
theorem
  for G being Tree-like _Graph, v1,v2 being Vertex of G, e being object
  for H being addEdge of G,v1,e,v2 st not e in the_Edges_of G holds
    H is non acyclic & for W1, W2 being Walk of H
      st W1 is Cycle-like & W2 is Cycle-like holds W1.edges() = W2.edges()
proof
  let G be Tree-like _Graph, v1,v2 be Vertex of G, e be object;
  let H be addEdge of G,v1,e,v2;
  assume A1: not e in the_Edges_of G;
  v2 in the_Vertices_of G;
  then v2 in G.reachableFrom v1 by GLIB_002:16;
  hence H is non acyclic by A1, GLIB_006:119;
  let W1, W2 be Walk of H;
  assume A2: W1 is Cycle-like & W2 is Cycle-like;
  A3: e in W1.edges()
  proof
    assume not e in W1.edges();
    then reconsider W7 = W1 as Walk of G by GLIB_006:109;
    W7 is Cycle-like by A2, GLIB_006:24;
    hence contradiction by GLIB_002:def 2;
  end;
  A4: e in W2.edges()
  proof
    assume not e in W2.edges();
    then reconsider W8 = W2 as Walk of G by GLIB_006:109;
    W8 is Cycle-like by A2, GLIB_006:24;
    hence contradiction by GLIB_002:def 2;
  end;
  e DJoins v1,v2,H by A1, GLIB_006:105;
  then A5: e Joins v1,v2,H by GLIB_000:16;
  consider W3 being Path of H such that
    A6: W3 is_Walk_from v1,v2 & W3.edges() = W1.edges() \ {e} and
    not e in H.loops() implies W3 is open by A2, A3, A5, Th32;
  consider W4 being Path of H such that
    A7: W4 is_Walk_from v1,v2 & W4.edges() = W2.edges() \ {e} and
    not e in H.loops() implies W4 is open by A2, A4, A5, Th32;
  e in {e} by TARSKI:def 1;
  then A8: not e in W3.edges() & not e in W4.edges() by A6, A7, XBOOLE_0:def 5;
  then reconsider W5 = W3 as Walk of G by GLIB_006:109;
  reconsider W6 = W4 as Walk of G by A8, GLIB_006:109;
  A9: W5 is_Walk_from v1,v2 & W5 is Path-like by A6, GLIB_001:19, GLIB_006:23;
  A10: W6 is_Walk_from v1,v2 & W6 is Path-like by A7, GLIB_001:19, GLIB_006:23;
  A11: W3.edges()
     = G.pathBetween(v1,v2).edges() by A9, HELLY:def 2, GLIB_001:110
    .= W4.edges() by A10, HELLY:def 2, GLIB_001:110;
  thus W1.edges() = W3.edges() \/ {e} by A3, A6, ZFMISC_1:116
    .= W2.edges() by A4, A7, A11, ZFMISC_1:116;
end;
