Theme Predication Vs Postposition: Embedded Fact Clause

Halliday & Matthiessen (2014: 125):

Now, one common type of these clauses is that where the postposed Subject is an embedded ‘fact’ clause. Here the pronoun substitute is always it:

it helps a lot to be able to speak the language
I don’t like it that you always look so tired

So if the postposed fact clause is introduced by that, and the matrix clause has the verb be plus a nominal, the result may look like a predicated Theme; for example:

it was a mistake that the school was closed down
it’s your good luck that nobody noticed

But these are not predicated Themes; the postposed Subject is not a relative clause, and there is no agnate form with the predication removed, proportional to it was his teacher who persuaded him to continue: his teacher persuaded him to continue. The last example is in fact ambiguous, and could be used to illustrate the difference: it’s your good luck (that) nobody noticed

(i) predicated Theme: agnate to
nobody noticed your good luck 

(ii) postposed Subject: agnate to
the fact that nobody noticed was your good luck