
theorem Th111:
  for G1 being _Graph, G2 being removeParallelEdges of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 holds v1 is isolated iff v2 is isolated
proof
  let G1 be _Graph, G2 be removeParallelEdges of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  hence v1 is isolated implies v2 is isolated by GLIB_000:83;
  assume v2 is isolated;
  then A2: v2.edgesInOut() = {} by GLIB_000:def 49;
  v1.edgesInOut() = {}
  proof
    assume v1.edgesInOut() <> {};
    then consider e0 being object such that
      A3: e0 in v1.edgesInOut() by XBOOLE_0:def 1;
    consider w being Vertex of G1 such that
      A4: e0 Joins v1,w,G1 by A3, GLIB_000:64;
    consider E1 being RepEdgeSelection of G1 such that
      A5: G2 is inducedSubgraph of G1, the_Vertices_of G1, E1 by Def7;
    consider e1 being object such that
      A6: e1 Joins v1,w,G1 & e1 in E1 and
      for e9 being object st e9 Joins v1,w,G1 & e9 in E1 holds e9 = e1
      by A4, Def5;
    the_Vertices_of G1 c= the_Vertices_of G1 &
      the_Edges_of G1 = G1.edgesBetween(the_Vertices_of G1) by GLIB_000:34;
    then A7: the_Vertices_of G1 = the_Vertices_of G2 &
      the_Edges_of G2 = E1 by A5, GLIB_000:def 37;
    then reconsider w as Vertex of G2;
    A8: e1 Joins v2,w,G2 by A1, A6, A7, GLIB_000:73;
    e1 is set by TARSKI:1;
    hence thesis by A2, A8, GLIB_000:64;
  end;
  hence thesis by GLIB_000:def 49;
end;
