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 Th22:
  for v,e being object, w being Vertex of G2, G1 being addEdge of G2,v,e,w
  for f1 being VColoring of G1, f2 being VColoring of G2, x being object
  st f1=f2+*(w.-->x) & v<>w & not x in rng f2 & f2 is proper holds f1 is proper
proof
  let v,e be object, w being Vertex of G2, G1 be addEdge of G2,v,e,w;
  let f1 be VColoring of G1, f2 be VColoring of G2, x be object;
  assume A1: f1 = f2 +* (w.-->x) & v <> w & not x in rng f2 & f2 is proper;
  per cases;
  suppose A2: not e in the_Edges_of G2;
    set G3 = the reverseEdgeDirections of G1, {e};
    A3: G3 is addEdge of G2,w,e,v by A2, GLIBPRE1:65;
    the_Vertices_of G1 = the_Vertices_of G3 by GLIB_007:4;
    then reconsider f3 = f1 as VColoring of G3;
    f3 is proper by A1, A3, Th21;
    hence thesis by Th18;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 11;
    hence thesis by A1, Th17;
  end;
end;
