
theorem Th81:
  for G1, G2 being _Graph st G1 == G2
  ex F being PGraphMapping of G1, G2 st F = id G1 & F is Disomorphism
proof
  let G1, G2 be _Graph;
  assume A1: G1 == G2;
  then reconsider F = id G1 as PGraphMapping of G1, G2 by GLIB_010:11;
  take F;
  thus F = id G1;
  dom F_V = the_Vertices_of G1 & dom F_E = the_Edges_of G1;
  then A2: F is total by GLIB_010:def 11;
  rng F_V = the_Vertices_of G2 & rng F_E = the_Edges_of G2
    by A1, GLIB_000:def 34;
  then A3: F is onto by GLIB_010:def 12;
  F_V is one-to-one & F_E is one-to-one;
  then A4: F is one-to-one by GLIB_010:def 13;
  now
    let e,v,w be object;
    assume A5: e in dom F_E & v in dom F_V & w in dom F_V;
    assume e DJoins v,w,G1;
    then (id G1)_E.e DJoins (id G1)_V.v,(id G1)_V.w,G1 by A5, GLIB_010:def 14;
    hence F_E.e DJoins F_V.v,F_V.w,G2 by A1, GLIB_000:88;
  end;
  then F is directed by GLIB_010:def 14;
  hence F is Disomorphism by A2, A3, A4;
end;
