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d5b7741 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | import OAI.Combinatorics.Crossing.DrawingDefinition
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Separation.Hausdorff
namespace OAI
namespace Zarankiewicz
theorem path_range_eq_of_subset {a b : Plane} (γ δ : Path a b)
(hγ : Function.Injective γ) (hsub : Set.range δ ⊆ Set.range γ) :
Set.range δ=Set.range γ := by
apply Set.Subset.antisymm hsub
rintro x ⟨t,rfl⟩
have hc : IsPreconnected (γ ⁻¹' Set.range δ) :=
(isPreconnected_range δ.continuous).preimage_of_isClosedMap hγ γ.continuous.isClosedMap hsub
have h0 : (0 : unitInterval) ∈ γ ⁻¹' Set.range δ := ⟨0,δ.source.trans γ.source.symm⟩
have h1 : (1 : unitInterval) ∈ γ ⁻¹' Set.range δ := ⟨1,δ.target.trans γ.target.symm⟩
exact hc.Icc_subset h0 h1 t.property
end Zarankiewicz
end OAI
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