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 e being object, v,w being Vertex of G2, G1 being addEdge of G2,v,e,w
  st v,w are_adjacent & G2 is c-tcolorable holds G1 is (c+`1)-tcolorable
proof
  let e be object, v,w be Vertex of G2, G1 be addEdge of G2,v,e,w;
  assume A1: v,w are_adjacent & G2 is c-tcolorable;
  per cases;
  suppose A2: not e in the_Edges_of G2;
    consider t being TColoring of G2 such that
      A3: t is proper & card((rng t_V)\/rng t_E) c= c by A1;
    set V = rng t_V, E = rng t_E;
    set y = V \/ E, g = t_E +* (e .--> y);
    reconsider t1 = [t_V, g] as TColoring of G1 by A2, Th141;
    not y in V \/ E & t1_V = t_V & t1_E = t_E +* (e .--> y);
    then A4: t1 is proper by A1, A2, A3, Th155;
    rng(e .--> y) = rng{[e,y]} by FUNCT_4:82
      .= {y} by RELAT_1:9;
    then V \/ rng g c= V \/ (E \/ {y}) by FUNCT_4:17, XBOOLE_1:9;
    then V \/ rng g c= (V \/ E) \/ {y} by XBOOLE_1:4;
    then A5: card(V \/ rng g) c= card((V \/ E) \/ {y}) by CARD_1:11;
    not y in V \/ E;
    then card((V\/E)\/{y}) = card(V \/ E) +` card {y} by ZFMISC_1:50, CARD_2:35
      .= card(V \/ E) +` 1 by CARD_1:30;
    then card((V \/ E) \/ {y}) c= c +` 1 by A3, CARD_2:84;
    hence thesis by A4, A5, XBOOLE_1:1;
  end;
  suppose e in the_Edges_of G2;
    then G1 == G2 by GLIB_006:def 11;
    then G1 is c-tcolorable by A1, Th167;
    hence thesis by Th161, CARD_2:94;
  end;
end;
