Fix integer ${ n \in \mathbb{Z} _{>0} }.$ Like here, consider positive part of ${ n-}$ball ${ (B ^n) ^{+} = \lbrace x \in \mathbb{R} ^{n} : \text{each } x _i > 0, \lVert x \rVert < 1 \rbrace }$ and positive part of ${ n -}$sphere ${ (S ^{n}) ^{+} = \lbrace x \in \mathbb{R} ^{n+1} : \text{each } x _i > 0, \lVert x \rVert = 1 \rbrace }.$
From ${ p = 2 }$ case of theorem ${ 1 }$ in the link, random vector ${ Z = \phi(Y) \in (S ^n) ^{+} }$ is uniform iff random vector ${ Y \in (B ^{n}) ^{+} }$ has density ${ f _{Y} (y) = \frac{\text{const}}{z _{n+1}} }$ ${ = \frac{\text{const}}{\sqrt{1 - \sum _{i=1} ^{n} y _i ^2 }} .}$
Consider the map ${ \psi : (B ^{n}) ^{+} \to (B ^{n-1}) ^{+} }$ sending ${ (y _1, \ldots, y _n) \mapsto (x _1, \ldots, x _{n-1}) = (y _1, \ldots, y _{n-1}) }.$
If ${ Y \in (B ^n) ^{+} }$ has density ${ f _{Y}(y) }$ then ${ X = \psi(Y) \in (B ^{n-1}) ^{+} }$ has density ${ f _{X} (x) = \int _{0} ^{\sqrt{1-\sum _{i=1} ^{n-1} x _i ^2} } f _{Y} (x _1, \ldots, x _{n-1}, u) \, du }.$
If ${ Z \in (S ^n) ^{+} }$ is uniform, the density of ${ Y = \phi ^{-1} (Z) \in (B ^n) ^{+} }$ is ${ f _{Y} (y) = \frac{\text{const}}{\sqrt{1 - \sum _{i=1} ^{n} y _i ^2 }} }$.
Hence the density of ${ X = \psi(Y) \in (B ^{n-1}) ^{+} }$ is ${ f _{X} (x) = \int _{0} ^{\sqrt{1-\sum _{i=1} ^{n-1} x _i ^2} } f _{Y} (x _1, \ldots, x _{n-1}, u) \, du }$ ${ = \int _{0} ^{L} \frac{\text{const}}{\sqrt{L ^2 - u ^2}}\, du }$ where ${ L = \sqrt{1-\sum _{i=1} ^{n-1} x _i ^2 } }.$
But integral ${ \int _{0} ^{L} \frac{du}{\sqrt{L ^2 - u ^2}} }$, on substituting ${ u = L \sin(\theta) ,}$ is the constant ${ \frac{\pi}{2} }.$ Hence above density of ${ X \in (B ^{n-1}) ^{+} }$ is constant. This gives:
Thm: Fix integer ${ n \geq 2 }.$ Say random vector ${ Z \in (S ^{n}) ^{+} }$ is uniform over ${ (S ^n) ^+ }.$ Then the random vector ${ X = (\psi \circ \phi ^{-1}) (Z) = (Z _1, \ldots, Z _{n-1}) }$ in ${ (B ^{n-1}) ^{+} }$ is uniform over ${ (B ^{n-1}) ^{+} .}$
Now by symmetry:
Thm: Fix integer ${ n \geq 2 }.$ Say random vector ${ Z }$ is uniform over ${ S ^n }.$ Then the random vector ${ X = (Z _1, \ldots, Z _{n-1}) }$ in ${ B ^{n-1} }$ is uniform over ${ B ^{n-1} .}$