 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;

theorem Th7:
  for GF be Ring,
      V be LeftMod of GF,
      A be Subset of V holds
  x in Lin(A) iff ex l being Linear_Combination of A st x = Sum(l)
proof
  let GF be Ring,
      V be LeftMod of GF,
      A be Subset of V;
  thus x in Lin(A) implies ex l being Linear_Combination of A st x = Sum(l)
  proof
    assume x in Lin(A);
    then x in the carrier of Lin(A) by STRUCT_0:def 5;
    then x in the set of all Sum(l) where l is Linear_Combination of A
      by Def2;
    hence thesis;
  end;
  given k being Linear_Combination of A such that
A1: x = Sum(k);
  x in the set of all Sum(l) where l is Linear_Combination of A by A1;
  then x in the carrier of Lin(A) by Def2;
  hence thesis by STRUCT_0:def 5;
end;
