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import TestGame.Metadata
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Data.Real.Basic -- definiert `ℝ
import Mathlib.Data.Fin.VecNotation -- Importiert Matrix/Vektor-Notation
--import Mathlib.LinearAlgebra.FinSupp -- contains `top_le_span_range_iff_forall_exists_fun`
import Mathlib.Tactic.FinCases
import Mathlib.Algebra.BigOperators.Finsupp -- default?
import Mathlib.LinearAlgebra.Span
Game "TestGame"
World "Module"
Level 6
Title "Hülle"
Introduction
"
Ein typischer Untervektorraum ist die Hülle `⟨M⟩`, oder `span`, also die Menge aller
`K`-Linearkombinationen von Elementen aus der Menge `M`.
In Lean ist dies `Submodule.span K M`.
"
local notation "ℝ²" => Fin 2 → ℝ
open Submodule Set Finsupp
Statement mem_span_of_mem
"Zeige, dass $x \\in M$ auch Element von der $K$-linearen Hülle
\\langle M \\rangle ist."
{V K : Type _} [Field K] [AddCommMonoid V]
[Module K V] (M : Set V) {x : V} (h : x ∈ M) :
x ∈ span K M := by
rw [mem_span]
intro p hp
specialize hp h
assumption