
theorem Th82:
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for v1 being Vertex of G1, v2 being Vertex of G2
  st v2 = F_V.v1 & F is isomorphism
  holds F_V.:G1.reachableFrom(v1) = G2.reachableFrom(v2)
proof
  let G1, G2 be _Graph, F be PGraphMapping of G1, G2;
  let v1 be Vertex of G1, v2 be Vertex of G2;
  assume A1: v2 = F_V.v1 & F is isomorphism;
  then A2: F_V.:G1.reachableFrom(v1) c= G2.reachableFrom(v2) by Th80;
  the_Vertices_of G1 = dom F_V by A1, GLIB_010:def 11;
  then G2.reachableFrom(v2) c= F_V.:G1.reachableFrom(v1) by A1, Th81;
  hence thesis by A2, XBOOLE_0:def 10;
end;
