 reserve x,y for object, X,Y,Z for set;
 reserve GF for commutative
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
 reserve GF for commutative non degenerated almost_left_invertible
     Abelian add-associative right_zeroed right_complementable
     associative well-unital distributive non empty doubleLoopStr;
 reserve a,b for Element of GF;
 reserve V for scalar-distributive vector-distributive
   scalar-associative scalar-unital add-associative right_zeroed
     right_complementable Abelian non empty ModuleStr over GF;
 reserve v,v1,v2,u for Vector of V;
 reserve A,B,C for Subset of V;
 reserve T for finite Subset of V;
 reserve l for Linear_Combination of A;
 reserve f,g for Function of V, GF;
reserve l0 for Linear_Combination of {}(the carrier of V);
reserve x for set,
  R for Ring,
  V for LeftMod of R,
  v,v1,v2 for Vector of V,
  A, B for Subset of V;
reserve R for domRing,
  V for LeftMod of R,
  A,B for Subset of V,
  l for Linear_Combination of A,
  f,g for Function of the carrier of V, the carrier of R;

theorem
  for V being strict LeftMod of R, A,B being Subset of V st Lin(A) = V &
  A c= B holds Lin(B) = V
proof
  let V be strict LeftMod of R, A,B be Subset of V;
  assume Lin(A) = V & A c= B;
  then V is Subspace of Lin(B) by Th13;
  hence thesis by VECTSP_4:25;
end;
