reserve G for _Graph;

theorem Th20:
  for H being reverseEdgeDirections of G
  holds VertexDomRel(H) = (VertexDomRel(G))~
proof
  let H be reverseEdgeDirections of G;
  now
    let w,v be object;
    hereby
      assume [w,v] in (VertexDomRel(G))~;
      then [v,w] in VertexDomRel(G) by RELAT_1:def 7;
      then consider e being object such that
        A1: e DJoins v,w,G by Th1;
      e in the_Edges_of G by A1, GLIB_000:def 14;
      then e DJoins w,v,H by A1, GLIB_007:7;
      hence [w,v] in VertexDomRel(H) by Th1;
    end;
    assume [w,v] in VertexDomRel(H);
    then consider e being object such that
      A2: e DJoins w,v,H by Th1;
    e in the_Edges_of H by A2, GLIB_000:def 14;
    then e in the_Edges_of G by GLIB_007:4;
    then e DJoins v,w,G by A2, GLIB_007:7;
    then [v,w] in VertexDomRel(G) by Th1;
    hence [w,v] in (VertexDomRel(G))~ by RELAT_1:def 7;
  end;
  hence thesis by RELAT_1:def 2;
end;
