parshat Vayishlach: Rachel's Triplets?

I mentioned a midrash last week about Rachel having triplets, giving a (probably incorrect or at least insufficient) explanation on how this is derived from the text of Chumash. Now I'd like to return to consider it further.

The verses in question are in Bereishit 35:15-16:

טז וַיִּסְעוּ מִבֵּית אֵל, וַיְהִי-עוֹד כִּבְרַת-הָאָרֶץ לָבוֹא אֶפְרָתָה; וַתֵּלֶד רָחֵל, וַתְּקַשׁ בְּלִדְתָּהּ.	16 And they journeyed from Beth-el; and there was still some way to come to Ephrath; and Rachel travailed, and she had hard labour.
יז וַיְהִי בְהַקְשֹׁתָהּ, בְּלִדְתָּהּ; וַתֹּאמֶר לָהּ הַמְיַלֶּדֶת אַל-תִּירְאִי, כִּי-גַם-זֶה לָךְ בֵּן.	17 And it came to pass, when she was in hard labour, that the mid-wife said unto her: 'Fear not; for this also is a son for thee.'

and the midrash is predicated on the words וַיְהִי בְהַקְשֹׁתָהּ בְּלִדְתָּהּ, "And it came to pass, when she was in hard labour." There is a divergence between Rabbi Yehoshua and his students.

His students, who did not hear Rabbi Yehoshua's explanation, gave an explanation that is decidedly within the realm of peshat.

Perhaps {I will refine and correct this later} at issue is the problem of the role of the words וַיְהִי בְהַקְשֹׁתָהּ בְּלִדְתָּהּ, when we already saw in the previous verse וַתְּקַשׁ בְּלִדְתָּהּ. Is she is labour more than once? Rabbi Yehoshua's students explain that while the role of the vav hahipuch, the va followed by the gemination of next letter, is to advance the narrative, here this is not what is happening. Rather, the vav hahipuch is attached to the word וַיְהִי, and the role is similar to the vav hachibbur.
(Flashback to Bereishit, where
בְּרֵאשִׁית, בָּרָא אֱלֹהִים, אֵת הַשָּׁמַיִם, וְאֵת הָאָרֶץ
וְהָאָרֶץ, הָיְתָה תֹהוּ וָבֹהוּ, וְחֹשֶׁךְ, עַל-פְּנֵי תְהוֹם; וְרוּחַ אֱלֹהִים, מְרַחֶפֶת עַל-פְּנֵי הַמָּיִם
וַיֹּאמֶר אֱלֹהִים, יְהִי אוֹר; וַיְהִי-אוֹר
where the middle verse, with vav hachibbur {=ve} serves to stay in place, such that we have "In the beginning of God's creation of heaven and earth, when the earth was formless and void..., God said let there be light and there was light.)

Here, the narrative is not being advanced, such that we are now finding out that she is having difficulty in labour (for we already know that), but rather, when she was in the aforementioned difficult labour, the midwife comforted her. So is the way they comfort women in travail: do not fear for you have given birth to a son. {Etz Yosef: for it is a

The sardyot who captured the students of Rabbi Yehoshua did not like their answer, and remarked that Rabbi Yehoshua had not explained it so, but rather:

each shevet had a twin born with him. Like this that Abba Chalfoi said: An extra twin was born with Binyamin.

I think I am correct in interpreting this as triplets. Perhaps it does mean that Binyamin, just as his brothers, was born with a single twin. But I think triplets are intended.

How do we know that each brother was born with a twin? I thought it was the `et {lit. "with"} in all the וַתִּקְרָא אֶת-שְׁמוֹ, though `et is occassionally missing. There is no `et here. Something to look up.

If I am correct, then what Rabbi Yehoshua is doing is saying:
וַתְּקַשׁ בְּלִדְתָּהּ = first daughter
וַיְהִי בְהַקְשֹׁתָהּ בְּלִדְתָּהּ = second daughter.
I would add at this point that each of these has a mapik heh, "her labour," which could possibly be interpreted as "the labour of her" = "the labour of her daughter."
Furthermore, we have the statement אַל-תִּירְאִי כִּי-גַם-זֶה לָךְ בֵּן, "fear not; for this also is a son for thee," we may interpret this as her having given birth twice already, and still has a son within her. For a total of one son and two daughters in one birth.

A bit of refining. Recall that the vav hahippuch advances the narrative. If so, it is nonsense to say
וַתֵּלֶד רָחֵל
וַתְּקַשׁ בְּלִדְתָּהּ
If she has already given birth (=וַתֵּלֶד רָחֵל), why should she be in labour (= וַתְּקַשׁ בְּלִדְתָּהּ)? Now forget that derasha I tried to make from the mapik heh. Rather:
וַתֵּלֶד רָחֵל = Rachel gave birth.
וַתְּקַשׁ בְּלִדְתָּהּ = Rachel was in labour. Then, she must have given birth, for we read:
וַיְהִי בְהַקְשֹׁתָהּ, בְּלִדְתָּהּ; וַתֹּאמֶר לָהּ הַמְיַלֶּדֶת = Rachel is in labour and the midwife tells Rachel something.

Now, the midwife tells Rachel, who is in labour, that she has yet a son in her. This means that Rachel is currently in labour with the son. This son is Binyamin).
Since Rachel only gave birth to two sons (Yosef and Binyamin), and the fact that the midwife is saying that in addition to what you gave birth to, you are also having a son, the first two births must have been daughters.

Going back, the students of Rabbi Yehoshua might have only (or additionally) been commenting on the midwife's statement (which is how the meforshim seem to understand it). That is, she is not telling her that there is an additional child in her, but gam is used asseveratively (="indeed"). She is telling Rachel that the single child that she is giving birth to is a son. Note that the students do not use the word gam in their explanation.

I do not think that this is not a simple matter of darshening the word gam to include, thus making 2 daughter (or one additional daughter), as the Matnot Kehuna suggests.