reserve G for _Graph;

theorem Th32:
  for v, w being object
  holds [v,w] in VertexAdjSymRel(G) iff ex e being object st e Joins v,w,G
proof
  let v, w be object;
  hereby
    assume [v,w] in VertexAdjSymRel(G);
    then per cases by XBOOLE_0:def 3;
    suppose [v,w] in VertexDomRel(G);
      then consider e being object such that
        A1: e DJoins v,w,G by Th1;
      take e;
      thus e Joins v,w,G by A1, GLIB_000:16;
    end;
    suppose [v,w] in (VertexDomRel(G))~;
      then consider e being object such that
        A2: e DJoins w,v,G by Th2;
      take e;
      thus e Joins v,w,G by A2, GLIB_000:16;
    end;
  end;
  given e being object such that
    A3: e Joins v,w,G;
  per cases by A3, GLIB_000:16;
  suppose e DJoins v,w,G;
    then [v,w] in VertexDomRel(G) by Th1;
    hence thesis by XBOOLE_0:def 3;
  end;
  suppose e DJoins w,v,G;
    then [v,w] in (VertexDomRel(G))~ by Th2;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
