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

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 LS(A, b) has a unique solution for every choice of the constant vector b. Construct this proof without using Theorem REMEF or Theorem RREFU.