Sentence examples for preceding propositions from inspiring English sources

If a proposed proposition was unconnected with any of the preceding propositions, the respondent would respond with agreement, rejection or doubt depending on what he understood the actual state of affairs in the world to be.

The subsequent examples apply the preceding proposition.

The preceding proposition has the following two complementary implications.

It follows from the preceding proposition that (uinoperatorname{End}(T)). □.

The following result follows easily from the preceding proposition.

The next result is an immediate consequence of the preceding proposition.

According to the preceding proposition, the following system for the undetermined coefficients is obtained.

Then, limn→∞||x n -p|| exists, and hence, It follows from the preceding proposition that.

Remark 2 Note that the continuity of φ is not used in the preceding proposition.

Based on these facts as well as the preceding Proposition 3.4, we can now prove Theorem 1.1 as follows.

This ad hoc method may be replaced by a decomposition of the type as described in the preceding proposition with (s(x)=1).