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 Th141:
  for e being object, v,w being Vertex of G2, G1 being addEdge of G2,v,e,w
  for t being TColoring of G2, y being object st not e in the_Edges_of G2
  holds [t_V, t_E +* (e .--> y)] is TColoring of G1
proof
  let e be object, v,w be Vertex of G2, G1 be addEdge of G2,v,e,w;
  let t be TColoring of G2, y be object;
  assume not e in the_Edges_of G2;
  then A1: the_Vertices_of G1 = the_Vertices_of G2 &
    the_Edges_of G1 = the_Edges_of G2 \/ {e} by GLIB_006:def 11;
  dom(t_E +* (e .--> y)) = dom t_E \/ dom(e .--> y) by FUNCT_4:def 1
    .= the_Edges_of G2 \/ dom(e .--> y) by PARTFUN1:def 2
    .= the_Edges_of G2 \/ dom{[e,y]} by FUNCT_4:82
    .= the_Edges_of G1 by A1, RELAT_1:9;
  then t_E +* (e .--> y) is EColoring of G1 by RELAT_1:def 18, PARTFUN1:def 2;
  hence thesis by A1, Def9;
end;
