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 v,e,x being object, w being Vertex of G2
  for G1 being addAdjVertex of G2,v,e,w, f being VColoring of G2
  st not e in the_Edges_of G2 & not v in the_Vertices_of G2
  holds f +* (v .--> x) is VColoring of G1
proof
  let v,e,x be object, w be Vertex of G2, G1 be addAdjVertex of G2,v,e,w;
  let f be VColoring of G2;
  assume not e in the_Edges_of G2 & not v in the_Vertices_of G2;
  then the_Vertices_of G1 = the_Vertices_of G2 \/ {v} by GLIB_006:def 12
    .= dom f \/ {v} by PARTFUN1:def 2
    .= dom f \/ dom{[v,x]} by RELAT_1:9
    .= dom f \/ dom(v .--> x) by FUNCT_4:82
    .= dom(f +* (v .--> x)) by FUNCT_4:def 1;
  hence thesis by RELAT_1:def 18, PARTFUN1:def 2;
end;
