reserve G for _Graph;

theorem Th2:
  for v,w being object holds [v,w] in (VertexDomRel(G))~ iff
    ex e being object st e DJoins w,v,G
proof
  let v,w be object;
  thus [v,w] in (VertexDomRel(G))~ implies ex e being object st e DJoins w,v,G
  proof
    assume [v,w] in (VertexDomRel(G))~;
    then [w,v] in VertexDomRel(G) by RELAT_1:def 7;
    then consider e being object such that
      A1: e DJoins w,v,G by Th1;
    take e;
    thus thesis by A1;
  end;
  thus (ex e being object st e DJoins w,v,G)implies [v,w] in (VertexDomRel(G))~
  proof
    assume ex e being object st e DJoins w,v,G;
    then [w,v] in VertexDomRel(G) by Th1;
    hence thesis by RELAT_1:def 7;
  end;
end;
