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 Th20:
  for v,w being Vertex of G2, e being object, G1 being addEdge of G2,v,e,w
  for f1 being VColoring of G1, f2 being VColoring of G2
  st f1 = f2 & v,w are_adjacent & f2 is proper holds f1 is proper
proof
  let v,w be Vertex of G2, e be object, G1 be addEdge of G2,v,e,w;
  let f1 be VColoring of G1, f2 be VColoring of G2;
  assume A1: f1 = f2 & v,w are_adjacent & f2 is proper;
  per cases;
  suppose A2: not e in the_Edges_of G2;
    now
      let e9,v9,w9 be object;
      assume A3: e9 Joins v9,w9,G1;
      then per cases by GLIB_006:72;
      suppose e9 Joins v9,w9,G2;
        hence f1.v9 <> f1.w9 by A1, Th10;
      end;
      suppose not e9 in the_Edges_of G2;
        then A4: (v9 = v & w9 = w)or(v9 = w & w9 = v) by A2, A3, GLIB_006:107;
        consider e1 being object such that
          A5: e1 Joins v,w,G2 by A1, CHORD:def 3;
        e1 Joins v9,w9,G2 by A4, A5, GLIB_000:14;
        hence f1.v9 <> f1.w9 by A1, Th10;
      end;
    end;
    hence thesis by Th10;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 11;
    hence thesis by A1, Th16;
  end;
end;
