The assume/flex take off takes into consideration of an engine failure and its climb performance.
So there is no need to add additional thrust in the event of an engine failure after takeoff unless you really need!!
In a turbo prob maybe this might be the case because the dead engine will produce alot of drag and props are not as powerful as a turbine engine.
But from a jet point of view with an engine failue you rotate at a slower rate and pitch attitude usually around 12.5 degrees (B737).
Assuming your maintaining runway heading, at your designated accel altitude you select flaps up speed and accelerate selecting the flaps up on schedule. Once the flaps are up and your still climbing you would ask for level change (B737) and ask for max continuous thrust.
Thats the way I have been taught, some suggest getting the autopilot in straight away to reduce workload, some say dont bother till level flight so you can trim the aircraft easily instead if doing it during a climb and constantly re-trimming as you speed up.