
theorem
  for G1 being _Graph, G2 being GraphComplement of G1
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v1 = v2 & 4 c= G1.order()
  holds v1 is endvertex implies v2 is non endvertex
proof
  let G1 be _Graph, G2 be GraphComplement of G1;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2 & 4 c= G1.order();
  assume v1 is endvertex;
  then consider x,y,z being Vertex of G1 such that
    A2: v1 <> x & v1 <> y & v1 <> z & x <> y & x <> z & y <> z and
    A3: v1,x are_adjacent & not v1,y are_adjacent & not v1,z are_adjacent
    by A1, GLIBPRE0:96;
  reconsider u = y, w = z as Vertex of G2 by Th99;
  v2,u are_adjacent & v2,w are_adjacent by A1, A2, A3, Th99;
  hence v2 is non endvertex by A2, GLIBPRE0:94;
end;
