
theorem Th19:
  for X be RealNormSpace,
      x be Point of X,
      M be non empty Subspace of X holds
   { x .|. y where y is Point of DualSp X :
      y in Ort_Comp M & ||.y.|| <= 1 }
      is non empty bounded_above real-membered set 
proof
  let X be RealNormSpace,
      x be Point of X,
      M be non empty Subspace of X;
  set B = {x .|. y where y is Point of DualSp X :
    y in Ort_Comp M & ||.y.|| <= 1};
  set z = 0.(DualSp X);
A1: z in Ort_Comp M by RLSUB_1:17;
  ||.z.|| = 0; then
A2P: x .|. z in B by A1;
  B c= REAL
  proof let r be object;
    assume r in B; then
    ex y be Point of DualSp X
    st r = x .|. y & y in Ort_Comp 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 DualSp X such that
 A3: r = x .|. y & y in Ort_Comp M & ||.y.|| <= 1;
    reconsider y0=y as Lipschitzian linear-Functional of X
    by DUALSP01:def 10;
 A4: |.y0.x.| <= ||.y.|| * ||.x.|| by DUALSP01:26;
    ||.y.|| * ||.x.|| <=1*||.x.|| by A3,XREAL_1:64; then
 A5: |.y0.x.| <= ||.x.|| by A4,XXREAL_0:2;
    y0.x <= |.y0.x.| by ABSVALUE:4;
    hence r <= r0 by A5,XXREAL_0:2,A3;
  end;
  hence thesis by A2P;
end;
