Local existence and uniqueness theorem simplified
The theorem can be stated simply as follows. For the equation and initial value problem: $$ y' = F(x,y)\,,\quad y_0 = y(x_0) $$ if $F$ and $∂F/∂y$ are continuous in a closed rectangle $$ R=[x_0-a,x_0+a]\times [y_0-b,x_0+b] $$ in the $x-y$ plane, where $a$ and $b$ are real (symbolically: $a, b ∈ ℝ$) and $×$ denotes the cartesian product, square brackets denote closed intervals, then there is an interval $$ I = [x_0-h,x_0+h] \subset [x_0-a,x_0+a] $$ for some $h ∈ ℝ$ where the solution to the above equation and initial value problem can be found. That is, there is a solution and it is unique.
It seems to say this version of local existence and uniqueness of solution is a simplification of some other version.
So I wonder if it is a special case of Picard–Lindelöf theorem?