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, {?,m i?1 } where m i?1 is the (i ? 1) th message broadcasted by p, except p that sets Mes p [p] to m i where m i is the i th message it broadcasts
, {?,m k?1 } where m k?1 is the (k ? 1) th message broadcasted by p, except p that sets Mes p [p] to m k where m k is the k th message it broadcasts
, Mes p [p] is set to m when p starts its k th broadcast. Then, (i) Mes p [p] remains equal to m until p terminates the broadcast and, from the termination of the broadcast to the start of the next one
, And so, it has suspected q or it has wait a Decide, ?? message or a Start message from q. If q is suspected in the (i ? 1) th R-PROPOSED(?) by the failure detector of p, q remains suspected in the i th R-PROPOSED(?) and so p does not send messages to q. If it has wait a Decide, ?? message or a Start message from q, by induction assumption part 2 and 3 this message can only send by q during the (i ? 1) th R-PROPOSED(?) And so q has finished all j th R-PROPOSED with j < i ? 1 and has executed the i ? 1 th R-PROPOSED(?) until at least Line 13, contradicting the assumption. To conclude the proof of the first part we have to show that no message R-x, ?? sent by process p while executed the first R-PROPOSED(v)