Since $\(\mathrm{A}, \mathrm{B}, \mathrm{C}\)$, and D are positive integers and $\(B=\mathrm{A}-4\)$, this implies $\(\mathrm{A} \geq 5\)$.

Also, it is given that $\(\mathrm{C}=\mathrm{A}-1\)$ and $\(A+B+C=D+1\)$.

Substituting the values of $\(B\)$ and $\(C\)$ in terms of $\(A\)$, in the given equation, we get $\(A+A-4+A-1=D+1 \Rightarrow 3 A=D+6 \Rightarrow D=3 A-6\)$. If $\(A=5, D=9\)$ and if $\(A=6, D=12\)$, which is not a single digit number.

Therefore, $\(A=5, D=9, B=1\)$, and $\(C=4\)$.

Hence, the required four - digit number and the correct answer is \(5149\).