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 Th67:
  x in Lin(A) iff ex l st x = Sum(l)
proof
  thus x in Lin(A) implies ex l st x = Sum(l)
  proof
    assume x in Lin(A);
    then x in the carrier of Lin(A) by STRUCT_0:def 5;
    then x in the set of all Sum(l)  by Def14;
    hence thesis;
  end;
  given k being Linear_Combination of A such that
A1: x = Sum(k);
  x in the set of all Sum(l) by A1;
  then x in the carrier of Lin(A) by Def14;
  hence thesis by STRUCT_0:def 5;
end;
