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;
