Let $M$ be a von Neumann algebra. If $x$ is positive, then Lemma 2.1(3) of the paper "Order Isomorphisms of Operator Intervals in von Neumann Algebras" (Mori, Integral Equations and Operator Theory, 2019, https://arxiv.org/abs/1811.01647) states that if $p$ is the support projection of $x$, then the order interval $[0,p]$ is order isomorphic to $[0,x]$ through the map $y \mapsto x^{1/2} y x^{1/2}$. The proof here is claimed to be "easy to see".

But in the earlier paper "Order isomorphisms of operator intervals" (Semrl, Integral Equations and Operator Theory, 2017, https://www.fmf.uni-lj.si/~semrl/preprints/orderoperatorintervals.pdf), Semrl proves this in the special case where $M = B(H)$ and $x$ is injective, i.e., its support projection is $I$. His proof takes 1.5 pages (pages 38-39). I do not understand why Mori claims that his more general result is "easy to see". I need this in one of my papers and so I proved this myself based on characterising the inverse as the map $y \mapsto \lim_n f_n(x) y f_n(x)$ (weak*-limit) where $f_n(t) := t^{-1/2}$ for $t \geq 1/n$ and 0 elsewhere. But this proof is not trivial, one has to show, amongst other things, that this limit always exists if $0 \leq y \leq x$.

Am I missing a simple argument showing that $y \mapsto x^{1/2} y x^{1/2}$ is an order isomorphism between $[0,p]$ and $[0,x]$?