 reserve x for set,
         n,m,k for Nat,
         r for Real,
         V for RealLinearSpace,
         v,u,w,t for VECTOR of V,
         Av for finite Subset of V,
         Affv for finite affinely-independent Subset of V;

theorem Th3:
  V is finite-dimensional iff (Omega).V is finite-dimensional
proof
  set O=(Omega).V;
  thus V is finite-dimensional implies O is finite-dimensional;
  assume O is finite-dimensional;
  then consider A be finite Subset of O such that
   A1: A is Basis of O by RLVECT_5:def 1;
  A2: the RLSStruct of V=O by RLSUB_1:def 4;
  then reconsider a=A as finite Subset of V;
  Lin(A)=O by A1,RLVECT_3:def 3;
  then A3: Lin(a)=O by RLVECT_5:20;
  A is linearly-independent by A1,RLVECT_3:def 3;
  then a is linearly-independent by RLVECT_5:14;
  then a is Basis of V by A2,A3,RLVECT_3:def 3;
  hence thesis by RLVECT_5:def 1;
end;
