilinearOperator of X,Y,Z by Def9;
      set z = (the carrier of [:X,Y:] ) --> 0.Z;
      reconsider z as Function of the carrier of [:X,Y:],the carrier of Z;
      assume
      A44: ||.f.|| = 0;
      now
        let t be VECTOR of X,s be VECTOR of Y;
        A45: [t,s] is Point of [:X,Y:];
        ||.g.(t,s).|| <= ||.f.|| *||.t.||*||.s.|| by Th32; then
        ||.g.(t,s).|| = 0 by A44;
        hence g.(t,s) = 0.Z by NORMSP_0:def 5
         .= z.(t,s) by A45,FUNCOP_1:7;
      end; then
      g=z by BINOP_1:2;
      hence f=0.R_NormSpace_of_BoundedBilinearOperators(X,Y,Z) by Th31;
    end;
    hence thesis by A1,A9,A20;
  end;
