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
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 & for x be VECTOR of X, v be VECTOR of V
                  st x=v holds f.x <= ||.v.|| )
holds
  ex g be Lipschitzian linear-Functional of V, G be Point of DualSp V
   st g = G & g|(the carrier of X) = f
    & for x be VECTOR of V holds g.x <= ||.x.|| & ||.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
A11: f=F and
A12: for x be VECTOR of X, v be VECTOR of V st x=v
  holds f.x <= ||.v.||;
  consider g be Lipschitzian linear-Functional of V,
    G be Point of DualSp V such that
A2: g=G &
    g|the carrier of X=f & ||.G.||=||.F.|| by A11,Th44;
  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;
  now let r be Real;
    assume r in PreNorms f; then
    consider t be VECTOR of X such that
C1: r = |.f.t.| & ||.t.|| <= 1;
    reconsider td=t as VECTOR of V by B1;
C7: ||.t.|| = ((the normF of V) | (the carrier of X)).td by DefSubSP
           .= ||.td.|| by FUNCT_1:49;
C5: -(f.t) = (-1)* (f.t)
             .= f.((-1)*t) by HAHNBAN:def 3;
    reconsider t0=t as VECTOR of X0;
D6: X0 is Subspace of V by B1,RLSUB_1:def 2;
    (-1)*td = -td & (-1)*t = -t by RLVECT_1:16; then
    (-1)*td = (-1)*t by RLSUB_1:15,D6; then
    f.((-1)*t) <= ||.(-1)*td.|| by A12; then
    f.((-1)*t) <= ||.-td.|| by RLVECT_1:16; then
    -(f.t) <= ||.td.|| by C5,NORMSP_1:2; then
    - (||.td.||) <= f.t by XREAL_1:26; then
    |.f.t.| <= ||.td.|| by A12,ABSVALUE:5;
    hence r <= 1 by C1,C7,XXREAL_0:2;
  end; then
  upper_bound PreNorms f <= 1 by SEQ_4:45; then
A3: ||.G.|| <= 1 by A2,A11,Th30;
  for x be VECTOR of V holds g.x <= ||.x.||
  proof
    let x be VECTOR of V;
C1: g.x <= |.g.x.| by ABSVALUE:4;
    |.g.x.| <= ||.G.|| * ||.x.|| by A2,Th32; then
C2: g.x <= ||.G.|| * ||.x.|| by C1,XXREAL_0:2;
    ||.G.|| * ||.x.|| <= 1 * ||.x.|| by A3,XREAL_1:64;
    hence thesis by C2,XXREAL_0:2;
  end;
  hence thesis by A2;
end;
