reserve G, G2 for _Graph, V, E for set,
  v for object;

theorem Th20:
  for G2, E for G1 being reverseEdgeDirections of G2, E
  for v1 being Vertex of G1, v2 being Vertex of G2 st v1 = v2
  holds G1.reachableFrom(v1) = G2.reachableFrom(v2)
proof
  let G2, E;
  let G1 be reverseEdgeDirections of G2, E;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v1 = v2;
  for v being object holds
    v in G1.reachableFrom(v1) iff v in G2.reachableFrom(v2)
  proof
    let v be object;
    hereby
      assume v in G1.reachableFrom(v1);
      then ex W being Walk of G1 st W is_Walk_from v1,v by GLIB_002:def 5;
      then ex W being Walk of G2 st W is_Walk_from v2,v by A1, Th19;
      hence v in G2.reachableFrom(v2) by GLIB_002:def 5;
    end;
    assume v in G2.reachableFrom(v2);
    then ex W being Walk of G2 st W is_Walk_from v2,v by GLIB_002:def 5;
    then ex W being Walk of G1 st W is_Walk_from v1,v by A1, Th19;
    hence v in G1.reachableFrom(v1) by GLIB_002:def 5;
  end;
  hence thesis by TARSKI:2;
end;
