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How i may prove what is stated? That the topology generated using the theorem D as in the image, is equal to the lower upper bound of a non-empty family of topologies for a non empty set $X$. 

Definitions: 
 The lower upper bound of a non-empty family of topologies of $X$ is the topology whose arise from the intersection of all topologies stronger than each topology in the family. 
A topology T is stronger than R if R is cointained in T.

I tried to prove that one set is cointained in the other by proving that every open set from one is in the another, but i could not, in special because the union of the topologies. One there know how to prove it or have a hint? Thank you very much.