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In this sentence:

> 石の上にも三年とは、冷たい石でも三年間座り続ければ暖まる **ように** 、何事にも忍耐強さが大切だという意味です。

Is it saying that even in a cold stone, if something continues to fall/sit on
top of it,in order to warm the stone you have to persevere in everything?  
I don't know how to interpret ように correctly in that position.
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meaning
ように at the end of a relative clause
Is it saying that even in a cold stone, if someone continues to sit on top of
it, he or she could warm the stone in three years. It values a sprit of
patience.

"・・・暖まるように、" = "・・・暖まる。この例が示すように、"

---I added more-----------------------------------------------------------------

石の上にも三年とは、冷たい石でも三年間座り続ければ暖まるように、何事にも忍耐強さが大切だという意味です。

石の上にも三年とは、何事にも忍耐強さが大切だという意味です。 "Perseverance preails: Staying on the stone for
three years" means "Patience is important at everything."

冷たい石でも三年間座り続ければ暖まるように、何事にも忍耐強さが大切だ。 If someone continues to sit on top of it,
he or she could warm the stone in three years. (This is the meaning of
"石の上にも三年")

As this Perseverance preails shows, Patience is important at everything.

冷たい石でも三年間座り続ければ暖まる（と言います。この例が示すように）、何事にも忍耐強さが大切だ。
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Technically, this ように is not in a relative clause because it does not modify a
following noun. It's in an adverbial subordinate clause. For instance,
"because I sing", "if I sing" and "until I sing" are adverbial subordinate
clauses.

ように can mean "in order to ～", but it means something different here. This ～ように
is "as (in) ～", "like (in) ～" used to present an example or something that
resembles.

> 石の上にも三年とは、[冷たい石でも三年間座り続ければ暖まるように、]何事にも忍耐強さが大切だという意味です。
>
> 石の上にも三年 means patience is important in everything, [just as even a cold
> stone can be warmed if someone keeps sitting on it for three years].
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「石の上にも三年とは、冷たい石でも三年間座り続ければ暖まるように、何事にも忍耐強さが大切だという意味です。」 "[Three years on a
rock] means that **just as** even cold rocks will get warm if you sit on it
for 3 years, persistence is important no matter what it is."

In this case, 「ように」 is used to mean "just as ~"

A link for your reference:
<http://thejapanesepage.com/grammar/chapter_five/youni>

A common usage is 「～のように」, but the 「の」 is not necessary, depending on the
structure of the sentence.
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