
theorem
  for G1 being _Graph, v being object, G2 being addAdjVertexAll of G1,v
  for G3 being GraphComplement of G1, G4 being addVertex of G3,v
  st not v in the_Vertices_of G1 & the_Edges_of G2 misses the_Edges_of G3
  holds G4 is GraphComplement of G2
proof
  let G1 be _Graph, v be object, G2 be addAdjVertexAll of G1, v;
  let G3 be GraphComplement of G1, G4 be addVertex of G3, v;
  assume A1: not v in the_Vertices_of G1 &
    the_Edges_of G2 misses the_Edges_of G3;
  the_Vertices_of G1 c= the_Vertices_of G1;
  then consider G9 being
      addAdjVertexAll of G3,v,the_Vertices_of G1 \ the_Vertices_of G1 such that
    A2: G9 is GraphComplement of G2 by A1, Th105;
  the_Vertices_of G1 \ the_Vertices_of G1 = {} by XBOOLE_1:37;
  then G9 is addVertex of G3,v by GLIB_007:55;
  then G4 == G9 by GLIB_006:77;
  hence thesis by A2, Th97;
end;
