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 Th92:
  for G1 being addVertices of G2, V
  for g1 being EColoring of G1, g2 being EColoring of G2 st g1 = g2
  holds g2 is proper implies g1 is proper
proof
  let G1 be addVertices of G2, V;
  let g1 be EColoring of G1, g2 be EColoring of G2;
  assume A1: g1 = g2 & g2 is proper;
  now
    let v be Vertex of G1;
    v in the_Vertices_of G1;
    then v in the_Vertices_of G2 \/ V by GLIB_006:def 10;
    then per cases by XBOOLE_0:def 3;
    suppose v in the_Vertices_of G2;
      then reconsider w = v as Vertex of G2;
      v.edgesInOut() = w.edgesInOut() by GLIBPRE0:45;
      hence g1 | v.edgesInOut() is one-to-one by A1;
    end;
    suppose not v in the_Vertices_of G2 & v in V;
      then v in V \ the_Vertices_of G2 by XBOOLE_0:def 5;
      then v.edgesInOut() = {} by GLIB_000:def 49, GLIB_006:88;
      hence g1 | v.edgesInOut() is one-to-one;
    end;
  end;
  hence thesis;
end;
