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import TestGame.Metadata
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import Mathlib.Algebra.Module.Submodule.Lattice
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import Mathlib.LinearAlgebra.Span
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open Submodule
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Game "TestGame"
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World "Module2"
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Level 5
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Title "Lineare Abbildung"
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Introduction
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"
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Die interne Summe von Untermodulen wird in Lean mit `⊔` (`\\sup`) geschrieben.
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"
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Statement
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"
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Zeige, dass `V₁ ⊔ V₂` tatsächlich die interne Summe `⟨V₁ ∪ V₂⟩` ist.
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"
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{K V : Type _} [Field K] [AddCommMonoid V] [Module K V] (V₁ V₂ : Submodule K V) :
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V₁ ⊔ V₂ = span K ( (V₁ : Set V) ∪ V₂ ) := by
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rw [span_union]
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simp
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