
theorem
  for G3, G4 being _Graph, V being set
  for G1 being addLoops of G3, V, G2 being addLoops of G4, V
  st G3 == G4 holds G2 is G1-Disomorphic
proof
  let G3, G4 be _Graph, V be set;
  let G1 be addLoops of G3, V, G2 be addLoops of G4, V;
  assume A1: G3 == G4;
  per cases;
  suppose A2: V c= the_Vertices_of G3;
    consider F0 being PGraphMapping of G3, G4 such that
      A3: F0 = id G3 & F0 is Disomorphism by A1, GLIBPRE0:75;
    A4: F0_V | V is one-to-one by A3, FUNCT_1:52;
    A5: dom(F0_V | V) = dom F0_V /\ V by RELAT_1:61
      .= V by A2, A3, XBOOLE_1:28;
    A6: rng(F0_V | V) = (id G3)_V.:V by A3, RELAT_1:115
      .= V by A2, 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
      F0 is isomorphism implies F is isomorphism and
      A7: F0 is Disomorphism implies F is Disomorphism
      by A2, A4, A5, A6, Th29;
    thus thesis by A3, A7, GLIB_010:def 24;
  end;
  suppose not V c= the_Vertices_of G3;
    then G1 == G3 & not V c= the_Vertices_of G4 by A1, Def5, GLIB_000:def 34;
    then G1 == G4 & G2 == G4 by A1, Def5, GLIB_000:85;
    hence thesis by GLIB_000:85,GLIBPRE0:76;
  end;
end;
