Question about the CLRS Linear Programming chapter
Lemma 29.4 mentions an LP, because that is what they need it for, but it is really a statement in linear algebra about the system of equations $Ax=b$ and a linear form $\phi(x)=c\cdot x$, which in the LP is the objective function.

Lemma 29.4:
If the variables $x_i,i\in B$ are basic variables, i.e. if the corresponding columns of $A$ are a maximal set of linearly independent columns, then you can solve for those variables in terms of the rest of the variables $x_i,i\in N$ and the polynomials obtained are unique. In addition, if you write $\phi(x)=\psi(x_i,i\in N)$, the polynomial $\psi$ is also unique.

(Well, this version of the lemma includes the assertion that one can solve for those variables. This is due to Gaussian elimination.)
Now, for the uniqueness, as they did, you assume that you have two expressions of the basic variables in terms of the non-basic ones
$(x_i)_{i\in B}=(b_i)_{i\in B}+S(x_j)_{j\in N}$ for some matrix $S$ and $(x_i)_{i\in B}=(b_i')_{i\in B}+S'(x_j)_{j\in N}$.
So the polynomials $(b_i)_{i\in B}+S(x_j)_{j\in N}$ and $(b_i')_{i\in B}+S'(x_j)_{j\in N}$ are the same.
By definition $(b_i)_{i\in B}=(b_i')_{i\in B}$ and $S=S'$.
There is no real need to talk about values of the variables, since all the statements are purely algebraic, in the ring of polynomials in the variables $x_i$. So, there is no need for Lemma 29.3. Of course, the equations written have to be thought as equations between polynomials and not as equations between values of functions that are satisfied for all values of the variables. Ignore this paragraph if it causes confusion.