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'further demonstration' is a correct and usable phrase in written English.
You can use it when you want to describe a situation in which an idea or concept is given a more in-depth explanation or presentation. For example, "This further demonstration of the complex physics experiment successfully proved the hypothesis."
Exact(60)
The use of his cells in making mice walk is a further demonstration of their versatility.
Further demonstration by supporters of the Palestinians are planned for Saturday in Paris.
Finally, a 500 W SPSV prototype is built for further demonstration.
A further demonstration case of the co-sintering of two particles of different sizes is provided.
"This measure is a further demonstration of how the UK takes its international environmental responsibilities seriously".
Illustrative examples give a further demonstration for the effectiveness of the proposed neural network.
All the finale brought was late false hope and a further demonstration of Liverpool's enduring reliance on their captain.
The trip to Hawr Rajab was to be a further demonstration that the group's days there were numbered.
This is acting of the highest order, and the film is a further demonstration of Mike Leigh's unsentimental compassion.
"It's a further demonstration of our core strategy of 'national brands delivered locally', and it's very important to us.
So there needs to be a further demonstration that the "deep-reading brain" is something worth valuing.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com