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 Th24:
  for v,e,w being object, G1 being addAdjVertex of G2,v,e,w
  for f1 being VColoring of G1, f2 being VColoring of G2, x being object
  st not w in the_Vertices_of G2 & f1 = f2 +* (w .--> x) & x <> f2.v
  holds f2 is proper implies f1 is proper
proof
  let v,e,w be object, G1 be addAdjVertex of G2,v,e,w;
  let f1 be VColoring of G1, f2 be VColoring of G2, x be object;
  assume that A1: not w in the_Vertices_of G2 & f1 = f2 +* (w .--> x) and
    A2: x <> f2.v & f2 is proper;
  set G3 = the reverseEdgeDirections of G1, {e};
  per cases;
  suppose not e in the_Edges_of G2;
    then A3: G3 is addAdjVertex of G2,w,e,v by GLIBPRE1:66;
    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, A2, A3, Th23;
    hence thesis by Th18;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 12;
    then the_Vertices_of G2 = the_Vertices_of G1 by GLIB_000:def 34
      .= dom f1 by PARTFUN1:def 2
      .= dom f2 \/ dom(w .--> x) by A1, FUNCT_4:def 1
      .= dom f2 \/ dom{[w,x]} by FUNCT_4:82
      .= dom f2 \/ {w} by RELAT_1:9
      .= the_Vertices_of G2 \/ {w} by PARTFUN1:def 2;
    hence thesis by A1, ZFMISC_1:39; :: by contradiction
  end;
end;
