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 g1 being EColoring of G1, g2 being EColoring of G2, x being object
  st g1 = g2 +* (e .--> x) & not x in rng g2 &
    not e in the_Edges_of G2 & not w in the_Vertices_of G2
  holds g2 is proper implies g1 is proper
proof
  let v be Vertex of G2, e,w be object, G1 be addAdjVertex of G2,v,e,w;
  let g1 be EColoring of G1, g2 be EColoring of G2, x be object;
  assume that A1: g1 = g2 +* (e .--> x) & not x in rng g2 and
    A2: not e in the_Edges_of G2 & not w in the_Vertices_of G2 and
    A3: g2 is proper;
  consider G9 being addVertex of G2,w such that
    A4: G1 is addEdge of G9,v,e,w by A2, GLIB_006:125;
  A5: the_Edges_of G9 = the_Edges_of G2 by GLIB_006:def 10;
  then reconsider g9 = g2 as EColoring of G9;
  g9 is proper by A3, Th92;
  hence thesis by A1, A2, A4, A5, Th93;
end;
