
theorem Th165:
  for G1, G2 being _Graph
  for G3 being removeLoops of G1, G4 being removeLoops of G2
  for F0 being one-to-one PGraphMapping of G1, G2
  ex F being one-to-one PGraphMapping of G3, G4
  st F = F0 | G3 &
    (F0 is weak_SG-embedding implies F is weak_SG-embedding) &
    (F0 is isomorphism implies F is isomorphism) &
    (F0 is Disomorphism implies F is Disomorphism)
proof
  let G1, G2 be _Graph, G3 be removeLoops of G1, G4 be removeLoops of G2;
  let F0 be one-to-one PGraphMapping of G1, G2;
  consider F being one-to-one PGraphMapping of G3, G4 such that
    A1: F = F0 | G3 and
    A2: F0 is total implies F is total and
    A3: F0 is onto implies F is onto and
    A4: F0 is directed implies F is directed and
    F0 is semi-Dcontinuous implies F is semi-Dcontinuous by Th164;
  take F;
  thus F = F0 | G3 by A1;
  thus F0 is weak_SG-embedding implies F is weak_SG-embedding by A2;
  thus F0 is isomorphism implies F is isomorphism by A2, A3;
  thus F0 is Disomorphism implies F is Disomorphism by A2, A3, A4;
end;
