
theorem
  for G1, G2 being _Graph, F being PGraphMapping of G1, G2
  for H being Subgraph of G1
  for F9 being PGraphMapping of H, rng(F | H) st F9 = F | H holds
    (F is weak_SG-embedding implies F9 is weak_SG-embedding) &
    (F is strong_SG-embedding implies F9 is isomorphism) &
    (F is directed strong_SG-embedding implies F9 is Disomorphism)
proof
  let G1, G2 be _Graph;
  let F be PGraphMapping of G1, G2, H being Subgraph of G1;
  let F9 be PGraphMapping of H, rng(F | H);
  assume A1: F9 = F | H;
  hereby
    assume F is weak_SG-embedding;
    then F9 is total one-to-one by A1, Th112;
    hence F9 is weak_SG-embedding;
  end;
  hereby
    assume A2: F is strong_SG-embedding;
    then F9 is total by A1, Th112;
    then F9 is total one-to-one continuous onto by A1, A2, Th112;
    hence F9 is isomorphism;
  end;
  hereby
    assume A3: F is directed strong_SG-embedding;
    then F9 is total by A1, Th112;
    then F9 is directed total one-to-one continuous onto by A1, A3, Th112;
    hence F9 is Disomorphism;
  end;
end;
