An HS125-800XP First Officer reported that during climb the master caution light illuminated followed by the number 1 HP air overheat annunciator. The checklist was completed and flight continued to its destination on a single pack.

Narrative

We were climbing through about FL190 when the red-master-warning-system light began to flash. The Captain canceled it. After about 10 seconds; the No. 1 HP air overheat annunciator light illuminated. We followed memory items (closing the affected main air valve and flight deck valve) and opened the checklist. It said to turn the affected main air valve from OFF to LP ONLY. We did this. About 10 seconds later; the No. 1 HP overheat annunciator light illuminated again; so we turned the left main air valve off. The annunciator extinguished and did not re-light. Since we were en route we discussed returning; diverting to an en route station or continuing. Things seemed stable and the problem contained so we continued. I put on my oxygen mask. The Captain briefed the passengers about the situation. After crossing a stretch of ocean and nearing the coast the flight phone was working. The Captain called the company and discussed the situation with an Assistant Chief pilot. He agreed with the course of action. We decided the fly closer to the coast since we only had one bleed-air source to power the air cycle machine and pressurize the cabin. We called ATC and got routing along the coast. The Captain called Dispatch and relayed the routing. Both the FMS and Dispatch agreed the new route was easily workable with our fuel on board. The rest of the flight was uneventful. I assume this was mechanical failure of some sort. The system worked for the first 10-minutes of flight thus there is no way to have found it at a prior time. It would have been helpful if the checklist said continue flight or land or gave other guidance.