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 Th17:
  for f1 being VColoring of G1, f2 being VColoring of G2
  for v being Vertex of G1, x being object
  st G1 == G2 & f2 = f1 +* (v .--> x) & not x in rng f1 & f1 is proper
  holds f2 is proper
proof
  let f1 be VColoring of G1, f2 be VColoring of G2;
  let v be Vertex of G1, x be object;
  assume A1: G1 == G2 & f2 = f1 +* (v .--> x) & not x in rng f1 & f1 is proper;
  now
    let e,u,w be object;
    assume e Joins u,w,G2;
    then A2: e Joins u,w,G1 by A1, GLIB_000:88;
    u in the_Vertices_of G1 & w in the_Vertices_of G1 by A2, GLIB_000:13;
    then u in dom f1 & w in dom f1 by PARTFUN1:def 2;
    then A3: f1.u in rng f1 & f1.w in rng f1 by FUNCT_1:3;
    per cases;
    suppose u <> v & w <> v;
      then f2.u = f1.u & f2.w = f1.w by A1, FUNCT_4:83;
      hence f2.u <> f2.w by A1, A2, Th10;
    end;
    suppose u = v & w <> v;
      then f2.u = x & f2.w = f1.w by A1, FUNCT_4:83, FUNCT_4:113;
      hence f2.u <> f2.w by A1, A3;
    end;
    suppose u <> v & w = v;
      then f2.u = f1.u & f2.w = x by A1, FUNCT_4:83, FUNCT_4:113;
      hence f2.u <> f2.w by A1, A3;
    end;
    suppose u = v & w = v;
      then G1 is non loopless by A2, GLIB_000:18;
      hence f2.u <> f2.w by A1; :: by contradiction
    end;
  end;
  hence thesis by Th10;
end;
