demonstrate (n) 证明
{Demonstrate} is a verb.
demonstrate (v) 证明

I demonstrate that I can swam.
{Can} is a model verb. It should be followed by the original form of a verb.
I want(verb) [to demonstrate] (infinitive/noun) that (conjunction) [I can swim].

I (subj) demonstrate (verb) my swimming (adj) skills (obj/noun).

I hate to demonstrate thing.
{Thing} is a countable noun. Use plural form here.
I hate to demonstrate things.

That demonstrate to me.
We always say {demonstrate something to sb.}.
They will demonstrate it to me.