How to prove the inequality marked in red?
In order to prove $t_2\leq t_1$, we need to verify that $\tau(\chi_{(t_1,\infty)}(A_1))\leq 1$.
Since $\|A_1\|_e=0$( $\|A_1\|_e $ is the essential norm of $A_1$), we have for any $\epsilon＞0$, $\tau(\chi_{(\epsilon,\infty)}(A_1))＜\infty$. How to check that $\tau(\chi_{(t_1,\infty)}(A_1))\leq 1$?
Since $(I-E)$ is a projection that commutes with $A$, it is easy to see that $f(A(I-E))=f(A)\,(I-E)$ for any Borel function $f$. Thus $$ \tau\big(1_{(s,\infty)}(A(I-E))\big)=\tau\big(1_{(s,\infty)}(A)\,(I-E)\big) \leq\tau(1_{(s,\infty)}(A)). $$ So $\mu_1(A_1)\leq\mu_1(A)$.