I'd really like to make diagrams to go along with this proof, but that would take time, so I'll just go ahead and write it out. (@MartinEnder)
1. Where can the 0-2 and 1-0 crossing go? There are precisely two possible locations, to be demonstrated by excluding all other possible locations. 1) The crossing cannot be at the left border, as a wire would then go outside of the bounding box. 2) The crossing cannot be on the top or bottom row (with one exception), then either there is a wire going out of the bounding box (up or down) or it is connected to a location that is neither 0 nor 1. The …