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;

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,RLVECT_2:21;
  assume Sum(l) = 0.V;
  then Carrier(L) = {} by A2;
  hence thesis;
end;
