We present a multitasking operating system kernel, called KIT, written in the machine language of a uni-processor von Neumann computer. The kernel is proved to implement, on this shared computer, a fixed number of conceptually distributed communicating processes. In addition to implementing processes, the kernel provides the following verified services: process scheduling, error handling, message passing, and an interface to asynchronous devices. The problem is stated in the Boyer-Moore logic, and the proof is mechanically checked with the Boyer-Moore theorem prover.