The client doesn't have to change the HTTP request for it to be sent to a proxy. It has to change the TCP headers.
The screenshot below shows a HTTP request sent from my browser to a proxy, as you can see nothing in the HTTP request itself specifies the proxy.
How this works is the browser/client will issue a HTTP GET request, which will then be forwarded to the TCP/IP stack and wrapped in a TCP header. The TCP header is where the destination is specified (proxy or otherwise).
