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証拠立てる is virtually never used in mathematical articles. I would translate it like so: すべてのxに対してP(x)が真であることを示せ。 すべてのxについてP(x)が成り立つことを証明せよ。


The sentence-final copula である ("be") is almost always omitted because it's obvious in definitions, leaving the sentences looking like ending with nouns. Both もの and こと are frequently used nominalizers translating "what do ~" and "doing ~" respectively. すなわち、1. (...) 2. (...) となるもの。 i.e. what satisfies 1. (...) and 2. (...). f:S→T が全射であるとは、f(S)=T ...

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