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
  n <= dim V implies n Subspaces_of V is non empty
proof
  assume n <= dim V;
  then ex W being strict Subspace of V st dim W = n by Lm2;
  hence thesis by Def3;
end;
