Intercooled cooling air with plural heat exchangers

A gas turbine engine comprising;

• a main compressor section having a high pressure compressor with a downstream most end, and more upstream locations;
• a turbine section having a high pressure turbine;
• a first tap tapping air from at least one of said more upstream locations in said main compressor section, passing said tapped air through a first heat exchanger and then to a cooling compressor;
• a second tap tapping air from a location in said main compressor section closer to said downstream most end than the location of said first tap, and said first and second taps mixing together and being delivered into said high pressure turbine;
• said cooling compressor being positioned downstream of said first heat exchanger, and upstream of a location where air from said first and second taps mix together;
• wherein there is a second heat exchanger downstream of said cooling compressor, and air passes through said second heat exchanger before being delivered to said high pressure turbine;
• wherein air is tapped from a third tap downstream of said second heat exchanger, and delivered to a component on the gas turbine engine to be cooled;
• and
• wherein the location of said third tap is upstream of a location where said first and second taps mix together.