Sentence examples for corollary without from inspiring English sources

A simple consequence of Theorem 4 is stated in the following corollary without proof.

Now, using Theorem 2.1, we can also give the following corollary without proof.

Some of these formulas are stated in the following three corollaries without proofs.

Now the Theorems ?3.1 and ?3.2 in [9] is stated as the following corollaries without the assumption of normality, and by Lemma ?2.10 and Remarks ?2.11, ?2.12 we have the same theorems.

Which brings us to a corollary: Amend without end.

The following result follows directly from Theorem 2.1 and Corollary 2.2 without proof.

Hence, we give the modified error bound for (GVVI) in Corollary 3.3 without Lipschitz assumption.

If (boldsymbol{X}_{0}=theta boldsymbol{D}boldsymbol{X}), then by Corollary 3.1 without tedious calculations, (boldsymbol{X}_{0}(boldsymbol{X}'boldsymbol{X}+kboldsymbol{I}_{p})^{-1}boldsymbol{X}'boldsymbol{y}sim boldsymbol{delta}).

As it stands, the reasons for rodents not being subject to the proposed causative processes in humans constitutes an unstated corollary— one without empirical support or plausible basis to the theories of human leukemogenesis of formaldehyde.

Since much of the war was fought in this state, we New Jerseyans should never forget the motivating slogan "No taxation without representation" and its corollary, "No representation without taxation".

So that made me think, Well, as weird as my story is, it isn't entirely without corollary in the real world.