reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p, p1, p2 for Path of G,
  vs, vs1, vs2 for FinSequence of the carrier of G,
  e, X for set,
  n, m for Nat;

theorem Th15:
  vs is_vertex_seq_of c & e in rng c implies (the Target of G).e
  in rng vs & (the Source of G).e in rng vs
proof
  assume that
A1: vs is_vertex_seq_of c and
A2: e in rng c;
  c is FinSequence of the carrier' of G by MSSCYC_1:def 1;
  then
A3: rng c c= the carrier' of G by FINSEQ_1:def 4;
  then reconsider e9 = e as Element of the carrier' of G by A2;
  reconsider t = (the Target of G).e, s = (the Source of G).e as Vertex of G
  by A2,A3,FUNCT_2:5;
  e9 in rng c by A2;
  then t in G-VSet rng c & s in G-VSet rng c;
  hence thesis by A1,A2,GRAPH_2:31,RELAT_1:38;
end;
