reserve G for _Graph;

theorem Th1:
  for v,w being object holds [v,w] in VertexDomRel(G) iff
    ex e being object st e DJoins v,w,G
proof
  let v,w be object;
  reconsider S = the_Source_of G
    as Relation of the_Edges_of G, the_Vertices_of G;
  hereby
    assume [v,w] in VertexDomRel(G);
    then consider e being object such that
      A1: [v,e] in S~ & [e,w] in the_Target_of G by RELAT_1:def 8;
    take e;
    [e,v] in S by A1, RELAT_1:def 7;
    then A2: e in dom the_Source_of G & (the_Source_of G).e = v by FUNCT_1:1;
    (the_Target_of G).e = w by A1, FUNCT_1:1;
    hence e DJoins v,w,G by A2, GLIB_000:def 14;
  end;
  given e being object such that
    A3: e DJoins v,w,G;
  e in the_Edges_of G by A3, GLIB_000:def 14;
  then A4: e in dom the_Source_of G & e in dom the_Target_of G
    by FUNCT_2:def 1;
  (the_Source_of G).e = v & (the_Target_of G).e = w by A3, GLIB_000:def 14;
  then [e,v] in the_Source_of G & [e,w] in the_Target_of G by A4, FUNCT_1:1;
  then [v,e] in S~ & [e,w] in the_Target_of G by RELAT_1:def 7;
  hence [v,w] in VertexDomRel(G) by RELAT_1:def 8;
end;
