Assembly of a sealing system on an internal supporting element of the casing of a rotary machine cooled with a gaseous medium, between the casing and the shaft of the rotary machine, including a first sealing element striking during use against the internal supporting element and positioned towards an internal side of the casing, an intermediate ring connected during use with the internal supporting element, but positioned towards an external side of the casing and a seal box for gaseous medium, supported by the intermediate ring and positioned during use radially within the internal supporting element; where the sealing system is assembled in sequence, leapfrogging, via a device including : a first and a second half-ring which may be coupled together circumferentially to form a ring-shaped assembly support, including a sleeve portion which may be coupled to slide with the shaft and a flange portion which may be coupled with a rear element of the seal box; and a spacer half-ring with a first end which has an internal diameter greater than the diameter of the shaft and a collar for coaxial coupling with the first sealing element, and a second end equipped with a flange which may be coupled in contact with the outside of the sleeve portion of the assembly support.