  Quasi-binary reaction of the deuteron breakup $p+d\to (pp)+n$ with the final
proton-proton pair $(pp)$ in the
  $^1S_0$ state is analyzed at initial energies $0.5 - 2$ GeV in the kinematics
of backward elastic pd-scattering $pd\to dp$. On the basis of the main
mechanisms of the $pd\to dp$ process, including initial and final state
interaction, we show that unpolarized cross section and spin observables of
this reaction exhibit important properties of the half-off-shell
$pp(^1S_0)$-scattering amplitude, which are relevant to the nucleon-nucleon
interaction at short distances.
