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If a Ftr1/ Wiz 5 / EK 1 takes the second level in Eldritch Knight, does he gain two new spells "automatically"?

The PRD says that when a Wizard gains a level, he does learn two new spells:

At each new wizard level, he gains two new spells of any spell level or levels that he can cast (based on his new wizard level) for his spellbook.

The PRD says something different for the Eldritch Knight:

Spells per Day: At the indicated levels, an eldritch knight gains new spells per day as if he had also gained a level in an arcane spellcasting class he belonged to before adding the prestige class. He does not, however, gain any other benefit a character of that class would have gained, except for additional spells per day, spells known (if he is a spontaneous spellcaster), and an increased effective level of spellcasting.

Emphasis mine.

That seems to me that the wizard learns two new spells when he gains a level but the Eldritch Knight based on a wizard does not, since a wizard is not a spontaneous spellcaster. Is that an error in the text? Or is that by design?

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A prestige class that advances spellcasting usually does not also add spells to a class's spellbook

The FAQ says

Question: Does a wizard (or other character that uses a spellbook), receive bonus spells to add to his spellbook when he gains a level in a prestige class that grants an increase to spellcasting? Answer: No. The increase to his spellcasting level does not grant any other benefits, except for spells per day, spells known (for spontaneous casters), and an increase to his overall caster level. He must spend time and gold to add new spells to his spellbook.

An eldritch knight who enters that prestige class as a wizard, for example, receives no additional spells in his spellbook for advancing as an eldritch knight.

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