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The exclusive reaction $^{12}$C$+p\to ~^{10}A+pp+N$, where the $^{12}$C beam at energy of 4 GeV/nucleon interacts with the proton target and as a result one proton is knock-outed from a short-range correlated $NN$-pair in $^{12}$C, was measured recently by the BM@N Collaboration. This reaction is considered here %in the distorted-wave impulse approximation by analogy with theory of quasi-elastic knockout of fast deuterons from nuclei $(p,Nd)$ using the translationally-invariant shell model with short-range $NN$-correlations. The initial and final state interaction effects are estimated in the eikonal approximation using the Glauber model for elastic $N-~^{10}$A scattering.