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 Th130:
  for G2 being edgeless _Graph, v being Vertex of G2, e,w being object
  for G1 being addAdjVertex of G2,v,e,w
  st not w in the_Vertices_of G2 holds G1.eChromaticNum() = 1
proof
  let G2 be edgeless _Graph, v be Vertex of G2, e,w be object;
  let G1 be addAdjVertex of G2,v,e,w;
  assume A1: not w in the_Vertices_of G2;
  now
    let c;
    assume A2: G1 is c-ecolorable;
    assume not 1 c= c;
    then c in {0} by CARD_1:49, ORDINAL1:16;
    then A3: c = 0 by TARSKI:def 1;
    e in the_Edges_of G2 \/ {e} by TARSKI:def 1;
    then e in the_Edges_of G1 by A1, GLIB_006:def 12;
    hence contradiction by A2, A3;
  end;
  hence thesis by Th122;
end;
