Is the prayer of a traveler who intended to make up Friday prayer as four rak'ahs of Dhuhr, then prayed it again with the same intention, valid?

If the worshipper enters the prayer with the intention of Dhuhr instead of Jumu'ah, and then three people join him and pray four rak'ahs with him, their prayer is valid.

However, if the worshipper's intention upon entering the prayer was two rak'ahs to make up for Jumu'ah, and then he changed his intention to Dhuhr when the other worshippers joined him, his prayer is invalid; due to the invalidity of the first intention and the second intention not being present from the beginning of the prayer.

The prayer of those who prayed behind him is valid; due to the hidden nature of the reason for invalidity from them.

Changing the intention from a Sunnah prayer to making up for Jumu'ah a second time invalidates it.

The outcome is that the prayer is not valid and must be repeated, unless the intention upon entering the first prayer was Dhuhr instead of Jumu'ah.