x_{1} (function) is the restriction of function x_{2} to domain set x_{3}

x_{3} should normally be a (strict) subset of the domain set of x_{1}; however, for generality, any set is allowed, but x_{1} will be undefined everywhere except on/in the intersection of x_{3} with the domain set of x_{2}. Outside of x_{3}, x_{1} is undefined, even if x_{2} is. In/on x_{3}, x_{1} and x_{2} are identically equal everywhere. x_{2} is a specific extension of x_{1}. For the sense of restriction in which, instead of being undefined outside of x_{3}, x_{1} is identically equal to the zero element of the space on the set exclusion of the domain of x_{2} lacking x_{3} (and which is undefined everywhere not in the domain set of x_{2}), try to use a product of a zdeltakronekre function with x_{2}, which will be defined on the intersection of their two domain sets and will identically be 0 or x_{2} as specified. Be careful, concerning x_{3}, when the domain set of x_{2} is the Cartesian product (pi'u) of sets; for example, R is not a subset of R^{2}, even if it is isomorphic to one (actually, uncountably infinitely many in this case).