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 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;
