At least in the terminology I'm used to, of mathematical proof, an assumption is a part of the context under which a thing is proven. So having more assumptions weakens the claim (and there is an associated weakening rule [1]).
In other words, the claim "Assuming Q, I prove P" does not mean (to me) that Q must hold in order for P to hold, but rather that one way to show that P is true is to show that Q is true.
In other words, the claim "Assuming Q, I prove P" does not mean (to me) that Q must hold in order for P to hold, but rather that one way to show that P is true is to show that Q is true.
[1]: https://en.wikipedia.org/wiki/Structural_rule