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 Th27:
  V is finite-dimensional implies W is finite-dimensional
proof
  set A = the Basis of W;
  consider I being Basis of V such that
A1: A c= I by Th16;
  assume V is finite-dimensional;
  then I is finite by Th23;
  hence thesis by A1;
end;
