
theorem Th68:
  for G1 being _Graph, G2 being removeLoops of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & v1 is endvertex holds v2 is endvertex
proof
  let G1 be _Graph, G2 be removeLoops of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2 & v1 is endvertex;
  then consider e being object such that
    A2: v1.edgesInOut() = {e} & not e Joins v1,v1,G1 by GLIB_000:def 51;
  reconsider e as set by TARSKI:1;
  e in {e} by TARSKI:def 1;
  then consider w1 being Vertex of G1 such that
    A3: e Joins v1,w1,G1 by A2, GLIB_000:64;
  A4: not e in G1.loops() by A2, A3, Th46;
  A5: the_Vertices_of G2 = the_Vertices_of G1 &
    the_Edges_of G2 = the_Edges_of G1 \ G1.loops() by GLIB_000:53;
  then reconsider w2 = w1 as Vertex of G2;
  e in the_Edges_of G1 by A3, GLIB_000:def 13;
  then e in the_Edges_of G2 by A4, A5, XBOOLE_0:def 5;
  then e Joins v2,w2,G2 by A1, A3, GLIB_000:73;
  then e in v2.edgesInOut() by GLIB_000:64;
  hence thesis by A1, GLIB_000:def 49, GLIB_000:84;
end;
