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 Th143:
  for v,e being object, w being Vertex of G2, G1 being addAdjVertex of G2,v,e,w
  for t being TColoring of G2, x,y being object
  st not e in the_Edges_of G2 & not v in the_Vertices_of G2
  holds [t_V +* (v .--> x), t_E +* (e .--> y)] is TColoring of G1
proof
  let v,e be object, w be Vertex of G2, G1 be addAdjVertex of G2,v,e,w;
  let t be TColoring of G2, x,y be object;
  assume A1: not e in the_Edges_of G2 & not v in the_Vertices_of G2;
  then consider G3 being addVertex of G2,v such that
    A2: G1 is addEdge of G3,v,e,w by GLIB_006:126;
  reconsider t3 = [t_V +* (v .--> x), t_E] as TColoring of G3 by Th140;
  A3: v is Vertex of G3 & w is Vertex of G3 by GLIB_006:68, GLIB_006:94;
  not e in the_Edges_of G3 by A1, GLIB_006:def 10;
  then [t3_V, t3_E +* (e .--> y)] is TColoring of G1 by A2, A3, Th141;
  hence thesis;
end;
