We apply More on Morphisms, Theorem \ref{more-morphisms-theorem-stein-factorization-Noetherian} to get a Stein factorization $X \to X' \to Y$. Then $X \to X'$ satisfies all the hypotheses of the lemma (some details omitted). Thus after replacing $Y$ by $X'$ we may in addition assume that $f_*\mathcal{O}_X = \mathcal{O}_Y$ and that the fibres of $f$ are geometrically connected. \medskip\noindent Assume that $f_*\mathcal{O}_X = \mathcal{O}_Y$ and that the fibres of $f$ are geometrically connected. Note that $y \in Y$ is a closed point as $f$ is closed and $E$ is closed. The restriction $f^{-1}(Y \setminus \{y\}) \to Y \setminus \{y\}$ of $f$ is a finite morphism (More on Morphisms, Lemma \ref{more-morphisms-lemma-characterize-finite}). Hence this restriction is an isomorphism since $f_*\mathcal{O}_X = \mathcal{O}_Y$ since finite morphisms are affine. To prove that $\mathcal{O}_{Y, y}$ is regular of dimension $2$ we consider the isomorphism $$ \mathcal{O}_{Y, y}^\wedge \longrightarrow \lim H^0(X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n), \mathcal{O}) $$ of Cohomology of Schemes, Lemma \ref{coherent-lemma-formal-functions-stalk}. Let $E_n = nE$ as in Lemma \ref{lemma-exceptional-first-kind-local}. Observe that $$ E_n \subset X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^n) $$ because $E \subset X_y = X \times_Y \Spec(\kappa(y))$. On the other hand, since $E = f^{-1}(\{y\})$ set theoretically (because the fibres of $f$ are geometrically connected), we see that the scheme theoretic fibre $X_y$ is scheme theoretically contained in $E_n$ for some $n > 0$. Namely, apply Cohomology of Schemes, Lemma \ref{coherent-lemma-power-ideal-kills-sheaf} to the coherent $\mathcal{O}_X$-module $\mathcal{F} = \mathcal{O}_{X_y}$ and the ideal sheaf $\mathcal{I}$ of $E$ and use that $\mathcal{I}^n$ is the ideal sheaf of $E_n$. This shows that $$ X \times_Y \Spec(\mathcal{O}_{Y, y}/\mathfrak m_y^m) \subset E_{nm} $$ Thus the inverse limit displayed above is equal to $\lim H^0(E_n, \mathcal{O}_n)$ which is a regular two dimensional local ring by Lemma \ref{lemma-exceptional-first-kind-local}. Hence $\mathcal{O}_{Y, y}$ is a two dimensional regular local ring because its completion is so (More on Algebra, Lemma \ref{more-algebra-lemma-completion-regular} and \ref{more-algebra-lemma-completion-dimension}). \medskip\noindent We still have to prove that $f : X \to Y$ is the blowup $b : Y' \to Y$ of $Y$ at $y$. We encourage the reader to find her own proof. First, we note that Lemma \ref{lemma-exceptional-first-kind-local} also implies that $X_y = E$ scheme theoretically. Since the ideal sheaf of $E$ is invertible, this shows that $f^{-1}\mathfrak m_y \cdot \mathcal{O}_X$ is invertible. Hence we obtain a factorization $$ X \to Y' \to Y $$ of the morphism $f$ by the universal property of blowing up, see Divisors, Lemma \ref{divisors-lemma-universal-property-blowing-up}. Recall that the exceptional fibre of $E' \subset Y'$ is an exceptional curve of the first kind by Lemma \ref{lemma-blowup}. Let $g : E \to E'$ be the induced morphism. Because for both $E'$ and $E$ the conormal sheaf is generated by (pullbacks of) $a$ and $b$, we see that the canonical map $g^*\mathcal{C}_{E'/Y'} \to \mathcal{C}_{E/X}$ (Morphisms, Lemma \ref{morphisms-lemma-conormal-functorial}) is surjective. Since both are invertible, this map is an isomorphism. Since $\mathcal{C}_{E/X}$ has positive degree, it follows that $g$ cannot be a constant morphism. Hence $g$ has finite fibres. Hence $g$ is a finite morphism (same reference as above). However, since $Y'$ is regular (and hence normal) at all points of $E'$ and since $X \to Y'$ is birational and an isomorphism away from $E'$, we conclude that $X \to Y'$ is an isomorphism by Varieties, Lemma \ref{varieties-lemma-modification-normal-iso-over-codimension-1}.