Assume that a survival time (T) has a constant hazard (lambda) over time interval (left[tau_{j}, tau_{j+1} ight))

Assume that a survival time \(T\) has a constant hazard \(\lambda\) over time interval \(\left[\tau_{j}, \tau_{j+1}\right)\) with \(\log \lambda=\mathbf{x}^{\top} \boldsymbol{\beta}\). Prove \(1-p=\exp \left(-\exp \left(\mathbf{x}^{\top} \boldsymbol{\beta}\right)\right)\), where \(p=\operatorname{Pr}\left(T<\tau_{j+1} \mid T \geq \tau_{j}\right)\).