It is proved that in a pure resistance circuit, the total power consumed on the series circuit P = UI is equal to the thermal power consumed on each resistor

It is proved that in a pure resistance circuit, the total power consumed on the series circuit P = UI is equal to the thermal power consumed on each resistor

Because in a series circuit, the current at each point is equal, and the voltage is distributed according to the resistance, and the total voltage is equal to each resistance
Sum of voltage
P total = u total I P1 = U1 * I P2 = u2i because u total = U1 + U2 I = I1 = I2
So u total I = U1 * I + U2 * I, so p total = P1 + P2

In a series circuit, there is resistance in the consumer, so why is the current equal everywhere?

Do you remember the most basic definition of current: the amount of charge flowing through the cross section per unit time
It should be easier to understand a series circuit than a water pipe connection
Different resistance can be understood as different thickness of interconnected water pipes
However, as long as the water pipe is filled with water, the amount of water flowing through different parts of the water pipe is the same in the same time. This is equivalent to the amount of electric charge flowing through the same part of the water pipe in the same time. Of course, it is the same in unit time (the time of flowing through the water is the same), which means that the electric current is equal everywhere
It can only be said that the water speed of different parts is different, that is, the charge moving speed of finer parts is different, but the current is measured by quantity, not by charge moving speed