Interesting that you ask this Jet.  I recently felt compelled to gain clarity with this same word even though I have been familiar with Mrs. Eddy's usage, and have used the term many times.  I found when looking up in the dictionary definition an idea that caught my attention that demonstrate includes "prove", elaborating with "to show by reasoning", "to make clear", in order to make it my own.

How this term has been used and understood by me is the idea that to demonstate is to become clear about whatever may be attempting to have me believe that all is not omnipresent good.  This is done not through my own understanding, but rather is gained in clarity through my studying, listening, allowing angel thoughts of understanding to flow into consciousness. Reasoning from the solution rather than starting with the problem.  Demonstration sounds like an action verb, yet it happens more as a gentle change of thought. As thought lets go of problem, it disappears.

Further with Mrs. Eddy's statement about "demonstrate your way out of wanting or needing…", may be the work of a lifetime or a moment.  I love the idea that as I "become conscious for a single moment that life is spiritual, neither in nor of matter, (I) will find myself suddenly well….."

Building on that idea, the ideas in Hargreave's "As I See It" that I read later that same day, leaped out of the page to me with greater clarity and understanding: "…recognizing all to be consciousness, we find that everything is now. A memory is a present state of thought calling itself a past.  A fear is a present state calling itself a future."

This all said, my mental starting point is not an acceptance of lack to be attached to the journey, followed by an attempt to alleviate it. Omnipresence doesn't have to go somewhere to be omnipresent. These ideas wash over me with clarity…in the moment.  I am slowly learning to yield and let go of the belief that there was ever a moment this wasn't true…in the moment.

Whether this was Mrs. Eddy's meaning, I know not, but this is my understanding of it.  This is the perfect question to ask the divine as you read for what flows as yours.