Explanation on exponent of cardinal (Set Theory and Metric Spaces - Kaplanski)

Question

I think I understand most of the following discussion. However, the highlighted phrase was too opaque for me to follow

Could you please explain:

1. What do the notations $\pi$₁f and $\pi$₂f refer to?

2. Are H(E, D₁) and H(E, D₂) the coordinates of the plane H(E, D₁) x H(E, D₂)?

3. Why is the ordered pair ($\pi$₁f, $\pi$₂f) an element of H(E, D₁) x H(E, D₂)?

Thank you in advance.

Answer 1

An example of ($\pi_1$f, $\pi_2$f).

x(t) = sin t, y(t) = cos t is a parametric equation of a circle.
Restating the same parametric equation
f(t) = (sin t, cos t)
= ($\pi_1$(sin t, cos t), $\pi_2$(sin t, cos t)
= ($\pi_1$f(t), $\pi_2$f(t))

So when f is a function into a product of two sets,
($\pi_1$f, $\pi_2$f) is explicating the coordinate values of f.
(Note that f can be a function of multiple variables.)