Piano pedaling with events in both hands

[Joseph Austin]

I'm working on a piano score (from a lesson book) in which the pedaling points 
are split between RH and LH parts.
In the example measure 1, the Pedal occurs on RH quarter beats with no LH note, 
but in meausre 2 it occurs on LH quarter-beats with no RH note.
Furthermore, I'm getting two separate pedal lines.
It seems the RH and LH pedaling should be merged in a piano staff, but I don't 
know how to specify this.

PedalPositioning.ly
Description: Binary data

PedalPositioning.pdf
Description: Adobe PDF document