Kiddushin, Chapter 4, Mishnah 10. Our Mishnah addresses the question of proving lineage: in which circumstances must a person bring evidence that his wife and children are of valid lineage, and in which circumstances what our eyes see is enough.
The text of the Mishnah:
"Mi sheyatza hu v'ishto limdinas hayam, uva hu v'ishto uvanav, v'amar: ishah sheyatzas imi limdinas hayam harei hi zo v'eilu baneha - eino tzarich l'havi ra'ayah lo al ha'ishah v'lo al habanim" - one who went overseas with his wife and returned with his wife and children, and said: the woman who went overseas with me is this one, and these are her children - he need not bring evidence, neither about the woman nor about the children.
The case is a man who married a woman, investigated her lineage and found it valid, and afterward traveled overseas with her. Now he has returned with her and with children who were born there, and he says: this is the woman who went out with me, and these are her children. The Mishnah rules that he is not required to bring any evidence at all, neither about the woman nor about the children.
The reason for this ruling:
Regarding the woman - this is the very same woman who went out with him, and her lineage was already investigated at the time of their marriage.
Regarding the children - although they did not yet exist when the couple set out, the Gemara explains that the case is one of small children who are circling around her and clinging to the hem of her garment. From the very sight before us it is clear that these are her children, and consequently, if she is of valid lineage, so are they.
If the woman died:
The Mishnah continues: "Meisah, v'eilu baneha - meivi ra'ayah al habanim, v'eino meivi ra'ayah al ha'ishah" - if she died, and he says these are her children, he must bring evidence about the children, but he need not bring evidence about the woman. In this case the man returns with the children alone and says that his wife died and that these are her children. Here he must bring evidence about the children - witnesses who testify that they saw that these are that woman's children, since the mother is not before us and the sight itself proves nothing. About the woman, however, he need not bring evidence, since he already investigated her lineage when he married her. And once there is someone who testifies that these are her children, they have the status of Jews of valid lineage.
In summary: our Mishnah teaches that a woman whose lineage was investigated at the time of the marriage does not require a second investigation when she returns, while regarding the children everything depends on what can be seen: as long as the mother is with them and they are clinging to her garment, the sight itself serves as proof and no further evidence is needed; but once the mother has died, explicit testimony is required that these are her children, and from that testimony their valid status follows.