
theorem Th31:
  for G1 being _Graph, G2 being LGraphComplement of G1
  for G being GraphUnion of G1, G2, v, w being Vertex of G
  ex e being object st e Joins v,w,G
proof
  let G1 be _Graph, G2 be LGraphComplement of G1;
  let G be GraphUnion of G1, G2, v, w be Vertex of G;
  the_Edges_of G1 misses the_Edges_of G2 by GLIB_012:def 7;
  then A1: G1 tolerates G2 by Th12;
  the_Vertices_of G1 = the_Vertices_of G2 by GLIB_012:def 7;
  then the_Vertices_of G1 = the_Vertices_of G1 \/ the_Vertices_of G2;
  then A2: v is Vertex of G1 & w is Vertex of G1 by A1, Th25;
  per cases;
  suppose ex e1 being object st e1 Joins v,w,G1;
    then consider e1 being object such that
      A3: e1 Joins v,w,G1;
    take e1;
    thus e1 Joins v,w,G by A3, GLIB_006:70;
  end;
  suppose not ex e1 being object st e1 Joins v,w,G1;
    then consider e2 being object such that
      A4: e2 Joins v,w,G2 by A2, GLIB_012:def 7;
    take e2;
    G is Supergraph of G2 by A1, Th26;
    hence e2 Joins v,w,G by A4, GLIB_006:70;
  end;
end;
