Question

According to which theorem/principle the sum of pressure head, velocity head, and gravity head is constant:

Answer 1

The height of a liquid column that corresponds to a specific pressure exerted by the liquid column on the base of its container is known as the pressure head in fluid mechanics. It's also known as a static pressure head or just a static head (but not static head pressure).

Complete step by step solution:
pressure head: The height of a liquid column that corresponds to a specific pressure exerted by the liquid column on the base of its container is known as the pressure head in fluid mechanics. It's also known as a static pressure head or just a static head.
velocity head: The bulk motion of fluid causes the velocity head (kinetic energy).
gravity head: gravitational potential energy per unit weight.
A hydraulic head, also known as a piezometric head, is a device that measures liquid pressure above a vertical reference point. At the entrance (or bottom) of a piezometer, it's usually expressed as a liquid surface elevation in units of length. It can be calculated in an aquifer using the depth to water in a piezometric well (a specialized water well) and given the elevation and pressure of the piezometer.
The total energy (the sum of the pressure energy, potential energy, and kinetic energy) per unit mass remains constant at every cross-section throughout the streamlined flow of an ideal liquid, according to Bernoulli's theorem.
The principle was first published in 1738 by Daniel Bernoulli in his book Hydrodynamica. Although Bernoulli deduced that pressure decreases as flow speed increases, Leonhard Euler was the first to derive Bernoulli's equation in its current form in 1752. The principle only applies to isentropic flows, which are those in which irreversible processes (such as turbulence) and non-adiabatic processes (e.g. heat radiation) are small and can be neglected.
Thus, according to Bernoulli’s theorem/principle the sum of pressure head, velocity head, and gravity head is constant.

Note:
Isaac Newton's Second Law of Motion can also be used to derive Bernoulli's principle. There is more pressure behind than in front of a small volume of fluid flowing horizontally from a region of high pressure to a region of low pressure. This exerts a net force on the volume, causing it to accelerate along the streamline.