Sentence examples for preceding facts from inspiring English sources

The preceding facts yield the following result.

The preceding facts mentioned allow us to state the next useful result [3].

In connection with the preceding facts, we are now in a position to formulate a localization of Proposition 3 in the following form.

Taking into account the preceding facts we discuss the same properties in the case of projective Φ-contractions in the result below.

In connection with the preceding facts, we are now in a position to formulate a localization for lower edges contractions in the following form.

In connection with the preceding facts, based on Lemma 1 and Theorem 1, we are now in a position to formulate the following new statements as direct consequences of the Axiom of Infinite Choice.

Observe that the preceding fact shows a distinguishing feature of partial metric aggregation functions with respect to the metric ones.

The preceding fact is a special case of a general result for non-Archimedean spaces, that is, every continuous function from a connected space to a non-Archimedean space is constant.

Inspired by the preceding fact, we prove two fixed point theorems which provide the mathematical basis for a new technique to carry out asymptotic complexity analysis of algorithms via partial metrics.

Additional Fact: The preceding is not a fact.

Relevant facts preceding Period I are also succinctly presented.