A solar eruptive event SOL2010-06-13 observed with SDO/AIA has been discussed in the contexts of the CME gebesis and an associated EUV transient in terms of a shock driven by the apparent CME rim. We have revealed in this event an erupting flux rope, studied its properties, and detected wave signatures inside the developing CME. These findings have allowed us to establish new features in the genesis of the CME and associated EUV wave and to reconcile all of the episodes into a causally-related sequence. (1) A hot 11 MK flux rope developed from a compact filament, accelerated up to 3 km/s$^2$ 1 min before a hard X-ray burst and earlier than other structures, reached 420 km/s, and decelerated to 50 km/s. (2) The CME development was driven by the flux rope. Closed structures above the rope got sequentially involved in the expansion from below upwards, came closer together, and disappeared to reveal their envelope, the rim, which became the outer boundary of the cavity. The rim was associated with the separatrix surface of a magnetic domain, which contained the pre-eruptive filament. (3) The rim formation was associated with a successive compression of the upper active-region structures into the CME frontal structure (FS). When the rim was formed, it resembled a piston. (4) The disturbance responsible for the CME formation was excited by the flux rope inside the rim, and then propagated outward. EUV structures at different heights started to accelerate, when their trajectories in the distance-time diagram were crossed by the front of this disturbance. (5) Outside the rim and FS, the disturbance propagated like a blast wave, manifesting in a type II radio burst and a leading part of the EUV transient. Its main, trailing part was the FS, which consisted of swept-up 2 MK loops enveloping the rim. The wave decelerated and decayed soon, being not driven by the trailing piston, which slowed down.
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