I am referring to Proposition 1.6 of CHAPTER 5 in Milne's notes on Class Field Theory, it claims that given a modulus $\mathfrak m$ of $K$, and its ray class group $\mathcal C_{\mathfrak m}$,
Every class in $\mathcal C_{\mathfrak m}$ is represented by an integral ideal $\mathfrak a$, and two integral ideals $\mathfrak a$ and $\mathfrak b$ represent the same class in $\mathcal C_\mathfrak m$ if and only if there exit nonzero $a,b \in \mathcal O_K$ such that $$a \equiv b \equiv 1 \mod{\mathfrak m_0},$$ and $a$ and $b$ have the same sign for every real prime dividing $\mathfrak m$.
My question is this:
For a proof, taking an element $\mathfrak a$ of $I^{S(\mathfrak m)}$ and writing it as $\mathfrak b \mathfrak c^{-1}$ with $\mathfrak b$ and $\mathfrak c$ integral ideals in $I^{S(\mathfrak m)}$, he chooses a nonzero $c \in \mathfrak c \cap K^S$, (which exists buy CRT) and notes that $c \mathfrak a$ is integral. What guarantees that $c \in K_{\mathfrak m,1}$, since $\mathcal C_{\mathfrak m}$ is defined to be $I^{S(\mathfrak m)}/K_{\mathfrak m,1}$? From the construction suggested, it only follows that the prime factoisation of $c$ doesn't involve any finite primes of $S(\mathfrak m)$. Indeed, we can apply CRT after requiring that $c$ be $1 \equiv \wp^{\mathfrak m(\wp)}$ for all finite primes $\wp$ dividing $\mathfrak m$, but it still doesn't guarantee that $c$ is totally positive.