Let $(X,\mathscr O_X)$ be a scheme which admits an open covering of subschemes $\{(U_i,\mathscr O_{X \mid U_i})\}$, i.e. $X= \cup_{i \in I} U_i$ with $U_i$ being affine scheme. Consider $\varphi_i$ as the isomorphism between $U_i$ and some $\operatorname{Spec}A_i$. I know that for any $p \in U_i \cap U_j$, $\mathscr O_{X,p} = \mathscr O_{X \mid U_i,p} = \mathscr O_{X \mid U_j,p}$ by showing that there is an one-to-one correspondence with germs.
But I try to break it down and figure it out algebraically. So $\mathscr O_{X \mid U_i,p} = \mathscr O_{\operatorname{Spec}A_i,\varphi_i(p) = A_{i_{\varphi_i(p)}}}$ and $\mathscr O_{X \mid U_j,p} = \mathscr O_{\operatorname{Spec}A_j,\varphi_j(p) = A_{j_{\varphi_j(p)}}}$. Then I try to show that $A_{i_{\varphi_i(p)}} = A_{j_{\varphi_j(p)}}$, and I am stuck...
How can I proceed? The "isomorphism" thing really bothers me...