Why is the following true? (Conditional expectation)

Question

In the book I'm reading, it says, :

consider a gambler who plays a sequence of fair games. Let X_i be the amount the gambler wins on the ith game (X_i is negative if the gambler loses), and let Z_i be the gambler’s total winnings at the end of the ith game. Because each game is fair, E[X_i] = 0 and E[ Z_i+1 |X₁,X₂,...,X_i]=Z_i +E[X_i+1]= Z_i.

Since game is fair, E[X_i] = 0 is clear to me.

Intuitively, E[ Z_i+1 |X₁,X₂,...,X_i]=Z_i +E[X_i+1]= Z_i makes sense as winnings after i+1th game is simply the sum of earnings till ith game, plus expected earning in i+1th round. But how do I formally prove this? Note that this proves Z_i is a martingale.