payment practice

In May I received a translation job from a German company (as@uebersetzungen-patente.eu  /    www.stang-translations.com)
while I attended some conferences in Europe.
First, that company wanted to a quote for a partial translation of 5 patents.
I gave them that quote (in Yen) and they ordered the partial translation.
After completing that translation I consulted them regarding the bill and as a result of that consultation sent them a bill (using the same template I used to issue a bill for them 3 years earlier), applying the calculation / rate I DID offer them before.

They DID accept that bill - no complaints at all - and immediately paid it.
Two weeks later, they contacted me, telling me the client was pleased with the work and would like to have a complete translation of said 5 patents - ca. 120 pages.
So, I started work - NATURALLY expecting to be paid the same rate.

AFTER completion of the work, they told me to send a bill over an amount 20% less than what they accepted as OK with the first bill. This 20% less rate is based on THEIR rate from 3 years ago.
The fact, that the exchange rate has changed significantly apparently does NOT interest them at all.
THEY assert, THAT is the rate to be applied and a rate based on Yen has never (???) been agreed upon.
Funny!! My quote was in Yen and the bill they accepted in Yen too.

They NEVER responded to my request for the reason why they accepted that first bill.
It was their freely made decision - and I assume they simply did not check the bill / calculation.
But that is THEIR fault, isn't it?

Why do I have to pay for their mistake?

None of the sites I asked for legal advice good give me any such "advice".

Is this good business practice?
It looks much more like the Chinese style of business to me.

So, if ever asked, beware of above mentioned company - or else get a WRITTEN contract before starting to work.