reserve x,y for object, X,Y,Z for set;
reserve a,b for Real;
reserve k for Element of NAT;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
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,L1,L2 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve F,G,H for FinSequence of the carrier of V;
reserve f,g for Function of the carrier of V, REAL;
reserve p,q,r for FinSequence;
reserve M for non empty set;
reserve CF for Choice_Function of M;

theorem Th15:
  x in A implies x in Lin(A)
proof
  deffunc F(Element of V)=In(0,REAL);
  assume
A1: x in A;
  then reconsider v = x as VECTOR of V;
  consider f being Function of the carrier of V, REAL such that
A2: f.v = jj and
A3: for u st u <> v holds f.u = F(u) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:8;
  ex T st for u st not u in T holds f.u = 0
  proof
    take T = {v};
    let u;
    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 RLVECT_2:def 3;
A4: Carrier(f) c= {v}
  proof
    let x be object;
    assume x in Carrier(f);
    then consider u such that
A5: x = u and
A6: f.u <> 0;
    u = v by A3,A6;
    hence thesis by A5,TARSKI:def 1;
  end;
  then reconsider f as Linear_Combination of {v} by RLVECT_2:def 6;
A7: Sum(f) = 1 * v by A2,RLVECT_2:32
    .= v by RLVECT_1:def 8;
  {v} c= A by A1,ZFMISC_1:31;
  then Carrier(f) c= A by A4;
  then reconsider f as Linear_Combination of A by RLVECT_2:def 6;
  Sum(f) = v by A7;
  hence thesis by Th14;
end;
