
theorem
  for G1, G2 being _Graph, F being one-to-one PGraphMapping of G1, G2
  st F is isomorphism holds F*(F") = id G2 & (F")*F = id G1
proof
  let G1, G2 be _Graph, F be one-to-one PGraphMapping of G1, G2;
  assume A1: F is isomorphism;
  thus F*(F") = [id rng F_V, F_E * (F_E")] by FUNCT_1:39
    .= [id rng F_V, id rng F_E] by FUNCT_1:39
    .= [id the_Vertices_of G2, id rng F_E] by A1, Def12
    .= id G2 by A1, Def12;
  thus (F")*F = [id dom F_V, (F_E") * F_E] by FUNCT_1:39
    .= [id dom F_V, id dom F_E] by FUNCT_1:39
    .= [id the_Vertices_of G1, id dom F_E] by A1, Def11
    .= id G1 by A1, Def11;
end;
