reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring of G;
reserve g for EColoring of G;

theorem Th107:
  for G2 being c-ecolorable _Graph, v,e,w being object
  for G1 being addAdjVertex of G2,v,e,w holds G1 is (c+`1)-ecolorable
proof
  let G2 be c-ecolorable _Graph, v,e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  per cases;
  suppose v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then consider G9 being addVertex of G2,w such that
      A1: G1 is addEdge of G9,v,e,w by GLIB_006:125;
    thus G1 is (c+`1)-ecolorable by A1, Th106;
  end;
  suppose not v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    then consider G9 being addVertex of G2,v such that
      A2: G1 is addEdge of G9,v,e,w by GLIB_006:126;
    thus G1 is (c+`1)-ecolorable by A2, Th106;
  end;
  suppose not((v in the_Vertices_of G2 & not w in the_Vertices_of G2 &
      not e in the_Edges_of G2)or(not v in the_Vertices_of G2 &
      w in the_Vertices_of G2 & not e in the_Edges_of G2));
    then G1 == G2 by GLIB_006:def 12;
    then G1 is c-ecolorable by Th103;
    hence thesis by Th99, CARD_2:94;
  end;
end;
