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;
reserve t for TColoring 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.tChromaticNum() = 3
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;
  not e in the_Edges_of G2;
  then G1 is non edgeless by A1, GLIB_006:132, GLIB_008:50;
  then A2: 3 c= G1.tChromaticNum() by Th196;
  G1 is 3-tcolorable by Th174;
  then G1.tChromaticNum() c= 3 by Th189;
  hence thesis by A2, XBOOLE_0:def 10;
end;
