I can confirm Clive's assessment:
The HP turbine cooling air came from the 5th stage of the 7 stage HP compressor. Incidently 5th stage air required for cooling and sealing the compressor bearings needed a heat exchanger. ref SAE 670316 Fig.11. So it seems the lowest pressure source possible was used for the job and no heat exchanger was needed either.
I read somewhere that there was no compromising the compressor design for the M2 cruise condition so as well as the definition of blade angles,etc. that would include minimizing the air taken from the gas path to the internal (secondary) air system for cooling,etc.
See
https://en.wikipedia.org/wiki/Jet_en...e#Introduction for non-technical explanations on sources of waste (entropy) generation in jet engines including secondary air system. This article is a recent revamp of the previous which was considered too technical.