Secant said:
There is nothing wrong in asking for proof.
In the sense you are asking for, yes there is. There are in fact many things wrong with what you are asking for.
First off, you are asking for an experiment that in all likelihood can never be performed. Sans a force field that rivals the imagination of science fiction writers, there is no way to send a probe to the center of the Earth. Rejecting the premise that the weight is zero at the center of the Earth implicitly makes an
argumentum ad ignorantiam, an appeal to ignorance: That a premise is false because it has not been proven true.
There is no such thing as proof in science. Scientific theories cannot be proven to be true. They can only be proven to be false. Dave C's example was a good one. We do not know, with absolute certainty, how much a 10kg sack of potatoes will weigh on the Moon until someone has done the experiment. Even then, we won't know with absolute certainty whether that sack will have a different weight the next time someone does the experiment.
Science, like the law, does not work on the basis of absolute certainty. Both science and the law work on the basis of reasonable doubt. With this metric, there is no doubt, within reason, what that sack of potatoes will weigh on the Moon (or atop Mt. Everest) without even doing the experiment. The physics and the mathematics are very well understood and there is no reason to think otherwise.
The same goes for the weight of something at the center of the Earth. Assuming that our models of how gravity works are correct, the mathematics (and the underlying math can be proven) says that the weight at the center of the Earth will be zero. One of these assumptions is that the gravitational attraction toward some object is the cumulative gravitational attraction toward the elementary particles that comprise the object. To say that the gravitational attraction at the center of the Earth is non-zero contradicts this basic assumption.
The gravitational attraction at the surface of the Earth is directed inward everywhere on the surface of the Earth. If gravitational attraction is a continuous function, then mathematically there must exist a point somewhere inside the surface at which the gravitational attraction is zero.
Finally, gravity is similar in form to electrostatic force. Gauss' law for gravity and Gauss' law for electrical flux are very similar in form. The latter says that the electric flux through any closed surface is proportional to the enclosed electric charge while the former says the gravitational flux through any closed surface is proportional to the enclosed mass. Gauss' law for electrical flux has been demonstrated and is the underlying reason why you are told to get inside your car in a lightning storm.
But you said it yourself it is a THEOREM, not fact.
You need to read up on the difference between theorems and theories.