Geez are we going along the old 'Your spelling is rubbish route' I hope (2 mistakes and a typo big deal, we will ignore your typo
). The air is cooled because it is heated because its been HIGHLY compressed. We are not removing any of the potential energy the air at altitude holds because we are leaving at altitude. There is no net gain due to the change of potential (ie altitude) the net gain is due to the compression.Thats the basic level of thermodynamics we need to understand.
We shall ignore the rest of your basic aerodynamic errors then shall we..
As for not having INFORMED comment....if you read my post it is highly informed! I guess an ATPL and 8000hours jet plus 3000 turboprop counts for little informed comment.