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;
reserve V for finite-dimensional RealLinearSpace,
  W, W1, W2 for Subspace of V,
  u, v for VECTOR of V;

theorem Th30:
  dim V = dim (Omega).V
proof
  consider I being finite Subset of V such that
A1: I is Basis of V by Def1;
A2: (Omega).V = the RLSStruct of V by RLSUB_1:def 4
    .= Lin(I) by A1,RLVECT_3:def 3;
  card I = dim V & I is linearly-independent by A1,Def2,RLVECT_3:def 3;
  hence thesis by A2,Th29;
end;
