A theorem on continua

The purpose of this paper is to prove Theoreme: Of two concentric circles ${\displaystyle C_1}$ and ${\displaystyle C_2}$, let ${\displaystyle C_1}$ be the smaller. Denote by H the point set which is the sum of ${\displaystyle C_1}$, ${\displaystyle C_2}$, and the annular domain bounded by ${\displaystyle C_1}$ and ${\displaystyle C_2}$. Let M be a continuum which contains a point A interior to ${\displaystyle C_1}$ and a point B exterior to ${\displaystyle C_2}$. If N is any connected subset of M containing A and B, N will contain at least one point of some continuum which is a subset of M and H, and which has at least one point in common with each of the circles ${\displaystyle C_1}$ and ${\displaystyle C_2}$.