Fundamental Theorem of Calculus, Part II in One Dimension

Let $I$ be an interval in the sense of Interval in the Real Line, let $[a,b]\subseteq I$, let $f:I\to\mathbb{R}$ be continuous on $[a,b]$ in the sense of Continuity on a Closed Interval, and let $F:I\to\mathbb{R}$ be an antiderivative of $f$ on $I$ in the sense of Antiderivative on an Interval. Then $\int_a^b f(x)\,dx = F(b)-F(a),$ where the integral is the Riemann integral of The Riemann Integral on a Closed Interval via Lower and Upper Integrals.