reserve G for _Graph;

theorem
  for E being set, H being reverseEdgeDirections of G, E
  holds VertexAdjSymRel(H) = VertexAdjSymRel(G)
proof
  let E be set, H be reverseEdgeDirections of G, E;
  now
    let v,w be object;
    hereby
      assume [v,w] in VertexAdjSymRel(G);
      then consider e being object such that
        A1: e Joins v,w,G by Th32;
      e Joins v,w,H by A1, GLIB_007:9;
      hence [v,w] in VertexAdjSymRel(H) by Th32;
    end;
    assume [v,w] in VertexAdjSymRel(H);
    then consider e being object such that
      A2: e Joins v,w,H by Th32;
    e Joins v,w,G by A2, GLIB_007:9;
    hence [v,w] in VertexAdjSymRel(G) by Th32;
  end;
  hence thesis by RELAT_1:def 2;
end;
