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According to prof Lavelle's "Wave Properties of Electrons and the DeBroglie Equation" module, that experiment supports the hypothesis because the only time a diffraction pattern can be observed is when whatever is being used is acting like a wave. In this case, light diffracts as it becomes a wavelet passing through slits. When the wavelets interfere with each other, it is constructive interference, resulting in the bands seen in a diffraction pattern.
When considering the wave-particle duality of matter, the wave part comes from the experiment showing that electrons are capable of diffraction because diffraction occurs when there is constructive and/or destructive interference of waves. If an electron is capable of diffraction, then that implies that there must have been some constructive or destructive interference to make that diffraction happen, which occurs to waves, showing that an electron and therefore all matter holds some wave-like behavior. Hope that helps.
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