
theorem Th6:
  for V being RealUnitarySpace holds (0).V is finite-dimensional
proof
  let V be RealUnitarySpace;
  reconsider V9= (0).V as strict RealUnitarySpace;
  reconsider I = {}(the carrier of V9) as finite Subset of V9;
  the carrier of V9 = {0.V} by RUSUB_1:def 2
    .= {0.V9} by RUSUB_1:4
    .= the carrier of (0).V9 by RUSUB_1:def 2;
  then
A1: V9 = (0).V9 by RUSUB_1:26;
  I is linearly-independent & Lin(I) = (0).V9 by RLVECT_3:7,RUSUB_3:3;
  then I is Basis of V9 by A1,RUSUB_3:def 2;
  hence thesis;
end;
