
theorem
  for G1 being _Graph, G2 being DGraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & 3 c= G1.order()
  holds v1 is endvertex implies v2 is non endvertex
proof
  let G1 be _Graph, G2 be DGraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2 & 3 c= G1.order();
  assume v1 is endvertex;
  then consider u,w being Vertex of G1 such that
    A2: u<>v1 & w<>v1 & u <> w & u,v1 are_adjacent & not v1,w are_adjacent
    by A1, GLIBPRE0:95;
  not ex e being object st e DJoins v1,w,G1
  proof
    given e being object such that
      A3: e DJoins v1,w,G1;
    e Joins v1,w,G1 by A3, GLIB_000:16;
    hence contradiction by A2, CHORD:def 3;
  end;
  then consider e1 being object such that
    A4: e1 DJoins v1,w,G2 by A2, Th80;
  not ex e being object st e DJoins w,v1,G1
  proof
    given e being object such that
      A5: e DJoins w,v1,G1;
    e Joins v1,w,G1 by A5, GLIB_000:16;
    hence contradiction by A2, CHORD:def 3;
  end;
  then consider e2 being object such that
    A6: e2 DJoins w,v1,G2 by A2, Th80;
  A7: e1 <> e2 by A2, A4, A6, GLIB_000:125;
  reconsider w as Vertex of G2 by Th80;
  reconsider e1, e2 as set by TARSKI:1;
  e1 Joins v2,w,G2 & e2 Joins v2,w,G2 by A1, A4, A6, GLIB_000:16;
  then e1 in v2.edgesInOut() & e2 in v2.edgesInOut() by GLIB_000:64;
  then A8: {e1,e2} c= v2.edgesInOut() by ZFMISC_1:32;
  assume v2 is endvertex;
  then consider e9 being object such that
    A9: v2.edgesInOut() = {e9} & not e9 Joins v2,v2,G2 by GLIB_000:def 51;
  e1 = e9 & e2 = e9 by A8, A9, ZFMISC_1:20;
  hence contradiction by A7;
end;
