reserve x,y for set;
reserve N for PT_net_Str;

theorem Th5:
  for N being Pnet holds [x,y] in Flow N & x in the
  carrier' of N implies y in the carrier of N
proof
  let N be Pnet;
  assume that
A1: [x,y] in Flow N and
A2: x in the carrier' of N;
A3: not x in the carrier of N by A2,Th4;
  (Flow N) c= [:the carrier of N, the carrier' of N:] \/ [:the
  carrier' of N, the carrier of N:] by Def2;
  then [x,y] in [:the carrier of N, the carrier' of N:] or [x,y] in [:the
  carrier' of N, the carrier of N:] by A1,XBOOLE_0:def 3;
  hence thesis by A3,ZFMISC_1:87;
end;
