
theorem Th54:
  for C being non _trivial Cycle-like _Graph, v being Vertex of C
  for P being removeVertex of C,v holds P is _finite Path-like
proof
  let C be non _trivial Cycle-like _Graph, v be Vertex of C;
  let P be removeVertex of C,v;
  thus P is _finite;
  A1: the_Vertices_of P = the_Vertices_of C \ {v} by GLIB_000:47;
  now
    given W being Walk of P such that
      A2: W is Cycle-like;
    reconsider W9 = W as Walk of C by GLIB_001:167;
    W9 is Cycle-like by A2, GLIB_006:24;
    then the_Vertices_of C = W9.vertices() by Th39
      .= W.vertices() by GLIB_001:98;
    then v in W.vertices();
    hence contradiction by A1, ZFMISC_1:56;
  end;
  then A3: P is acyclic by GLIB_002:def 2;
  now
    let u,w be Vertex of P;
    consider C9 being Walk of C such that
      A4: C9 is Cycle-like by GLIB_002:def 2;
    the_Vertices_of P c= the_Vertices_of C;
    then the_Vertices_of P c= C9.vertices() by A4, Th39;
    then A5: u in C9.vertices() & w in C9.vertices() by TARSKI:def 3;
    v in {v} by TARSKI:def 1;
    then u <> v & w <> v by A1, XBOOLE_0:def 5;
    then consider W being Path of C such that
      A6: W is_Walk_from u,w & not v in W.vertices() by A4, A5, Th5;
    reconsider W as Walk of P by A6, GLIB_001:171;
    take W;
    thus W is_Walk_from u,w by A6, GLIB_001:19;
  end;
  then P is connected by GLIB_002:def 1;
  hence P is Tree-like by A3;
  let u be Vertex of P;
  reconsider w = u as Vertex of C by A1, XBOOLE_0:def 5;
  u.degree() <= w.degree() by GLIB_000:81;
  then u.degree() <= 2 by GLIB_016:def 4;
  hence u.degree() c= 2 by FIELD_5:3;
end;
