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;

theorem Th78:
  for H being Subgraph of G holds g | the_Edges_of H is EColoring of H
proof
  let H be Subgraph of G;
  set g9 = g | the_Edges_of H;
  the_Edges_of H c= the_Edges_of G;
  then the_Edges_of H c= dom g by PARTFUN1:def 2;
  then dom g9 = the_Edges_of H by RELAT_1:62;
  hence thesis by PARTFUN1:def 2;
end;
