
theorem
  for V being RealUnitarySpace, v being VECTOR of V ex W being Subspace
  of V st the carrier of W = {u where u is VECTOR of V : u .|. v = 0}
proof
  let V be RealUnitarySpace;
  let v be VECTOR of V;
  set M = {u where u is VECTOR of V : u.|.v = 0};
  for x being object st x in M holds x in the carrier of V
  proof
    let x be object;
    assume x in M;
    then ex u being VECTOR of V st x = u & u.|.v = 0;
    hence thesis;
  end;
  then reconsider M as Subset of V by TARSKI:def 3;
  0.V .|. v = 0 by BHSP_1:14;
  then
A1: 0.V in M;
  then reconsider M as non empty Subset of V;
  for x,y being VECTOR of V, a being Real st x in M & y in M
    holds (1-a)*x + a*y in M
  proof
    let x,y be VECTOR of V;
    let a be Real;
    assume that
A2: x in M and
A3: y in M;
    consider u2 being VECTOR of V such that
A4: y = u2 and
A5: u2 .|. v = 0 by A3;
    consider u1 being VECTOR of V such that
A6: x = u1 and
A7: u1 .|. v = 0 by A2;
    ((1-a)*u1 + a*u2) .|. v = ((1-a)*u1) .|. v + (a*u2).|. v by BHSP_1:def 2
      .= (1-a)*(u1.|.v) + (a*u2).|.v by BHSP_1:def 2
      .= a*0 by A7,A5,BHSP_1:def 2;
    hence thesis by A6,A4;
  end;
  then reconsider M as non empty Affine Subset of V by RUSUB_4:def 4;
  take Lin(M);
  thus thesis by A1,RUSUB_4:28;
end;
