Real numbers are dense #1287
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Real numbers are dense #1287
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| elim-exists | ||
| ( claim) | ||
| ( λ q (q∈ux , q∈ly) → | ||
| elim-exists | ||
| ( claim) | ||
| ( λ p (p<q , p∈ux) → | ||
| elim-exists | ||
| ( claim) | ||
| ( λ r (q<r , r∈ly) → | ||
| intro-exists (real-ℚ q) | ||
| ( intro-exists p (p∈ux , p<q) , | ||
| intro-exists r (q<r , r∈ly))) | ||
| ( forward-implication (is-rounded-lower-cut-ℝ y q) q∈ly)) | ||
| ( forward-implication (is-rounded-upper-cut-ℝ x q) q∈ux)) | ||
| where | ||
| claim : Prop (lsuc lzero ⊔ l1 ⊔ l2) | ||
| claim = ∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y) |
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