
theorem
  for G1 being _Graph, v being object, G2 being addVertex of G1,v
  for G3 being GraphComplement of G1 st not v in the_Vertices_of G1
  ex G4 being addAdjVertexAll of G3,v st G4 is GraphComplement of G2
proof
  let G1 be _Graph, v be object, G2 be addVertex of G1, v;
  let G3 be GraphComplement of G1;
  assume A1: not v in the_Vertices_of G1;
  the_Edges_of G1 misses the_Edges_of G3 by Th98;
  then A2: the_Edges_of G2 misses the_Edges_of G3 by GLIB_006:def 10;
  {} is Subset of the_Vertices_of G1 &
    G2 is addAdjVertexAll of G1,v,{} by XBOOLE_1:2, GLIBPRE0:51;
  then consider G4 being addAdjVertexAll of G3,v,the_Vertices_of G1 \ {}
    such that A3: G4 is GraphComplement of G2 by A1, A2, Th105;
  reconsider G4 as addAdjVertexAll of G3,v by Th98;
  take G4;
  thus thesis by A3;
end;
