Provide an alternative for the second half of the proof of Theorem NMUS, without appealing to properties of the reduced row-echelon form of the coefficient matrix. In other words, prove that if $A$ is nonsingular, then $\linearsystem{A}{\vect{b}}$ has a unique solution for every choice of the constant vector $\vect{b}$. Construct this proof without using Theorem REMEF or Theorem RREFU.