
theorem Th43:
  for C being Cycle-like _Graph, e being Edge of C
  for P being removeEdge of C,e holds P is _finite Path-like
proof
  let C be Cycle-like _Graph, e be Edge of C;
  let P be removeEdge of C,e;
  thus P is _finite;
  A1: the_Edges_of P = the_Edges_of C \ {e} by GLIB_000:51;
  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_Edges_of C = W9.edges() by Th39
      .= W.edges() by GLIB_001:110;
    then e in W.edges();
    hence contradiction by A1, ZFMISC_1:56;
  end;
  then A3: P is acyclic by GLIB_002:def 2;
  now
    consider W being Walk of C such that
      A4: W is Cycle-like by GLIB_002:def 2;
    take W;
    the_Edges_of C = W.edges() by A4, Th39;
    hence W is Cycle-like & e in W.edges() by A4;
  end;
  then P is connected by GLIB_002:5;
  hence P is Tree-like by A3;
  let v be Vertex of P;
  reconsider w = v as Vertex of C by GLIB_000:def 33;
  v.degree() <= w.degree() by GLIB_000:81;
  then v.degree() <= 2 by GLIB_016:def 4;
  hence v.degree() c= 2 by FIELD_5:3;
end;
