Let $f(x)$ be a "well behaved" function. Because the functions ${}_n$ that are solutions to a particular TISE form a complete orthonormal set of functions $($basis functions$),$ we can say:

$f(x)=c_1{}_1(x)+c_2{}_2(x)+c_3{}_3(x)+$dots

$={}_n=1^{}{c}_n{}_n(x)$

a$)$ Left multiply the left and right sides of eq $4\mathrm{.}1$ by ${}_j^{**},$ the complex conjugate of the $j^{th}$ solution to the TISE.

b$)$ Integrate all of the terms obtained in part a over the region $c_j{j}^{th}cf(x){\displaystyle \underset{-}{\overset{}{}}}f(x{)}^{**}f(x)dx=1f(x)|c_1{|}^2+|c_2{|}^2+|c_3{|}^2+$dots$={\displaystyle \underset{n=1}{\overset{}{}}}|c_n{|}^2=1|c_n{|}^2=c_n**c_n{\displaystyle \underset{-}{\overset{}{}}}{}_i^{**}{}_{j}dx={}_{ij}f(x)f(x)|c_n{|}^2{}_n^{th}-.$

$c$