"Then we suppose that this is her daughter."
Translation:Alors on suppose que c'est sa fille.
It would be gramatically correct, but it would mean something slightly different. In French,
"Alors on suppose qu'elle est sa fille": You see a man, you know he has daughters, you see him with a kid, then you "suppose qu'elle est sa fille".
"Alors on suppose que c'est sa fille". This one, taken alone, can definitely mean exactly the same one than the other, but can also have a broader meaning, if the context informed you already, because you can add a verb after: "Alors on suppose que c'est sa fille... (qui a fait ceci/cela).
Exemple: La voiture de John n'était pas chez lui hier soir. John n'a pas pris sa voiture hier soir. Alors on suppose que c'est sa fille.
Here the context makes it totally explicit that "on suppose que c'est sa fille... qui l'a prise" so you wouldn't repeat it, it wouldn't sound elegant.
Can Ceci not be used here? Alors nous supposons que ceci est sa fille. ?
Not quite anymore, you're already assuming that it's his/her daughter, which means that you're past the simple hypothesis. For more info: http://french.about.com/od/grammar/qt/subjunctive_supposer.htm
Cette is a demonstrative adjective; it needs to be attached to a noun. If you don't want to use a noun, then you must use ce (or celui (or its variations)). Note that ce can be a demonstrative pronoun AND a demonstrative adjective. In the case of it being a pronoun, it never inflects so is always ce (unless followed by est or était, in which case it becomes c').
It is a variation of ce, yes. Ce is the most neutral, ceci emphasises this more and cela emphasises that more (ça is a short form of cela). Ça often gets used as this though as there is no short form for ceci. Even with this said, whenever you're using est or était, you can never use ça, as far as I know.
"On" is often used to describe a group of undefined people "On dit qu'il a gagné" = "They say that he won". In that case you don't know exactly who said that, but some people said that. "Nous" is used as the english "we", meaning that you clearly refer to yourself and somebody (or even something in extreme cases) else, or yourself and many people. When you say "nous", it means that in your mind the people who are included in your pronoun are clearly defined.
However, in every day language, the pronoun "on" is widely used in almost all cases where we could also use "nous". I actually can't think of a moment where I would naturally choose to use "nous" rather than "on" as subject of a verb, unless I want to sound quite formal.
if you don't know this, look: ( Étymologiquement on signifie homme (du latin homo), ce mot latin "homo" a évolué de la façon suivante : hom, om, on. Ce pronom existe aussi en catalan, c'est le pronom "hom" dont il est facile de voir sa racine latine "homo".
- Et comme le nom "homme" (son origine), le pronom on peut aussi prendre l'article l' (l'homme, l'on) après certains mots : et, ou, où, qui, que, si.
Exemple: Où l'on va? À qui l'on va donner ces documents? Si l'on part ce soir, on arrivera demain à 7 heures.) http://www.edu365.cat/eso/muds/frances/on/index.htm