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 Th90:
  for g1 being EColoring of G1, g2 being EColoring of G2
  st G1 == G2 & g1 = g2 & g1 is proper holds g2 is proper
proof
  let g1 be EColoring of G1, g2 be EColoring of G2;
  assume A1: G1 == G2 & g1 = g2 & g1 is proper;
  let v be Vertex of G2;
  reconsider w = v as Vertex of G1 by A1, GLIB_000:def 34;
  v.edgesInOut() = w.edgesInOut() by A1, GLIB_000:96;
  hence g2 | v.edgesInOut() is one-to-one by A1;
end;
