Proof is for mathematicians. In science we just have hypotheses and evidence that supports or refutes them.
The e25 test uses antibodies (attached to the paper strip) to detect the coronavirus spike protein, the hallmark of intact, infectious virus. There is evidence that viral load is correlated with contagiousness. So, for practical purposes, a positive e25 test does indicate that the person is currently shedding viral particles.
The RT-PCR test that is the current standard detects viral genomes, or fragments thereof. It is more sensitive than an antibody-based test, but also gives positive results even after the person is no longer contagious, by detecting viral RNA fragments that have not been cleared.