| 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 | |