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 Th46:
  not (the carrier' of G) in rng p9 & vs = vs9 & vs9
  is_vertex_seq_of p9 implies vs is_vertex_seq_of p9
proof
  set G9 = AddNewEdge(v1, v2);
  set S = the Source of G;
  set T = the Target of G;
  set E = the carrier' of G;
  set S9 = the Source of G9;
  set T9 = the Target of G9;
  the carrier' of G9 = E \/ {E} by Def7;
  then
A1: rng p9 c= E \/ {E} by FINSEQ_1:def 4;
  assume
A2: not (the carrier' of G) in rng p9;
A3: rng p9 c= E
  proof
    let x be object;
    assume
A4: x in rng p9;
    then x in E or x in {E} by A1,XBOOLE_0:def 3;
    hence thesis by A2,A4,TARSKI:def 1;
  end;
  assume that
A5: vs = vs9 and
A6: vs9 is_vertex_seq_of p9;
  thus vs is_vertex_seq_of p9
  proof
    thus
A7: len vs = len p9 + 1 by A5,A6;
    let n be Nat;
    assume that
A8: 1<=n and
A9: n<=len p9;
    set e = p9.n;
    reconsider vn9 = vs9/.n, vn19 = vs9/.(n+1) as Vertex of G9;
    p9.n joins vs9/.n, vs9/.(n+1) by A6,A8,A9;
    then
A10: S9.e = vn9 & T9.e = vn19 or S9.e = vn19 & T9.e = vn9;
    reconsider vn = vs/.n, vn1 = vs/.(n+1) as Vertex of G;
    1 <= n+1 & n+1 <= len vs by A7,A9,NAT_1:11,XREAL_1:6;
    then
A11: n+1 in dom vs by FINSEQ_3:25;
    then
A12: vn1 = vs.(n+1) by PARTFUN1:def 6
      .= vn19 by A5,A11,PARTFUN1:def 6;
    n in dom p9 by A8,A9,FINSEQ_3:25;
    then e in rng p9 by FUNCT_1:def 3;
    then
A13: S9.e = S.e & T9.e = T.e by A3,Th35;
    len p9 <= len vs by A7,NAT_1:11;
    then n <= len vs by A9,XXREAL_0:2;
    then
A14: n in dom vs by A8,FINSEQ_3:25;
    then vn = vs.n by PARTFUN1:def 6
      .= vn9 by A5,A14,PARTFUN1:def 6;
    hence thesis by A10,A13,A12;
  end;
end;
