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Your battery is not being charged, when the battery voltage is low then the electronic ignition module will not work and you will not get a spark for the plugs. When you apply the brakes the brake light lowers the voltage (about 3 volts) this is enough drop in the battery voltage  so that it is below the level that the ignition module needs to function. You should invest in a digital multimeter ($10) so that you can read the battery voltage.
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