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 Th44:
for V be RealNormSpace, X be SubRealNormSpace of V,
    f be Lipschitzian linear-Functional of X,
    F be Point of DualSp X
 st f = F holds
  ex g be Lipschitzian linear-Functional of V, G be Point of DualSp V
    st g = G & g|(the carrier of X) = f & ||.G.||=||.F.||
proof
  let V be RealNormSpace, X be SubRealNormSpace of V,
  f be Lipschitzian linear-Functional of X,
  F be Point of DualSp X such that
A1:f=F;
  reconsider X0 = X as RealLinearSpace;
B1: the carrier of X0 c= the carrier of V
  & 0.X0 = 0.V
  & the addF of X0 = (the addF of V)||the carrier of X0
  & the Mult of X0 = (the Mult of V) | [:REAL, the carrier of X0:]
   by DefSubSP; then
  reconsider X0 as Subspace of V by RLSUB_1:def 2;
  reconsider fi0=f as linear-Functional of X0;
  deffunc F(Element of the carrier of V) = ||.F.|| * ||.$1.||;
D0:for v be Element of the carrier of V
   holds F(v) in REAL by XREAL_0:def 1;
  consider q be Function of the carrier of V,REAL such that
D1: for v be Element of the carrier of V holds q.v = F(v)
    from FUNCT_2:sch 8(D0);
  q is Banach-Functional of V
  proof
E0: q is subadditive
    proof
      let x,y be VECTOR of V;
E2:   q.x = ||.F.|| * ||.x.|| & q.y = ||.F.|| * ||.y.|| by D1;
      ||.F.|| * ||.x+y.|| <= ||.F.|| * (||.x.|| + ||.y.||)
                                          by XREAL_1:64,NORMSP_1:def 1;
      hence thesis by D1,E2;
    end;
    q is absolutely_homogeneous
    proof
      let x be VECTOR of V, r be Real;
E5:   ||.r*x.|| = |.r.| * ||.x.|| by NORMSP_1:def 1;
      q.(r*x) = ||.F.|| * ||.r*x.|| by D1
             .= |.r.| * (||.F.|| * ||.x.||) by E5;
      hence thesis by D1;
    end;
    hence thesis by E0;
  end; then
  reconsider q as Banach-Functional of V;
  for x be VECTOR of X0, v be VECTOR of V st x=v holds fi0.x <= q.v
  proof
    let x0 be VECTOR of X0, v be VECTOR of V;
    assume D21: x0=v;
    reconsider x=x0 as VECTOR of X;
D22: fi0.x0 <= |. fi0.x0 .| by ABSVALUE:4;
D23:|. fi0.x .| <= ||.F.|| * ||.x.|| by A1,Th32;
    ||.x.|| = ((the normF of V) | (the carrier of X)).v by D21,DefSubSP
           .= ||.v.|| by FUNCT_1:49,D21;
    then
    |. fi0.x0 .| <= q.v by D1,D23;
    hence thesis by D22,XXREAL_0:2;
  end; then
  consider g be linear-Functional of V such that
A3: g|the carrier of X0=fi0 &
   for x be VECTOR of V holds g.x <= q.x by HAHNBAN:22;
B4: for x be VECTOR of V holds |. g.x .| <= ||.F.|| * ||.x.||
  proof
    let x be VECTOR of V;
    g.x <= q.x by A3; then
A31: g.x <= ||.F.|| * ||.x.|| by D1;
A32: -(g.x) = (-1)* (g.x)
             .= g.((-1)*x) by HAHNBAN:def 3;
    g.((-1)*x) <= q.((-1)*x) by A3;
    then g.((-1)*x) <= ||.F.|| * ||.(-1)*x.|| by D1;
    then g.((-1)*x) <= ||.F.|| * ||.-x.|| by RLVECT_1:16;
    then -(g.x) <= ||.F.|| * ||.x.|| by A32,NORMSP_1:2;
    then - (||.F.|| * ||.x.||) <= g.x by XREAL_1:26;
    hence thesis by ABSVALUE:5,A31;
  end;
  then
  reconsider g as Lipschitzian linear-Functional of V by Def8;
  reconsider G=g as Point of DualSp V by Def9;
  now let r be Real;
    assume r in PreNorms g; then
    consider t be VECTOR of V such that
C1: r = |.g.t.| and
C2: ||.t.|| <= 1;
C3: |.g.t.| <= ||.F.|| * ||.t.|| by B4;
    ||.F.|| * ||.t.|| <= ||.F.|| * 1 by C2,XREAL_1:64;
    hence r <= ||.F.|| by C1,C3,XXREAL_0:2;
  end; then
  upper_bound PreNorms g
             <= (BoundedLinearFunctionalsNorm X).f by A1,SEQ_4:45;
  then
A41: (BoundedLinearFunctionalsNorm V).g
          <= (BoundedLinearFunctionalsNorm X).f by Th30;
  now let r be object;
    assume r in PreNorms f; then
    consider t be VECTOR of X such that
C1: r = |.f.t.| and
C2: ||.t.|| <= 1;
    reconsider td=t as VECTOR of V by B1;
    ||.t.|| = ((the normF of V) | (the carrier of X)).td by DefSubSP
           .= ||.td.|| by FUNCT_1:49;
    then
    r = |.g.td.| & ||.td.|| <= 1 by C1,A3,FUNCT_1:49,C2;
    hence r in PreNorms g;
  end; then
A42: PreNorms f c= PreNorms g;
  upper_bound PreNorms f
             <= upper_bound PreNorms g by A42,SEQ_4:48; then
  (BoundedLinearFunctionalsNorm X).f
          <= upper_bound PreNorms g by Th30; then
B4: (BoundedLinearFunctionalsNorm X).f
          <= (BoundedLinearFunctionalsNorm V).g by Th30;
  take g,G;
  thus thesis by A3,A1,A41,XXREAL_0:1,B4;
end;
