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;

theorem
  A c= B & B is linearly-independent implies A is linearly-independent
proof
  assume that
A1: A c= B and
A2: B is linearly-independent;
  let l be Linear_Combination of A;
  reconsider L = l as Linear_Combination of B by A1,Th19;
  assume Sum(l) = 0.V;
  then Carrier(L) = {} by A2;
  hence thesis;
end;
