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 Th3:
  for H being Subgraph of G holds f | the_Vertices_of H is VColoring of H
proof
  let H be Subgraph of G;
  set f9 = f | the_Vertices_of H;
  the_Vertices_of H c= the_Vertices_of G;
  then the_Vertices_of H c= dom f by PARTFUN1:def 2;
  then dom f9 = the_Vertices_of H by RELAT_1:62;
  hence thesis by PARTFUN1:def 2;
end;
