Are not fooled by: that’s transparent (3-7)

I believe the answer is:

see-through

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'transparent' is the definition.
(I know that transparent can be written as see-through)

'are not fooled by' is the wordplay.
'are not fooled' becomes 'see' (I can't explain this - if you can you should believe this answer much more).
'by' becomes 'through' (I've seen this before).
'see'+'through'='SEE-THROUGH'

'that's' acts as a link.

Can you help me to learn more?

(Other definitions for see-through that I've seen before include "Translucent" , "Transparent" .)

I've seen this clue in the Evening Standard.

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