I'm reading intel software developer manual (section 3.1). Here is said that
The IA-32 architecture supports three basic operating modes: protected mode, real-address mode, and system management mode.
and
Intel 64 architecture adds IA-32e mode. IA-32e mode has two sub-modes.
This two sub-modes is Compatibility mode and 64-bit mode. Thus Intel 64 architecture contains 4 operating modes (protected mode, real-address mode, system management mode, IA-32e mode). But I don't understand for what protected mode is needed in Intel 64? Compatibility sub-mode of IA-32e permit to run legacy 16- and 32-bits application without re-compilation to 64-bit OS and 64-bit sub-mode permits OS to run application which access to 64-bit linear address space. Where protected mode is used when I'm using 64-bit OS? Application running ensured by IA-32e mode.