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
  for G2 being edgeless _Graph, v,e being object, w being Vertex of G2
  for G1 being addAdjVertex of G2,v,e,w
  st not v in the_Vertices_of G2 holds G1.eChromaticNum() = 1
proof
  let G2 be edgeless _Graph, v, e be object, w be Vertex of G2;
  let G1 be addAdjVertex of G2,v,e,w;
  assume A1: not v in the_Vertices_of G2;
  set G3 = the reverseEdgeDirections of G1, {e};
  not e in the_Edges_of G2;
  then G3 is addAdjVertex of G2,w,e,v by GLIBPRE1:66;
  then G3.eChromaticNum() = 1 by A1, Th130;
  hence thesis by Th125;
end;
