
theorem Th24:
  for X be RealNormSpace,
      x be Point of DualSp X,
      M be non empty Subspace of X holds
  { y .|. x where y is Point of X :
     y in M & ||.y.|| <= 1 }
     is non empty bounded_above real-membered set
proof
  let X be RealNormSpace,
      x be Point of DualSp X,
      M be non empty Subspace of X;
  set B = {y .|. x where y is Point of X :
     y in M & ||.y.|| <= 1};
  set z = 0.X;
A1: z in M by RLSUB_1:17;
  ||.z.|| = 0; then
A2P: z .|. x in B by A1;
  B c= REAL
  proof let r be object;
    assume r in B; then
    ex y be Point of X st r = y .|. x & y in M & ||.y.|| <= 1;
    hence r in REAL;
  end; then
  reconsider B as real-membered set;
  B is bounded_above
  proof
    reconsider r0 = ||.x.|| as Real;
    take r0;
    let r be ExtReal;
    assume r in B; then
    consider y be Point of X such that
A3: r = y .|. x & y in M & ||.y.|| <= 1;
    reconsider x0=x as Lipschitzian linear-Functional of X
      by DUALSP01:def 10;
A4: |.x0.y.| <= ||.x.|| * ||.y.|| by DUALSP01:26;
    ||.x.|| * ||.y.|| <=||.x.||*1 by A3,XREAL_1:64; then
A5: |.x0.y.| <= ||.x.|| by A4,XXREAL_0:2;
    x0.y <= |.x0.y.| by ABSVALUE:4;
    hence r <= r0 by A5,XXREAL_0:2,A3;
  end;
  hence thesis by A2P;
end;
