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