theorem Th45:
  not (the carrier' of G) in rng p9 implies p9 is Path of G
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;
  p9 is Chain of G
  proof
    thus p9 is FinSequence of the carrier' of G by A3,FINSEQ_1:def 4;
    consider vs9 being FinSequence of the carrier of G9 such that
A5: vs9 is_vertex_seq_of p9 by MSSCYC_1:def 1;
    reconsider vs = vs9 as FinSequence of the carrier of G by Def7;
    take vs;
    thus vs is_vertex_seq_of p9
    proof
      thus
A6:   len vs = len p9 + 1 by A5;
      let n be Nat;
      assume that
A7:   1<=n and
A8:   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 A5,A7,A8;
      then
A9:   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 A6,A8,NAT_1:11,XREAL_1:6;
      then
A10:  n+1 in dom vs by FINSEQ_3:25;
      then
A11:  vn1 = vs.(n+1) by PARTFUN1:def 6
        .= vn19 by A10,PARTFUN1:def 6;
      n in dom p9 by A7,A8,FINSEQ_3:25;
      then e in rng p9 by FUNCT_1:def 3;
      then
A12:  S9.e = S.e & T9.e = T.e by A3,Th35;
      len p9 <= len vs by A6,NAT_1:11;
      then n <= len vs by A8,XXREAL_0:2;
      then
A13:  n in dom vs by A7,FINSEQ_3:25;
      then vn = vs.n by PARTFUN1:def 6
        .= vn9 by A13,PARTFUN1:def 6;
      hence thesis by A9,A12,A11;
    end;
  end;
  then reconsider p99 = p9 as Chain of G;
  p99 is one-to-one;
  hence thesis;
end;
