Remark 1

Every mapping $f$ of a subset $M$ of $X$ into $X'$, where $M$ belongs to $\mathfrak{F}$, is equivalent mod $S$ to a mapping $f_1$ of $X$ into $X'$ (which justifies the above terminology): it is sufficient to extend $f$ to $X$, e.g. by giving it a constant value on $X - M$.