 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 Th8:
  for GF be Ring,
      V be LeftMod of GF,
      A be Subset of V holds
  x in A implies x in Lin(A)
proof
  let GF be Ring,
      V be LeftMod of GF,
      A be Subset of V;
  deffunc F(set) = 0.GF;
  assume
A1: x in A;
  then reconsider v = x as Vector of V;
  consider f being Function of V, GF such that
A2: f.v = 1_GF and
A3: for u being Vector of V st u <> v holds f.u = F(u) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V, the carrier of GF) by
FUNCT_2:8;
  ex T being finite Subset of V st
     for u being Vector of V st not u in T holds f.u = 0.GF
  proof
    take T = {v};
    let u be Vector of V;
    assume not u in T;
    then u <> v by TARSKI:def 1;
    hence thesis by A3;
  end;
  then reconsider f as Linear_Combination of V by VECTSP_6:def 1;
A4: Carrier(f) c= {v}
  proof
    let x be object;
    assume x in Carrier(f);
    then consider u being Vector of V such that
A5: x = u and
A6: f.u <> 0.GF;
    u = v by A3,A6;
    hence thesis by A5,TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by VECTSP_6:def 4;
A7: Sum(f) = 1_GF * v by A2,VECTSP_6:17
    .= v;
  {v} c= A by A1,ZFMISC_1:31;
  then Carrier(f) c= A by A4;
  then reconsider f as Linear_Combination of A by VECTSP_6:def 4;
  Sum(f) = v by A7;
  hence thesis by Th7;
end;
