Both Uses of

consequence

in

The Demolished Man

Consequently, he belonged to "The League of Esper Patriots," an extreme right-wing political group within the Guild, dedicated to the preservation of the autocracy and incomes of the upper grade Espers.
Chpt 2 *

consequently = resultantly (as a result)

In consequence, Madame Beaumont was left unprotected.†
Chpt 8

consequence = result

Definitions:

(1) (consequence as in: a direct consequence of)
a result of something (often an undesired side effect)
(2) (consequence as in: of little consequence)
importance or relevance
(3) (meaning too rare to warrant focus)
In classic literature, consequential may refer to someone with too much feeling of self-importance as when Dickens wrote "Because he's a proud, haughty, consequential, turned-up-nosed peacock."

Self-consequence was used in a similar manner, but is more easily understood by modern readers since important is one of the modern senses of consequence.

Another classic sense of consequent that is similar to importance or significance refers to "material wealth or prominence" as when Jane Austen wrote: "They had each had money, but their marriages had made a material difference in their degree of consequence."