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 Th102:
  for H being Subgraph of G holds G is c-ecolorable implies H is c-ecolorable
proof
  let H be Subgraph of G;
  assume G is c-ecolorable;
  then consider g1 being proper EColoring of G such that
    A1: card rng g1 c= c;
  reconsider g2 = g1 | the_Edges_of H as EColoring of H by Th78;
  reconsider g2 as proper EColoring of H by Th89;
  card rng g2 c= card rng g1 by RELAT_1:70, CARD_1:11;
  hence thesis by  A1, XBOOLE_1:1;
end;
