
theorem
  for F being Graph-yielding Function holds
    (F is simple iff rng F is simple) &
    (F is Dsimple iff rng F is Dsimple) &
    (F is Tree-like iff rng F is Tree-like)
proof
  let F be Graph-yielding Function;
  hereby
    assume F is simple;
    then rng F is non-multi loopless by Th3;
    hence rng F is simple;
  end;
  hereby
    assume rng F is simple;
    then F is non-multi loopless by Th3;
    hence F is simple;
  end;
  hereby
    assume F is Dsimple;
    then rng F is non-Dmulti loopless by Th3;
    hence rng F is Dsimple;
  end;
  hereby
    assume rng F is Dsimple;
    then F is non-Dmulti loopless by Th3;
    hence F is Dsimple;
  end;
  hereby
    assume F is Tree-like;
    then rng F is acyclic connected by Th3;
    hence rng F is Tree-like;
  end;
  hereby
    assume rng F is Tree-like;
    then F is acyclic connected by Th3;
    hence F is Tree-like;
  end;
end;
