
theorem Th10:
  for V being finite-dimensional RealUnitarySpace holds dim V = dim (Omega).V
proof
  let V be finite-dimensional RealUnitarySpace;
  consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
A2: (Omega).V = the UNITSTR of V by RUSUB_1:def 3
    .= Lin(I) by A1,RUSUB_3:def 2;
  card I = dim V & I is linearly-independent by A1,Def2,RUSUB_3:def 2;
  hence thesis by A2,Th9;
end;
