let M be Pnet; :: thesis: ( (f_exit M) * ((f_exit M) \ (id (Elements M))) = {} & (f_enter M) * ((f_enter M) \ (id (Elements M))) = {} )
set S = id the carrier of M;
thus (f_exit M) * ((f_exit M) \ (id (Elements M))) = {} :: thesis: (f_enter M) * ((f_enter M) \ (id (Elements M))) = {}

proof

set R = (Flow M) | the carrier' of M;
A1: (id the carrier of M) * ((Flow M) | the carrier' of M) = {} by Th12;
(f_exit M) * ((f_exit M) \ (id (Elements M))) = (((Flow M) | the carrier' of M) \/ (id the carrier of M)) * ((Flow M) | the carrier' of M) by Th14
.= (((Flow M) | the carrier' of M) * ((Flow M) | the carrier' of M)) \/ ((id the carrier of M) * ((Flow M) | the carrier' of M)) by SYSREL:6
.= {} by A1, Th11 ;
hence (f_exit M) * ((f_exit M) \ (id (Elements M))) = {} ; :: thesis: verum

end;

set R = ((Flow M) ~) | the carrier' of M;
A2: (id the carrier of M) * (((Flow M) ~) | the carrier' of M) = {} by Th12;
(f_enter M) * ((f_enter M) \ (id (Elements M))) = ((((Flow M) ~) | the carrier' of M) \/ (id the carrier of M)) * (((Flow M) ~) | the carrier' of M) by Th14
.= ((((Flow M) ~) | the carrier' of M) * (((Flow M) ~) | the carrier' of M)) \/ ((id the carrier of M) * (((Flow M) ~) | the carrier' of M)) by SYSREL:6
.= {} by A2, Th11 ;
hence (f_enter M) * ((f_enter M) \ (id (Elements M))) = {} ; :: thesis: verum