It is not necessary to know Iz if method use is5.1 and 5.2

Thanks so much.

I do get (i think) When selecting a cable, dividing the (In or Ib) by the correction factors gives us the minimum tabulated rating required, so harsher installation conditions will push (It) upwards.When verifying a chosen cable, multiplying the tabulated rating by the correction factors gives the actual derated capacity, so the same harsh conditions drive Iz downwards.
They should roughtly do the same job, but the It≥ C is a quicker method

But I keep seeing Ib ≥ In ≥ Iz  as the ‘big thing’

Hopefully the other way around: Ib ≤ In ≤ Iz (a small, but very significant difference!), but yes (where overload protection is required) that's the fundamental requirement. That doesn't mean you have to go about calculating it in that direct manner though - as long as the result you come out with satisfies that requirement, all is well. It's rather like saying you need to show that 1 ≤ 2 ≤ 3, but you've only got the figures 10, 20 and 30 instead - you could divide everything through by 10 and prove it, but you can also see directly that the same relationship will hold true. In our case things are multiplied (or divided) by all the correction factors, rather than 10, but the same logic holds. If it's true that "how much current you need the cable to carry" ≤ Iz then it must also be true that  ("how much current you need the cable to carry" divided by C) ≤ (Iz divided by C) (for any positive value of C).

But it appears to me that  (Iz) is not really established

Indeed - with the 'alternative method' you only establish your minimum acceptable figure for Iz (which is the same as "how much current you need the cable to carry") you don't need to calculate Iz of the actual cable size you select. Because the cable size you actually select can't be smaller than the theoretical "just the right size" cable calculated, the actual Iz of the chosen cable can't possibly be smaller than your minimum acceptable figure, so you can be sure the requirement is met even without calculating the exact figure.

   - Andy.

But I keep seeing Ib ≥ In ≥ Iz  as the ‘big thing’

Hopefully the other way around: Ib ≤ In ≤ Iz (a small, but very significant difference!), but yes (where overload protection is required) that's the fundamental requirement. That doesn't mean you have to go about calculating it in that direct manner though - as long as the result you come out with satisfies that requirement, all is well. It's rather like saying you need to show that 1 ≤ 2 ≤ 3, but you've only got the figures 10, 20 and 30 instead - you could divide everything through by 10 and prove it, but you can also see directly that the same relationship will hold true. In our case things are multiplied (or divided) by all the correction factors, rather than 10, but the same logic holds. If it's true that "how much current you need the cable to carry" ≤ Iz then it must also be true that  ("how much current you need the cable to carry" divided by C) ≤ (Iz divided by C) (for any positive value of C).

But it appears to me that  (Iz) is not really established

Indeed - with the 'alternative method' you only establish your minimum acceptable figure for Iz (which is the same as "how much current you need the cable to carry") you don't need to calculate Iz of the actual cable size you select. Because the cable size you actually select can't be smaller than the theoretical "just the right size" cable calculated, the actual Iz of the chosen cable can't possibly be smaller than your minimum acceptable figure, so you can be sure the requirement is met even without calculating the exact figure.

   - Andy.