reserve x,y for set;
reserve a,b for Real;
reserve i,j for Integer;
reserve V for RealLinearSpace;
reserve W1,W2,W3 for Subspace of V;
reserve v,v1,v2,v3,u,w,w1,w2,w3 for VECTOR of V;
reserve A,B,C for Subset of V;
reserve L,L1,L2 for Linear_Combination of V;
reserve l,l1,l2 for Linear_Combination of A;

theorem Th12:
  x in A implies x in Z_Lin(A)
proof
  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 = In(1,REAL) and
A3: for u st u <> v holds f.u = H(u) from FUNCT_2:sch 6;
  reconsider f as Element of Funcs(the carrier of V, REAL) by FUNCT_2:8;
  ex T being finite Subset of V 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) = f.v * v by RLVECT_2:32
    .= v by A2,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;
  rng f c= INT
  proof
    let y be object;
    assume A8:y in rng f;
    consider x be object
    such that A9: x in the carrier of V & y=f.x by A8,FUNCT_2:11;
    reconsider z=x as VECTOR of V by A9;
    per cases;
    suppose z <> v; then
      f.z = 0 by A3;
      hence y in INT by A9,NUMBERS:17;
    end;
    suppose z = v;
      hence y in INT by A2,A9,NUMBERS:17;
    end;
  end;
  hence thesis by A7;
end;
