
theorem Th8:
  for G being complete _Graph st 3 c= G.order() holds G is non acyclic
proof
  let G be complete _Graph;
  set u = the Vertex of G;
  assume 3 c= G.order();
  then consider v,w being object such that
    A1: v in the_Vertices_of G & w in the_Vertices_of G and
    A2: v <> u & w <> u & v <> w by SCMYCIEL:5;
  reconsider v,w as Vertex of G by A1;
  consider e1 being object such that
    A3: e1 Joins u,v,G by A2, CHORD:def 3, CHORD:def 6;
  consider e2 being object such that
    A4: e2 Joins v,w,G by A2, CHORD:def 3, CHORD:def 6;
  consider e3 being object such that
    A5: e3 Joins w,u,G by A2, CHORD:def 3, CHORD:def 6;
  set W1 = G.walkOf(u,e1,v), W2 = W1.addEdge(e2), W3 = W2.addEdge(e3);
  A6: W1.first() = u & W1.last() = v by A3, GLIB_001:15;
  then A7: W2.first() = u & W2.last() = w by A4, GLIB_001:63;
  A9: W1 is open by A2, A6, GLIB_001:def 24;
  W1.vertices() = {u,v} by A3, GLIB_001:91;
  then not w in W1.vertices() by A2, TARSKI:def 2;
  then A10: W2 is Path-like by A4, A6, A9, GLIB_001:151;
  A11: W2 is open by A2, A7, GLIB_001:def 24;
  len W2 = len W1 + 2 by A4, A6, GLIB_001:64
    .= 3 + 2 by A3, GLIB_001:14;
  then W3 is Cycle-like by A5, A7, A10, A11, CHORD:33;
  hence G is non acyclic by GLIB_002:def 2;
end;
