
theorem
  for G1, G2 being _Graph, F being one-to-one 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"G2.reachableFrom(v2) = G1.reachableFrom(v1)
proof
  let G1, G2 be _Graph, F be one-to-one 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 the_Vertices_of G1 = dom F_V by GLIB_010:def 11;
  then A2: F"_V.v2 = v1 by A1, FUNCT_1:34;
  thus F_V"G2.reachableFrom(v2)
     = F_V".:G2.reachableFrom(v2) by FUNCT_1:85
    .= G1.reachableFrom(v1) by A1, A2, Th82, GLIB_010:75;
end;
