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 Th146:
  t is proper iff t_V is proper & t_E is proper &
    for e,v,w being object st e Joins v,w,G holds t_V.v <> t_E.e
proof
  hereby
    assume A1: t is proper;
    hence t_V is proper & t_E is proper;
    let e,v,w be object;
    assume A2: e Joins v,w,G;
    then reconsider u = v as Vertex of G by GLIB_000:13;
    A3: e in u.edgesInOut() by A2, GLIB_000:62;
    then e in the_Edges_of G;
    then e in dom(t_E) by PARTFUN1:def 2;
    then t_E.e in t_E.:u.edgesInOut() by A3, FUNCT_1:def 6;
    hence t_V.v <> t_E.e by A1;
  end;
  assume A4: t_V is proper & t_E is proper &
    for e,v,w being object st e Joins v,w,G holds t_V.v <> t_E.e;
  hence t_V is proper & t_E is proper;
  let v be Vertex of G;
  assume t_V.v in t_E.:v.edgesInOut();
  then consider e being object such that
    A5: e in dom t_E & e in v.edgesInOut() & t_V.v = t_E.e by FUNCT_1:def 6;
  consider w being Vertex of G such that
    A6: e Joins v,w,G by A5, GLIB_000:64;
  thus contradiction by A4, A5, A6;
end;
