 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);

theorem Th13:
  for GF be Ring,
      V be LeftMod of GF,
      A,B be Subset of V holds
  A c= B implies Lin(A) is Subspace of Lin(B)
proof
  let GF be Ring,
      V be LeftMod of GF,
      A,B be Subset of V;
  assume
A1: A c= B;
  now
    let v be Vector of V;
    assume v in Lin(A);
    then consider l being Linear_Combination of A such that
A2: v = Sum(l) by Th7;
    reconsider l as Linear_Combination of B by A1,VECTSP_6:4;
    Sum(l) = v by A2;
    hence v in Lin(B) by Th7;
  end;
  hence thesis by VECTSP_4:28;
end;
