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 v being Vertex of G2, e,w being object, G1 being addAdjVertex of G2,v,e,w
  for g being EColoring of G2, x being object
  st not e in the_Edges_of G2 & not w in the_Vertices_of G2
  holds g +* (e .--> x) is EColoring of G1
proof
  let v be Vertex of G2, e,w be object, G1 be addAdjVertex of G2,v,e,w;
  let g be EColoring of G2, x be object;
  assume A1: not e in the_Edges_of G2 & not w in the_Vertices_of G2;
  A2: dom(e.-->x) = dom {[e,x]} by FUNCT_4:82
    .= {e} by RELAT_1:9;
  dom(g+*(e.-->x)) = dom g \/ dom(e.-->x) by FUNCT_4:def 1
    .= the_Edges_of G2 \/ {e} by A2, PARTFUN1:def 2
    .= the_Edges_of G1 by A1, GLIB_006:def 12;
  hence thesis by RELAT_1:def 18, PARTFUN1:def 2;
end;
