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;
reserve l0 for Linear_Combination of {}(the carrier of V);

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)  by Def2;
      then ex l0 st v = Sum(l0) by A1;
      then v = 0.V by RLVECT_2:31;
      hence thesis by Lm2;
    end;
    assume v in (0).V;
    then v = 0.V by Lm2;
    hence v in A by RLSUB_1:17;
  end;
  hence thesis by RLSUB_1:31;
end;
