The phrase "demonstrate plainly" is correct and usable in written English.
It means to clearly or openly show or prove something. Example: The data from the study demonstrates plainly that increased exercise leads to improved overall health.
Most memorable was the gallery devoted to women and work, a room filled with diaries, photographs, and installations demonstrating plainly the realities of female existence.
"The city's regulation of highway billboards has more holes than Swiss cheese, demonstrating plainly that its asserted interest is nothing but a pretext for the city's true purpose for its regulation: to eliminate competition and make money for itself," lawyers for some of the companies wrote.
"The new and highly disturbing data we have on youth use demonstrates plainly that e-cigarettes are creating an epidemic of regular nicotine use among teens.
The evidence "plainly demonstrates a concerted effort to keep black prospective jurors off the jury".
Since the subdued stutter of "Seinfeld," "Curb Your Enthusiasm" and "The Office" have demonstrated so plainly to the contrary, why is loud still considered funny?
"Trump is saying: 'Well yeah, but' – and then he says, as plain as day, 'I didn't attack her.' And right after he says that he says: 'I was viciously attacked.' This demonstrates as plainly as anything that he's a person who lies reflexively".
Of course there is no due diligence (as the corrupt cases plainly demonstrate).
The writing signs plainly demonstrate its connection with the hieratic script, although the exact relationship is not yet clear.
The unfolding catastrophe in the Gulf of Mexico plainly demonstrates the limits of technology and the inevitability of human error.
As this week's miscarriage of justice plainly demonstrates, Spain still needs his help in keeping its judiciary fearless and independent.
