let N be e_net; ( e_adjac N c= [:(e_shore N),(e_shore N):] & e_flow N c= [:(e_shore N),(e_shore N):] )
A1:
( the entrance of N \/ the escape of N) \ (id the carrier of N) c= the entrance of N \/ the escape of N
by XBOOLE_1:36;
A2:
the escape of N c= [: the carrier of N, the carrier of N:]
by Def1;
A3:
the entrance of N c= [: the carrier of N, the carrier of N:]
by Def1;
then
the entrance of N ~ c= [: the carrier of N, the carrier of N:]
by SYSREL:4;
then A4:
( id the carrier of N c= [: the carrier of N, the carrier of N:] & ( the entrance of N ~) \/ the escape of N c= [: the carrier of N, the carrier of N:] )
by A2, RELSET_1:13, XBOOLE_1:8;
( id ( the carrier of N \ (rng the entrance of N)) c= id the carrier of N & id the carrier of N c= [: the carrier of N, the carrier of N:] )
by RELSET_1:13, SYSREL:15, XBOOLE_1:36;
then A5:
id ( the carrier of N \ (rng the entrance of N)) c= [: the carrier of N, the carrier of N:]
by XBOOLE_1:1;
the entrance of N \/ the escape of N c= [: the carrier of N, the carrier of N:]
by A3, A2, XBOOLE_1:8;
then
( the entrance of N \/ the escape of N) \ (id the carrier of N) c= [: the carrier of N, the carrier of N:]
by A1, XBOOLE_1:1;
hence
( e_adjac N c= [:(e_shore N),(e_shore N):] & e_flow N c= [:(e_shore N),(e_shore N):] )
by A5, A4, XBOOLE_1:8; verum