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 Th36:
  vs9 = vs & vs is_vertex_seq_of c implies vs9 is_vertex_seq_of c
proof
  assume that
A1: vs9 = vs and
A2: vs is_vertex_seq_of c;
  thus len vs9 = len c + 1 by A1,A2;
  let n be Nat;
  set T = the Target of G;
  set S = the Source of G;
  set v = c.n;
  set x = vs/.n;
  set y = vs/.(n+1);
  assume
A3: 1<=n & n<=len c;
  then c.n joins vs/.n, vs/.(n+1) by A2;
  then
A4: S.v = x & T.v = y or S.v = y & T.v = x;
  set G9 = AddNewEdge(v1, v2);
  set S9 = the Source of G9;
  set T9 = the Target of G9;
A5: the carrier of G = the carrier of G9 by Def7;
  c is FinSequence of the carrier' of G by MSSCYC_1:def 1;
  then
A6: rng c c= the carrier' of G by FINSEQ_1:def 4;
  n in dom c by A3,FINSEQ_3:25;
  then c.n in rng c by FUNCT_1:def 3;
  then S9.v = S.v & T.v = T9.v by A6,Th35;
  hence thesis by A1,A5,A4;
end;
