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 Th151:
  for H being Subgraph of G, t9 being TColoring of H
  st t9 = [ t_V | the_Vertices_of H , t_E | the_Edges_of H ] & t is proper
  holds t9 is proper
proof
  let H be Subgraph of G, t9 be TColoring of H;
  assume A1: t9 = [t_V|the_Vertices_of H, t_E|the_Edges_of H] & t is proper;
  then A2: t9_V is proper by Th15;
  A3: t9_E is proper by A1, Th89;
  now
    let e,v,w be object;
    assume A4: e Joins v,w,H;
    then e in the_Edges_of H & v in the_Vertices_of H by GLIB_000:def 13, 13;
    then A5: e in dom(t9_E) & v in dom(t9_V) by PARTFUN1:def 2;
    v is set & w is set by TARSKI:1;
    then e Joins v,w,G by A4, GLIB_000:72;
    then A6: t_V.v <> t_E.e by A1, Th146;
    t_V.v = t9_V.v & t_E.e = t9_E.e by A1, A5, FUNCT_1:47;
    hence t9_V.v <> t9_E.e by A6;
  end;
  hence thesis by A2, A3, Th146;
end;
