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 Th89:
  for H being Subgraph of G, g9 being EColoring of H
  st g9 = g | the_Edges_of H & g is proper holds g9 is proper
proof
  let H be Subgraph of G, g9 be EColoring of H;
  assume A1: g9 = g | the_Edges_of H & g is proper;
  now
    let v be Vertex of H, e1, e2 be object;
    assume A2: e1 in v.edgesInOut() & e2 in v.edgesInOut() & g9.e1 = g9.e2;
    the_Vertices_of H c= the_Vertices_of G & v in the_Vertices_of H;
    then reconsider w = v as Vertex of G;
    A3: v.edgesInOut() c= w.edgesInOut() by GLIB_000:78;
    g9.e1 = g.e1 & g9.e2 = g.e2 by A1, A2, FUNCT_1:49;
    hence e1 = e2 by A1, A2, A3, Th85;
  end;
  hence thesis by Th85;
end;
