Yeah, the key insight I think is that if you don't put any constraints on some variable then the intent is that the equation is true for any choice of that vari

Yeah, the key insight I think is that if you don't put any constraints on some variable then the intent is that the equation is true for any choice of that variable- so since x_i = y_i for any i, then it's logically the same as saying x0 = y0, x1 = y1, ... It was an aha moment when I realized what they were doing there. Very elegant actually.