
theorem
for X be RealNormSpace, L be non empty Subset of X st
  X is non trivial &
  ( for f be Point of DualSp X holds
      ex Y1 be Subset of REAL st
        Y1 = {|. f.x .| where x is Point of X : x in L}
      & sup Y1 < +infty ) holds
  ex Y be Subset of REAL st
    Y = {||.x.|| where x is Point of X : x in L}
  & sup Y < +infty
proof
   let X be RealNormSpace, L be non empty Subset of X;
   assume that
A1: X is non trivial and
A2: for f be Point of DualSp X holds
      ex Y1 be Subset of REAL st
        Y1 = {|. f.x .| where x is Point of X : x in L}
      & sup Y1 < +infty;
A3:for f be Point of DualSp X holds
     ex Kf be Real st
       0 <= Kf
     & for x be Point of X st x in L holds |. f.x .| <= Kf
   proof
    let f be Point of DualSp X;
    reconsider f1=f as
      Lipschitzian linear-Functional of X by DUALSP01:def 10;
    consider Y1 be Subset of REAL such that
B1:   Y1 = {|. f.x .| where x is Point of X : x in L}
      & sup Y1 < +infty by A2;
    reconsider r0=0 as Real;
    for r be ExtReal st r in Y1 holds r0 <= r
    proof
     let r be ExtReal;
     assume r in Y1; then
     ex x be Point of X st r = |. f.x .| & x in L by B1;
     hence r0 <= r by COMPLEX1:46;
    end; then
U5: r0 is LowerBound of Y1 by XXREAL_2:def 2; then
U3: r0 <= inf Y1 by XXREAL_2:def 4;
    consider x0 be object such that
B11:  x0 in L by XBOOLE_0:def 1;
    reconsider x0 as Point of X by B11;
    set y1=|. f.x0 .|;
    y1 in Y1 by B1,B11; then
U6: inf Y1 <= y1 & y1 <= sup Y1 by XXREAL_2:3,4; then
B2: sup Y1 in REAL by B1,U3,XXREAL_0:14;
    reconsider Kf=sup Y1 as Real by B2;
BX: for x be Point of X st x in L holds |. f.x .| <= Kf
    proof
     let x be Point of X;
     assume C0: x in L;
     reconsider r=|. f.x .| as Real;
     r in Y1 by C0,B1;
     hence |. f.x .| <= Kf by XXREAL_2:4;
    end;
    take Kf;
    thus thesis by U5,U6,BX,XXREAL_2:def 4;
   end;
   consider M be Real such that
D1: 0 <= M &
    for x be Point of X st x in L holds ||.x.|| <= M by A1,A3,Th75;
   set f = 0.(DualSp X);
   consider x0 be object such that
B11: x0 in L by XBOOLE_0:def 1;
   reconsider x0 as Point of X by B11;
   set y1=|. f.x0 .|;
   set Y = {||.x.|| where x is Point of X : x in L};
D2: ||.x0.|| in Y by B11;
   Y c= REAL
   proof
    let z be object;
    assume z in Y; then
    ex x be Point of X st z = ||. x .|| & x in L;
    hence z in REAL;
   end; then
   reconsider Y as non empty Subset of REAL by D2;
   for r be ExtReal st r in Y holds r <= M
   proof
    let r be ExtReal;
    assume r in Y; then
    ex x be Point of X st r = ||. x .|| & x in L;
    hence r <= M by D1;
   end; then
   M is UpperBound of Y by XXREAL_2:def 1; then
D3:sup Y <= M by XXREAL_2:def 3;
   take Y;
   M in REAL by XREAL_0:def 1;
   hence thesis by D3,XXREAL_0:2,9;
end;
