“Supported” vs. “demonstrated”

They differ mainly in degree of evidential establishment.

Supported

There is evidence pointing in favor of the claim, but the evidence may still leave significant room for doubt or alternative explanations.

Example:

Abolition contributed to changes in labor markets.

Suppose several historians, employment records, wage data, and demographic evidence support this. The proposition is supported.

Demonstrated

The evidence is sufficiently strong, consistent, and well-connected to the proposition that you can reasonably present it as an established conclusion.

So:

Supported = evidence favors the proposition.
Demonstrated = evidence establishes the proposition strongly enough to treat it as established.

The boundary is not mathematical. “Demonstrated” does not mean absolute certainty.