
theorem Th41:
  for G2 being _trivial edgeless _Graph, v being Vertex of G2, e being object
  for G1 being addEdge of G2,v,e,v holds G1 is Cycle-like
proof
  let G2 be _trivial edgeless _Graph, v be Vertex of G2, e be object;
  let G1 be addEdge of G2,v,e,v;
  A1: the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 11
    .= {e};
  now
    let e1,e2,v1,v2 be object;
    assume e1 DJoins v1,v2,G1 & e2 DJoins v1,v2,G1;
    then e1 in the_Edges_of G1 & e2 in the_Edges_of G1 by GLIB_000:def 14;
    hence e1 = e2 by A1, ZFMISC_1:def 10;
  end;
  then A2: G1 is non-Dmulti by GLIB_000:def 21;
  not e in the_Edges_of G2;
  then G1 is non loopless by GLIB_006:113;
  hence thesis by A2;
end;
