
theorem Th45:
  for C being non _trivial Cycle-like _Graph
  for v1, v2 being Vertex of C, e being Edge of C st e DJoins v1,v2,C
  ex P being non _trivial _finite Path-like _Graph
  st not e in the_Edges_of P & C is addEdge of P,v1,e,v2 &
    Endvertices P = {v1,v2}
proof
  let C be non _trivial Cycle-like _Graph;
  let v1, v2 be Vertex of C, e be Edge of C;
  assume A1: e DJoins v1,v2,C;
  then A2: v1 = (the_Source_of C).e & v2 = (the_Target_of C).e &
    e in the_Edges_of C by GLIB_000:def 14;
  take P = the removeEdge of C,e;
  the_Edges_of P = the_Edges_of C \ {e} by GLIB_000:51;
  hence A3: not e in the_Edges_of P by ZFMISC_1:56;
  reconsider w1 = v1, w2 = v2 as Vertex of P by GLIB_000:def 33;
  thus C is addEdge of P,v1,e,v2 by A2, GLIB_008:37;
  then A4: C is addEdge of P,w1,e,w2;
  A5: v1 <> v2 by A1, GLIB_000:136;
  1 + 1 = v1.degree() by GLIB_016:def 4
    .= w1.degree() +` 1 by A3, A4, A5, GLIBPRE0:47;
  then w1 is endvertex by GLIB_000:174;
  then A6: w1 in Endvertices P by GLIB_006:56;
  1 + 1 = v2.degree() by GLIB_016:def 4
    .= w2.degree() +` 1 by A3, A4, A5, GLIBPRE0:48;
  then w2 in Endvertices P by GLIB_000:174, GLIB_006:56;
  then A7: {w1,w2} c= Endvertices P by A6, ZFMISC_1:32;
  consider x,y being Vertex of P such that
    A8: x <> y & Endvertices P = {x,y} by Th36;
  (w1 = x or w1 = y) & (w2 = x or w2 = y) by A7, A8, ZFMISC_1:22;
  hence thesis by A5, A8;
end;
