
theorem
  for V being RealUnitarySpace, A being non empty Affine Subset of V st
  0.V in A holds A = the carrier of Lin(A)
proof
  let V be RealUnitarySpace;
  let A be non empty Affine Subset of V;
  assume 0.V in A;
  then
A1: A is Subspace-like by Th26;
  for x being object st x in the carrier of Lin(A) holds x in A
  proof
    let x be object;
    assume x in the carrier of Lin(A);
    then x in Lin(A);
    then
A2: ex l being Linear_Combination of A st x = Sum(l) by RUSUB_3:1;
    ( for v,u being VECTOR of V st v in A & u in A holds v + u in A)& for
    a being Real, v being VECTOR of V
       st v in A holds a * v in A by A1;
    then A is linearly-closed by RLSUB_1:def 1;
    hence thesis by A2,RLVECT_2:29;
  end;
  then
A3: the carrier of Lin(A) c= A;
  for x being object st x in A holds x in the carrier of Lin(A)
  by RUSUB_3:2,STRUCT_0:def 5;
  then A c= the carrier of Lin(A);
  hence thesis by A3;
end;
