support vs prove

support is a noun but prove is not a noun.

Word	Noun	Adjective	Verb	Adverb
support	Yes	No	Yes	No
prove	No	No	Yes	No

As verbs, prove is a hyponym of support; that is, prove is a word with a more specific, narrower meaning than support:

support: establish or strengthen as with new evidence or facts
prove: establish the validity of something, as by an example, explanation or experiment

Other hyponyms of support include back, back up, vouch, verify, demonstrate, establish, shew, show, document, validate.

support (noun)	prove (noun)
aiding the cause or policy or interests of
the act of bearing the weight of or strengthening
documentary validation
a military operation (often involving new supplies of men and materiel) to strengthen a military force or aid in the performance of its mission
the activity of providing for or maintaining by supplying with money or necessities
any device that bears the weight of another thing
supporting structure that holds up or provides a foundation
something providing immaterial assistance to a person or cause or interest
a musical part (vocal or instrumental) that supports or provides background for other musical parts
the financial means whereby one lives
financial resources provided to make some project possible

support (verb)	prove (verb)
establish or strengthen as with new evidence or facts	establish the validity of something, as by an example, explanation or experiment
put up with something or somebody unpleasant	obtain probate of
adopt as a belief	prove formally; demonstrate by a mathematical, formal proof
argue or speak in defense of	provide evidence for
be a regular customer or client of	take a trial impression of
be the physical support of; carry the weight of	cause to puff up with a leaven
play a subordinate role to (another performer)	increase in volume
support materially or financially	put to the test, as for its quality, or give experimental use to
give moral or psychological support, aid, or courage to	be shown or be found to be
be behind; approve of
support with evidence or authority or make more certain or confirm