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De Broglie's hypothesis relates to the wave-like behavior that matter portrays. The reason electron diffraction supports De Broglie's hypothesis is due to the fact that the electrons are portraying properties of waves by diffracting through a crystal. Light portrays wave-like properties for the same reason; a crystal diffracts lights and creates a rainbow.
Diffraction is a phenomenon that occurs with waves; diffraction is a result of waves interfering with each other. Thus, the fact that electrons are able to diffract means that they should have wavelike properties. De Broglie's equation basically states that everything has wavelike properties and that the waves are noticeable and measurable as the mass of an object decreases. Thus, electron diffraction supports the theory that electrons have wavelike properties.
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