assertions

is the cut off means e.g. there is a sale of 500 and it "claims" or "states" that sepparation exists between what small small figures make up the whole figure of 500.??

Cut-off is making sure revenue and expenses are recognised in the right period. It also means ensuring that if an item is included in closing inventory, then it has also bee recognised in purchases but not recognised in sales ie that its treatment is consistent.

for example, according to accruals basis, sales should be recognised when a product is shipped/service provided and expenses when theyre incurred. It checks that it is done properly, right? or not exactly?

Yes.

sir, could you tell me please, recording and assessing accounting systems: narrative notes, flowcharts and questionnaries are used by auditor or the client?

by auditor at interim audit right?

First question: the auditor. Second question: yes - at the interim audit so that tests of procedures can also be carried out then.

thank you. how can we establish test of control using Control Activity such as Arithmetic control?

Eg, has someone evidenced the arithmetic check by stamping the document something like 'Calculations checked'.