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;

theorem Th106:
  for G2 being c-ecolorable _Graph, v,e,w being object
  for G1 being addEdge of G2,v,e,w holds G1 is (c+`1)-ecolorable
proof
  let G2 be c-ecolorable _Graph, v,e,w be object;
  let G1 be addEdge of G2,v,e,w;
  per cases;
  suppose A1: v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2;
    consider g2 being proper EColoring of G2 such that
      A2: card rng g2 c= c by Def6;
    set y = rng g2, g1 = g2 +* (e .--> y);
    reconsider g1 as EColoring of G1 by A1, Th79;
    A3: rng(e .--> y) = rng{[e,y]} by FUNCT_4:82
      .= {y} by RELAT_1:9;
    not y in rng g2;
    then reconsider g1 as proper EColoring of G1 by A1, Th93;
    A4: card rng g1 c= card(rng g2 \/ {y}) by A3, FUNCT_4:17, CARD_1:11;
    not y in rng g2;
    then card(rng g2\/{y}) = card rng g2 +` card {y} by CARD_2:35, ZFMISC_1:50
      .= card rng g2 +` 1 by CARD_1:30;
    then card(rng g2 \/ {y}) c= c +` 1 by A2, CARD_2:84;
    hence thesis by A4, XBOOLE_1:1;
  end;
  suppose not(v in the_Vertices_of G2 & w in the_Vertices_of G2 &
      not e in the_Edges_of G2);
    then G1 == G2 by GLIB_006:def 11;
    then G1 is c-ecolorable by Th103;
    hence thesis by Th99, CARD_2:94;
  end;
end;
