Proof of Theorem SMEE

Let $n$ denote the size of $A$ and $B$. Since $A$ and $B$ are similar, there exists a nonsingular matrix $S$, such that $A=\similar{B}{S}$ (Definition SIM). Then \begin{align*} \charpoly{A}{x}&=\detname{A-xI_n}&&\knowl{./knowls/definition.CP.knowl}{\text{Definition CP}}\\ &=\detname{\similar{B}{S}-xI_n}&&\knowl{./knowls/definition.SIM.knowl}{\text{Definition SIM}}\\ &=\detname{\similar{B}{S}-x\similar{I_n}{S}}&&\knowl{./knowls/theorem.MMIM.knowl}{\text{Theorem MMIM}}\\ &=\detname{\similar{B}{S}-\inverse{S}xI_nS}&&\knowl{./knowls/theorem.MMSMM.knowl}{\text{Theorem MMSMM}}\\ &=\detname{\similar{\left(B-xI_n\right)}{S}}&&\knowl{./knowls/theorem.MMDAA.knowl}{\text{Theorem MMDAA}}\\ &=\detname{\inverse{S}}\detname{B-xI_n}\detname{S}&&\knowl{./knowls/theorem.DRMM.knowl}{\text{Theorem DRMM}}\\ &=\detname{\inverse{S}}\detname{S}\detname{B-xI_n}&&\knowl{./knowls/property.CMCN.knowl}{\text{Property CMCN}}\\ &=\detname{\inverse{S}S}\detname{B-xI_n}&&\knowl{./knowls/theorem.DRMM.knowl}{\text{Theorem DRMM}}\\ &=\detname{I_n}\detname{B-xI_n}&&\knowl{./knowls/definition.MI.knowl}{\text{Definition MI}}\\ &=1\detname{B-xI_n}&&\knowl{./knowls/definition.DM.knowl}{\text{Definition DM}}\\ &=\charpoly{B}{x}&&\knowl{./knowls/definition.CP.knowl}{\text{Definition CP}} \end{align*}