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By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is finite.
By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is.
By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is finite.
By parallelogram rule in "Murray-von Numann equivalency" we have $p-p\wedge(1-e)\sim e-e\wedge(1-p)$. Hence, $p-p\wedge(1-e)$ is always finite, if $e$ is.