
theorem
  for G3 being _Graph, V, E being set, G4 being reverseEdgeDirections of G3, E
  for G1 being addLoops of G3, V, G2 being addLoops of G4, V
  holds G2 is G1-isomorphic
proof
  let G3 be _Graph, V, E be set, G4 be reverseEdgeDirections of G3, E;
  let G1 be addLoops of G3, V, G2 be addLoops of G4, V;
  per cases;
  suppose A1: V c= the_Vertices_of G3;
    consider F0 being PGraphMapping of G3, G4 such that
      A2: F0 = id G3 & F0 is isomorphism by GLIBPRE0:77;
    A3: F0_V | V is one-to-one by A2, FUNCT_1:52;
    A4: dom(F0_V | V) = dom F0_V /\ V by RELAT_1:61
      .= V by A1, A2, XBOOLE_1:28;
    A5: rng(F0_V | V) = (id G3)_V.:V by A2, RELAT_1:115
      .= V by A1, FUNCT_1:92;
    consider F being PGraphMapping of G1, G2 such that
      F_V = F0_V & F_E | dom F0_E = F0_E and
      F0 is non empty  implies F is non empty and
      F0 is total implies F is total and
      F0 is onto implies F is onto and
      F0 is one-to-one implies F is one-to-one and
      F0 is directed implies F is directed and
      F0 is weak_SG-embedding implies F is weak_SG-embedding and
      A6: F0 is isomorphism implies F is isomorphism and
      F0 is Disomorphism implies F is Disomorphism
      by A1, A3, A4, A5, Th29;
    thus thesis by A2, A6, GLIB_010:def 23;
  end;
  suppose not V c= the_Vertices_of G3;
    then G1 == G3 & not V c= the_Vertices_of G4 by Def5, GLIB_007:4;
    then A7: G1 == G3 & G2 == G4 by Def5;
    then G2 is reverseEdgeDirections of G3, E by GLIB_007:2;
    then G3 is reverseEdgeDirections of G2, E by GLIB_007:3;
    then G1 is reverseEdgeDirections of G2, E by A7, GLIB_007:2;
    then G2 is reverseEdgeDirections of G1, E by GLIB_007:3;
    hence thesis by GLIBPRE0:78;
  end;
end;
