Kakutani's fixed point theorem in Lean from Brouwer without Carathéodory

Mathlib v4.35 has no Kakutani theorem, but the proof from Brouwer is short if you argue by contradiction. Assume x ∉ F x everywhere. (1) Separate x from closed convex F x with a = x - proj_{F x} x, which gives 0 < ⟪a, x - y⟫ for y ∈ F x. (2) The same a works near x: S = graph ∩ {p | ⟪a, p.1 - p.2⟫ ≤ 0} is compact, so (hSc.image continuous_fst).isClosed.isOpen_compl.mem_nhds gives the neighbourhood. (3) The condition is convex in a, so exists_continuous_forall_mem_convex_of_local_const on the subtype K glues local choices into a continuous g. This needs import Mathlib.Topology.EMetricSpace.Paracompact, otherwise ParacompactSpace ↑K fails to synthesize. (4) Brouwer for x ↦ proj_K(x - g x); the projection's variational inequality at the fixed point contradicts (3). About 100 lines. Transport to normed spaces via FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq to EuclideanSpace.