reserve V for RealLinearSpace,
  W for Subspace of V,
  x, y, y1, y2 for set,
  i, n for Element of NAT,
  v for VECTOR of V,
  KL1, KL2 for Linear_Combination of V,
  X for Subset of V;

theorem Th26:
  (0).V is finite-dimensional
proof
  reconsider V9= (0).V as strict RealLinearSpace;
  reconsider I = {}(the carrier of V9) as finite Subset of V9;
  the carrier of V9 = {0.V} by RLSUB_1:def 3
    .= {0.V9} by RLSUB_1:11
    .= the carrier of (0).V9 by RLSUB_1:def 3;
  then
A1: V9 = (0).V9 by RLSUB_1:32;
  I is linearly-independent & Lin(I) = (0).V9 by RLVECT_3:7,16;
  then I is Basis of V9 by A1,RLVECT_3:def 3;
  hence thesis;
end;
