Gemara

(Mishnah): Temidim are not Me'akev Musafim. Musafim are not Me'akev Temidim.

(Abaye): The Mishnah teaches that either may be offered first. The law that no Korban precedes the morning Tamid is only l'Chatchilah.

Berachos 26a (R. Yochanan): If one mistakenly omitted Ma'ariv, he prays (Shemoneh Esre in) Shacharis twice. If he mistakenly omitted Shacharis, he prays Minchah twice.

26b (Beraisa): If one mistakenly omitted Minchah on Shabbos, he prays two weekday Tefilos (Shemoneh Esreh) on Motza'i Shabbos. He says Havdalah in the first, but not in the second.

If he said Havdalah in the second and not in the first, he was Yotzei the second Tefilah, but not the first.

Inference: Since he did not say Havdalah in the first, it is as if he did not pray (the first).

Contradiction (Beraisa #2): If (in winter) one mistakenly omitted the mention of rain in the second Berachah or the request for rain in the ninth, he must go back;

If he mistakenly omitted Havdalah in the fourth Berachah, he does not go back, for he can say Havdalah later with wine. (Likewise, in Beraisa #1 the first Tefilah should be valid, for he can say Havdalah later!)

This is left difficult.

Rishonim

Rif (Berachos 18a): If he said Havdalah only in the second Shemoneh Esreh, only the latter was valid, for one must pray first Chovah (the current obligation). Here, he was Megaleh Da'as (showed his intent) that the first is Tashlumin and the second is Chovah. Therefore, he must pray again; so the Chovah will be first.

Rambam (Hilchos Tefilah 3:8): If one did not pray a Tefilah in its time, because he was Shogeg, Ones or distracted, he compensates for the missed Tefilah at the next time for Tefilah. First he prays the Chovah, and then the Tashlumin.

Rosh (Berachos 4:2): Rav Hai Gaon says that even though the Gemara left the question difficult, since it did not say 'Teyuvta' (refutation), we do not say that the Beraisa is mistaken. It is reasonable that since he said Havdalah in the second, he was Megaleh Da'as that the first is Tashlumin and the second is Chovah. One may not pray Tashlumin before Chovah. Therefore, he must repeat the Tashlumin. However, if he did not say Havdalah in either one, or he said Havdalah in both, he does not pray again, for there is no Giluy Da'as that the first was for Tashlumin. The Gemara's question supports this. It inferred 'since he did not say Havdalah in the first, it is as if he did not pray.' This implies that if it were as if he prayed, he was Yotzei both, even though he said Havdalah only in the second. The first counted for Chovah, and the second was Tashlumin.

Rashba (26b DH Im): If he made Havdalah only in the second, he was not Yotzei with the first. We hold like this. Even though the Gemara left this question difficult, perhaps they could have found an answer. We can say that when he was Megaleh Da'as that the first is for Tashlumin, since he should not do so, he must pray a third time. We view the first as if it were not, in order that Tashlumin not precede Chovah. However, if he did not say Havdalah in either one, or he said Havdalah in both, he does not pray again, for there is no Giluy Da'as that the first was not for Chovah. Even though he was not Mavdil, perhaps he erred. When he was Mavdil in both, perhaps he erred the second time and did not intend to say Havdalah. A Gaon says so.

Poskim

Shulchan Aruch (OC 108:1): If one did not pray Shacharis due to a mistake or Ones, he prays Minchah twice. The first Tefilah is Minchah and the second is Tashlumin. If he switched the order, he was not Yotzei Tashlumin, and he must pray Tashlumin again. The same applies whenever one must pray Tashlumin.

Darchei Moshe (1): The Gemara implies that if he intended that the first be for Tashlumin, he was not Yotzei.

Taz (2): The Shulchan Aruch learns from one who needed to pray Tashlumin for Minchah, and he said Havdalah only in the second Tefilah. He must pray a third time, for he showed that he intended that the first be Tashlumin. The same applies here. We need not say so. Perhaps there is different, for he did an act (said Havdalah only in the latter) to show that the first was Tashlumin. Here, his improper intent was only in the heart!

Magen Avraham (2): The Shulchan Aruch is difficult, for the Gemara left the matter difficult! The Rosh said that when there is a Giluy Da'as that he intended the first for Tashlumin, he was not Yotzei. If he said Havdalah in both or neither, there is no Giluy Da'as, so he was Yotzei. The Gemara's question implies that if it were as if he prayed, he was Yotzei both, even though he said Havdalah only in the second. This shows that only a Giluy Da'as disqualifies, but not mere intent. If not for the Darchei Moshe, we could say that also here he was Megaleh Da'as, e.g. he prayed first without Ashrei, and then said Ashrei and Shemoneh Esre, like we do for Minchah. In any case, this requires investigation, for it is a Giluy Da'as only if it is evident from the Tefilah itself, e.g. he did not say Havdalah.

Note: When he was not Mavdil in the first Tefilah, why is this called 'evident from the Tefilah itself?' If he will not be Mavdil also in the second Tefilah, he was Yotzei! (If the 'the Tefilah itself' means the words he said, including the second Tefilah, also Ashrei should be included!) Perhaps the Chazakah (assumption) is that one does not forget Havdalah, so it is evident from the omission in the first Tefilah that it was for Tashlumin, unless later it becomes clear that he forgot. (If he was not Mavdil either time, surely one time he forgot.)

Magen Avraham (2): The Rashba connotes like this, that if he said Havdalah in both, he was Yotzei even if he intended that the latter is for Chovah. We say that perhaps he erred and said Havdalah. If so, this is not Giluy Da'as.

Machatzis ha'Shekel (DH v'Chen): Why don't we ask him what he intended? We must say that his intent does not matter, because it was not evident from his Tefilah.

Eliyahu Rabah (1): The Levush says that Chovah may be first because Tadir always comes first. This is difficult, for in Menachos we say that this is only l'Chatchilah, but it is not Me'akev! Ma'adanei Melech (Berachos 4:11, and Malbushei Yom Tov 1) explain(s) that it is lest people pray Tashlumin even not at the time of the coming Tefilah, i.e before the time for the coming Tefilah (it is not yet a time for Tefilah). Based on what I wrote (89:3), that one may pray Shacharis until Minchah Gedolah, this is difficult. (Every moment of the day is time for Tefilah!) Rather, it is because Tefilah is accepted only along with the next Tefilah. If he prays beforehand, it is not accepted, for the time passed. Based on Sod (deep secrets), it is because one should not weaken the Midah (of Hash-m's influence) that applies at the time and cause another Midah to come earlier.

Kaf ha'Chayim (2): Based on the Ari Zal, at Shacharis there is Hamshachas Muchin (this sounds like drawing influence of intellect. I asked a great Mekubal, and he was unsure - PF) through inner Kelim. At Minchah it is through intermediate Kelim, and in Ma'ariv through outer Kelim. It is problematic to pray Tashlumin first, for Hamshachas Muchin is through the wrong Kelim. If he prays Chovah first, Hamshachas Muchin will come through the proper Kelim, and then he can pray also the Tashlumin.

Mishnah Berurah (7): If he prayed Tashlumin first he was not Yotzei, for Chachamim enacted Tashlumin only when one is engaged in the obligatory Tefilah, but not beforehand. Some say that he must pray again only if he explicitly was Megaleh Da'as in the Tefilah that the latter is Chovah and the first is Tashlumin, e.g. the case of Motza'ei Shabbos. The Pri Chodosh, Pri Megadim and Derech ha'Chaim say that we rely on the Mechaber. In any case it is better to intend when praying again 'if I am exempt, this is Nedavah.'

Kaf ha'Chayim (3): One should have a Chidush (new request) in the extra Tefilah to avoid concern for Berachos l'Vatalah. If one prayed twice Stam without intent for which is the Chovah and which is the Tashlumin, presumably he intended to do like the Halachah; Hash-m accepts the first for Chovah.

Mishnah Berurah (8): He was not Yotzei whether he switched the order b'Mezid or b'Shogeg.

Mishnah Berurah (9): If he prayed the first for Chovah and the second for Tashlumin, and erred in the first and omitted 'Ten Tal u'Matar' in a way that obligates praying the Chovah again, and did not realize until after the second Tefilah, the Tashlumin was valid. He prays again for the Chovah. We do not say that it is as if he did not pray at all (i.e. the Tashlumin preceded the Chovah), for he prayed Tashlumin at a time of (engaging in) Tefilah. The same applies to all mistakes that invalid Tefilah. Magen Giborim says that this is if he explicitly intended that the latter be Tashlumin. If not, it is better to consider the second to be for Chovah, and he prays again for Tashlumin, and he is Yotzei according to all opinions.

Bi'ur Halachah (DH v'Im): The Gra did not mention the Magen Avraham's question. It seems that he disagrees. If one intended Stam, all agree that he was Yotzei. If one began to pray Tashlumin (first), and in the middle remembered and finished it properly, i.e. for Chovah, I am unsure if this helps. Since the Berachos of Tefilah must be in order, perhaps it is like one matter, so uprooting his intent (for Tashlumin) helps. Even though in many respects the first three Berachos are considered like one, perhaps uprooting intent is different. Perhaps 107:1 proved this. If one began to pray and then realized that he already prayed, he stops, even in the middle of a Berachah. Rishonim explain that since he began with intent for Chovah, he cannot finish it for Nedavah. If so, the same applies here, and he must stop in the middle. Or, perhaps Chachamim were lenient about Tashlumin. This requires investigation.

Note: Also, perhaps if one began Stam (for Chovah) and finishes with a new intent (Nedavah), it is not as if he began with that intent, but if he finishes Stam (for Chovah, i.e. not for Tashlumin), this uproots his initial special intent (for Tashlumin). We find that vows can be uprooted retroactively. For Zechiyah (Reuven acquired on behalf of Shimon), if Shimon is pleased when he finds out, this helps retroactively only if it was (almost) certain that he would be pleased, i.e. it is purely beneficial. However, for Hechsher, if Shimon was pleased when he learned that his Peros became wet, it is as if he was pleased from the beginning.

(Taz 684:4): If one prayed Musaf before Minchah (even after the time of Minchah came), he was Yotzei (OC 286:4). This is unlike one who prayed Tashlumin before the current Tefilah. He was not Yotzei, for one must pray the current Tefilah first. There, it was not time for Tashlumin. He is like one who prayed not in the time of Tefilah.