
theorem Th26:
  for V being RealLinearSpace, A being non empty Affine Subset of
  V st 0.V in A holds A is Subspace-like & A = the carrier of Lin(A)
proof
  let V be RealLinearSpace;
  let A be non empty Affine Subset of V;
  assume
A1: 0.V in A;
A2: for x,y being Element of V, a being Real
    st x in A & y in A holds x + y in A & a * x in A
  proof
    let x,y be Element of V;
    let a be Real;
    assume that
A3: x in A and
A4: y in A;
    reconsider x,y as VECTOR of V;
A5: 2 * ( (1-1/2) * x + (1/2) * y ) = 2*( (1-1/2) * x) + 2*((1/2) * y) by
RLVECT_1:def 5
      .= ( 2*(1-1/2) ) * x + 2*((1/2) * y) by RLVECT_1:def 7
      .= ( 2 - 1 ) * x + ( 2*(1/2) ) * y by RLVECT_1:def 7
      .= x + 1 * y by RLVECT_1:def 8
      .= x + y by RLVECT_1:def 8;
    (1 - 1/2) * x + (1/2) * y in A by A3,A4,Def4;
    hence thesis by A1,A3,A5,Th25;
  end;
  hence A is Subspace-like by A1;
  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
A6: ex l being Linear_Combination of A st x = Sum(l) by RLVECT_3:14;
    ( 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 A2;
    then A is linearly-closed by RLSUB_1:def 1;
    hence thesis by A6,RLVECT_2:29;
  end;
  then
A7: the carrier of Lin(A) c= A;
  for x being object st x in A holds x in the carrier of Lin(A)
  by RLVECT_3:15,STRUCT_0:def 5;
  then A c= the carrier of Lin(A);
  hence thesis by A7;
end;
