An engine is flat rated up to a certain temp. This means it's performance is unchanged by a increase in temp up to it's flat rated temp. Eg. Engine A is a 600HP enigine flat rated to 30deg. This means it will provide 600HP up to and including 30deg. Thereafter it will loose power with a Temp rise.
Assumed Tempreture take off (Boeing) and Flex take off (Airbus) are the same thing. Let's say an engine is designed for 25000 pounds of thrust under ISA conditions. An a/c needs to take off from Heathrow (sea level) for a flight to Paris (short hop, little fuel, light a/c). This aircraft has a Max design take off weight of 70 000kg. On this day it only weighs 55 000kg though. The pilot calculates that at this weight the a/c could get airborne (legally) should the temp rise to say 65deg. We then limit the take off thrust N1 or EPR to provide the thrust that would only be available at 65deg (eg. 23500 pounds thrust). Max design thrust is available if the Thrust levers are advanced by the pilot.
I speak under correction regarding Derated take off thrust. The B747-400 can do a 10% or 25% derate take off. Here the engine is detuned via the Fuel control unit by the percentage requested.
In both derate and assumed temp the a/c must still comply with all take and climb performance requierments for that paticular runway and subsequent climb segments.