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 Th36:
  for v,e,w being object, G1 being addEdge of G2,v,e,w
  st v <> w & G2 is c-vcolorable holds G1 is (c+`1)-vcolorable
proof
  let v,e,w be object, G1 be addEdge of G2,v,e,w;
  assume A1: v <> w & G2 is c-vcolorable;
  per cases;
  suppose A2: not e in the_Edges_of G2 & v in the_Vertices_of G2 &
      w in the_Vertices_of G2;
    consider f2 being VColoring of G2 such that
      A3: f2 is proper & card rng f2 c= c by A1;
    :: given a proper coloring of the subgraph, we
    :: color one end of the new edge in a new color
    set h = w .--> rng f2, f1 = f2 +* h;
    reconsider f1 as VColoring of G2 by A2;
    the_Vertices_of G1 = the_Vertices_of G2 by A2, GLIB_006:102;
    then reconsider f1 as VColoring of G1;
    not rng f2 in rng f2;
    then A4: f1 is proper by A1, A2, A3, Th22;
    rng f2 \/ rng h = rng f2 \/ rng{[w,rng f2]} by FUNCT_4:82
      .= rng f2 \/ {rng f2} by RELAT_1:9;
    then A5: card rng f1 c= card(rng f2 \/ {rng f2}) by FUNCT_4:17, CARD_1:11;
    card(rng f2 \/ {rng f2}) c= card rng f2 +` card{rng f2} by CARD_2:34;
    then card(rng f2 \/ {rng f2}) c= card rng f2 +` 1 by CARD_1:30;
    then A6: card rng f1 c= card rng f2 +` 1 by A5, XBOOLE_1:1;
    card rng f2 +` 1 c= c +` 1 by A3, CARD_2:83;
    hence thesis by A4, A6, XBOOLE_1:1;
  end;
  suppose not(not e in the_Edges_of G2 & v in the_Vertices_of G2 &
      w in the_Vertices_of G2);
    then G1 == G2 by GLIB_006:def 11;
    then G1 is c-vcolorable by A1, Th32;
    hence thesis by Th27, CARD_2:94;
  end;
end;
