reserve R for Ring,
  V for RightMod of R,
  a,b for Scalar of R,
  x,y for set,
  p,q ,r for FinSequence,
  i,k for Nat,
  u,v,v1,v2,v3,w for Vector of V,
  F,G,H for FinSequence of V,
  A,B for Subset of V,
  f for Function of V, R,
  S,T for finite Subset of V;
reserve L,L1,L2,L3 for Linear_Combination of V;
reserve l for Linear_Combination of A;
reserve RR for domRing;
reserve VV for RightMod of RR;
reserve LL for Linear_Combination of VV;
reserve aa for Scalar of RR;
reserve uu, vv for Vector of VV;
reserve R for domRing;
reserve V for RightMod of R;
reserve L,L1,L2 for Linear_Combination of V;
reserve a for Scalar of R;
reserve x for set;
reserve R for Ring;
reserve V for RightMod of R;
reserve v,v1,v2 for Vector of V;
reserve A,B for Subset of V;
reserve R for domRing;
reserve V for RightMod of R;
reserve v,u for Vector of V;
reserve A,B for Subset of V;
reserve l for Linear_Combination of A;
reserve f,g for Function of the carrier of V, the carrier of R;

theorem
  Lin({}(the carrier of V)) = (0).V
proof
  set A = Lin({}(the carrier of V));
  now
    let v;
    thus v in A implies v in (0).V
    proof
      assume v in A;
      then
A1:   v in the carrier of A by STRUCT_0:def 5;
      the carrier of A = the set of all
Sum(l0) where l0 is Linear_Combination of {}(the
      carrier of V) by Def14;
      then
      ex l0 being Linear_Combination of {}(the carrier of V) st v = Sum(l0
      ) by A1;
      then v = 0.V by Th31;
      hence thesis by RMOD_2:35;
    end;
    assume v in (0).V;
    then v = 0.V by RMOD_2:35;
    hence v in A by RMOD_2:17;
  end;
  hence thesis by RMOD_2:30;
end;
