reserve G for _Graph;

theorem Th19:
  for G1, G2 being _Graph st G1 == G2
  holds VertexDomRel(G1) = VertexDomRel(G2)
proof
  let G1, G2 be _Graph such that A1: G1 == G2;
  now
    let v,w be object;
    hereby
      assume [v,w] in VertexDomRel(G1);
      then consider e being object such that
        A2: e DJoins v,w,G1 by Th1;
      e DJoins v,w,G2 by A1, A2, GLIB_000:88;
      hence [v,w] in VertexDomRel(G2) by Th1;
    end;
    assume [v,w] in VertexDomRel(G2);
    then consider e being object such that
      A3: e DJoins v,w,G2 by Th1;
    e DJoins v,w,G1 by A1, A3, GLIB_000:88;
    hence [v,w] in VertexDomRel(G1) by Th1;
  end;
  hence thesis by RELAT_1:def 2;
end;
