reserve E,V for set, G,G1,G2 for _Graph, c,c1,c2 for Cardinal, n for Nat;
reserve f for VColoring 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.vChromaticNum() = 2
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;
  now
    thus G1 is 2-vcolorable by Th38;
    let c;
    assume A2: G1 is c-vcolorable;
    assume not 2 c= c;
    then c in {0,1} by ORDINAL1:16, CARD_1:50;
    then per cases by TARSKI:def 2;
    suppose c = 0;
      hence contradiction by A2;
    end;
    suppose A3: c = 1;
      not e in the_Edges_of G2;
      then e DJoins v,w,G1 by A1, GLIB_006:132;
      then e in the_Edges_of G1 by GLIB_000:def 14;
      hence contradiction by A2, A3;
    end;
  end;
  hence thesis by Th58;
end;
