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;
reserve t for TColoring of G;

theorem
  for v being Vertex of G2, e,w being object, G1 being addEdge of G2,v,e,w
  for t1 being TColoring of G1, t2 being TColoring of G2, x,y being object
  st not e in the_Edges_of G2 & v <> w &
    t1_V = t2_V +* (w .--> x) & t1_E = t2_E +* (e .--> y) &
    {x,y} misses (rng t2_V)\/rng t2_E & x <> y & t2 is proper
  holds t1 is proper
proof
  let v be Vertex of G2, e,w be object, G1 be addEdge of G2,v,e,w;
  let t1 be TColoring of G1, t2 be TColoring of G2, x,y be object;
  assume that A1: not e in the_Edges_of G2 & v <> w and
    A2: t1_V = t2_V +* (w .--> x) & t1_E = t2_E +* (e .--> y) and
    A3: {x,y} misses (rng t2_V)\/rng t2_E & x <> y & t2 is proper;
  set G3 = the reverseEdgeDirections of G1, {e};
  A4: G3 is addEdge of G2,w,e,v by A1, GLIBPRE1:65;
  the_Vertices_of G1 = the_Vertices_of G3 by GLIB_007:4;
  then reconsider f3 = t1_V as VColoring of G3;
  the_Edges_of G1 = the_Edges_of G3 by GLIB_007:4;
  then reconsider g3 = t1_E as EColoring of G3;
  reconsider t3 = [f3,g3] as TColoring of G3;
  t3 is proper by A1, A2, A3, A4, Th156;
  hence thesis by Th153;
end;
