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 Th98:
  for F being PGraphMapping of G1,G, g9 being EColoring of G1
  st F is weak_SG-embedding & g9 = g*(F_E) & g is proper holds g9 is proper
proof
  let F be PGraphMapping of G1, G, g9 be EColoring of G1;
  assume A1: F is weak_SG-embedding & g9 = g*(F_E) & g is proper;
  then dom F_E = the_Edges_of G1 & F_E is one-to-one by GLIB_010:def 11;
  hence thesis by A1, Th97;
end;
