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In the last post we said bad connection or starter , But what you have told me now you have a connection problem causing an open condition in the cranking circuit.
Somewhere between the battery and the starter you have a bad connection witch is going completely open when a load is applied. It could be on the negative or positive side of the circuit.
You may be able to easily find it by having someone crank the engine while you watch (do this when it is real dark out or in the garage with the lights off ) look for sparks along the battery connections ground connection and at the starter.