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;
reserve G for finite Graph,
  v for Vertex of G,
  c for Chain of G,
  vs for FinSequence of the carrier of G,
  X1, X2 for set;
reserve G for Graph,
  v, v1, v2 for Vertex of G,
  c for Chain of G,
  p for Path of G,
  vs for FinSequence of the carrier of G,
  v9 for Vertex of AddNewEdge(v1, v2),
  p9 for Path of AddNewEdge(v1, v2),
  vs9 for FinSequence of the carrier of AddNewEdge(v1, v2);

theorem Th37:
  c is Chain of AddNewEdge(v1, v2)
proof
  set G9 = AddNewEdge(v1, v2);
  consider p being FinSequence of the carrier of G such that
A1: p is_vertex_seq_of c by GRAPH_2:33;
  c is FinSequence of the carrier' of G by MSSCYC_1:def 1;
  then
A2: rng c c= the carrier' of G by FINSEQ_1:def 4;
  the carrier' of G9 = (the carrier' of G) \/ {the carrier' of G} by Def7;
  then the carrier' of G c= the carrier' of G9 by XBOOLE_1:7;
  then rng c c= the carrier' of G9 by A2;
  hence c is FinSequence of the carrier' of G9 by FINSEQ_1:def 4;
  reconsider p9 = p as FinSequence of the carrier of G9 by Def7;
  take p9;
  thus thesis by A1,Th36;
end;
