reserve V for non empty RealLinearSpace;
reserve S for Real_Sequence;
reserve k,n,m,m1 for Nat;
reserve g,h,r,x for Real;

theorem Th37:
  for X be RealNormSpace, f,g be Point of DualSp X, a be Real holds
  ( ||.f.|| = 0 iff f = 0.(DualSp X) ) &
  ||.a*f.|| = |.a.| * ||.f.|| & ||.f+g.|| <= ||.f.|| + ||.g.||
proof
  let X be RealNormSpace;
  let f,g be Point of DualSp X;
  let a be Real;
A1: now
    assume
A2: f = 0.(DualSp X);
    reconsider g=f as Lipschitzian linear-Functional of X by Def9;
    set z = (the carrier of X) --> 0;
    reconsider z as Function of the carrier of X,REAL
      by FUNCOP_1:45,XREAL_0:def 1;
    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 such that
A8:    r=|.g.t.| and ||.t.|| <= 1;
     thus 0 <= r & r <=0 by A8,A6,COMPLEX1:44;
    end;
    then 0<=r0 by A3;
    then upper_bound PreNorms g = 0 by A7,A3,A4,SEQ_4:def 1;
    hence ||.f.|| = 0 by Th30;
  end;
A9: ||.f+g.|| <= ||.f.|| + ||.g.||
  proof
    reconsider f1=f, g1=g, h1=f+g as Lipschitzian linear-Functional of X
    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 such that
A12:  ||.t.|| <= 1;
      0 <= ||.f.|| & 0 <= ||.g.|| by Th33;
      then ||.f.||*||.t.|| <= ||.f.||*1
         & ||.g.||*||.t.|| <= ||.g.||*1 by A12,XREAL_1:64;
      then
A14:  ||.f.||*||.t.|| + ||.g.||*||.t.|| <= ||.f.||*1 + ||.g.||*1 by XREAL_1:7;
A15:  |.f1.t+g1.t.| <=|.f1.t.|+|.g1.t.| by COMPLEX1:56;
      |.g1.t.|<= ||.g.||*||.t.|| & |.f1.t.|<= ||.f.||*||.t.|| by Th32;
      then |.f1.t.|+|.g1.t.| <= ||.f.||*||.t.|| + ||.g.||*||.t.||
           by XREAL_1:7;
      then
A17:  |.f1.t.|+|.g1.t.| <= ||.f.|| + ||.g.|| by A14,XXREAL_0:2;
      |.h1.t.|= |.f1.t+g1.t.| by Th35;
      hence |.h1.t.| <= ||.f.|| + ||.g.|| by A15,A17,XXREAL_0:2;
    end;
    now
      let r be Real;
     assume r in PreNorms h1;
      then ex t be VECTOR of X st r=|.h1.t.| & ||.t.|| <= 1;
     hence r <= ||.f.|| + ||.g.|| by A11;
    end;
    hence thesis by A10,Th30;
  end;
A19: ||.a*f.|| = |.a.| * ||.f.||
  proof
    reconsider f1=f, h1=a*f as Lipschitzian linear-Functional of X by Def9;
A21: now
A22:  0 <= ||.f.|| by Th33;
      let t be VECTOR of X;
      assume ||.t.|| <= 1; then
A23:  ||.f.||*||.t.|| <= ||.f.||*1 by A22,XREAL_1:64;
      |.f1.t.|<= ||.f.||*||.t.|| by Th32; then
A24:  |.f1.t.| <= ||.f.|| by A23,XXREAL_0:2;
A25:  |.a*f1.t.| =|.a.|*|.f1.t.| by COMPLEX1:65;
A26:  0<= |.a.| by COMPLEX1:46;
      |.h1.t.|= |.a*f1.t.| by Th36;
      hence |.h1.t.| <= |.a.|*||.f.|| by A25,A24,A26,XREAL_1:64;
    end;
A27: now
      let r be Real;
      assume r in PreNorms h1;
      then ex t be VECTOR of X st r=|.h1.t.| & ||.t.|| <= 1;
      hence r <= |.a.|*||.f.|| by A21;
    end;
A28: now
      per cases;
      case
A29:    a <> 0;
A30:    now
A31:      0 <= ||.a*f.|| by Th33;
          let t be VECTOR of X;
          assume ||.t.|| <= 1; then
A32:      ||.a*f.||*||.t.|| <= ||.a*f.||*1 by A31,XREAL_1:64;
          |.h1.t.|<= ||.a*f.||*||.t.|| by Th32; then
A33:      |.h1.t.| <= ||.a*f.|| by A32,XXREAL_0:2;
          h1.t=a*f1.t by Th36; then
A34:      a"*h1.t = (a"* a)*f1.t
                 .= 1*f1.t by A29,XCMPLX_0:def 7
                 .= f1.t;
A35:      |.a".| =|.1*a".| .=|. 1/a.| by XCMPLX_0:def 9
            .=1/|.a.| by ABSVALUE:7
            .=1*|.a.|" by XCMPLX_0:def 9
            .=|.a.|";
A36:      0<= |.a".| by COMPLEX1:46;
          |.a"*h1.t.| =|.a".|*|.h1.t.| by COMPLEX1:65;
          hence |.f1.t.| <= |.a.|"*||.a*f.|| by A34,A33,A36,A35,XREAL_1:64;
        end;
A37:    now
          let r be Real;
          assume r in PreNorms f1;
          then ex t be VECTOR of X st r=|.f1.t.| & ||.t.|| <= 1;
          hence r <= |.a.|"*||.a*f.|| by A30;
        end;
A38:    ( 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;
A39:    0 <= |.a.| by COMPLEX1:46;
        ||.f.|| <=|.a.|"*||.a*f.|| by A37,A38,Th30;
        then |.a.|*||.f.|| <=|.a.|*(|.a.|"*||.a*f.||) by A39,XREAL_1:64;
        then
A40:    |.a.|*||.f.|| <=(|.a.|*|.a.|")*||.a*f.||;
        |.a.| <>0 by A29,COMPLEX1:47;
        then |.a.|*||.f.|| <=1*||.a*f.|| by A40,XCMPLX_0:def 7;
        hence |.a.|* ||.f.|| <=||.a*f.||;
      end;
      case
A41:    a=0;
        reconsider fz=f as
        VECTOR of R_VectorSpace_of_BoundedLinearFunctionals X;
A42:    a*f = a*fz
          .= 0.R_VectorSpace_of_BoundedLinearFunctionals X by A41,RLVECT_1:10
          .= 0.(DualSp X);
        thus |.a.|* ||.f.|| =0 * ||.f.|| by A41,ABSVALUE:2
          .=||.a*f.|| by A42,Th34;
      end;
    end;
    (BoundedLinearFunctionalsNorm X).(a*f) = upper_bound PreNorms h1 by Th30;
    then ||.a*f.|| <= |.a.|*||.f.|| by A27,SEQ_4:45;
    hence thesis by A28,XXREAL_0:1;
  end;
  now
    reconsider g=f as Lipschitzian linear-Functional of X by Def9;
    set z = (the carrier of X) --> 0;
    reconsider z as Function of the carrier of X,REAL
      by FUNCOP_1:45,XREAL_0:def 1;
    assume
A43: ||.f.|| = 0;
    now
      let t be VECTOR of X;
      |.g.t.| <= ||.f.|| *||.t.|| by Th32;
      then |.g.t.| = 0 by A43,COMPLEX1:46;
      hence g.t = 0 by COMPLEX1:45
               .= z.t;
    end;
    then g=z;
    hence f=0.(DualSp X) by Th31;
  end;
  hence thesis by A1,A19,A9;
end;
