remark 7.88 $\pi$ elsewhere in this treatise

Rests on

• depends_on corollary 7.81 The decimal expansion of $\pi$ ¶
  • depends_on proposition 7.79 The polygon recursion ¶
    • depends_on definition 7.74 $\pi$ ¶
      • depends_on axiom 7.1 Completeness of $\R$ ¶
      • depends_on lemma 7.73 $\cos 2 < 0$ ¶
        • depends_on equation 7.53 eq:ana-cos-sin-series ¶
        • depends_on proposition 7.52 Alternating series test ¶
        • proves proof ch:05-real-analysis@proof-48 ¶
      • depends_on theorem 7.23 Intermediate value theorem ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶ ↺
        • depends_on definition 7.20 Continuity at a point ¶
        • proves proof ch:05-real-analysis@proof-8 ¶
      • depends_on theorem 7.51 Termwise differentiation ¶
        • depends_on proposition 7.47 Comparison; absolute convergence ¶
        • depends_on proposition 7.27 Differentiable implies continuous ¶
        • depends_on proposition 7.48 Ratio test ¶
        • depends_on proposition 7.6 Algebra of limits ¶
        • depends_on theorem 7.50 Power series; radius of convergence ¶
        • proves proof ch:05-real-analysis@proof-32 ¶
    • depends_on lemma 7.78 Chord, arc, tangent ¶
      • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on theorem 7.35 Mean value theorem ¶
        • proves proof ch:05-real-analysis@proof-20 ¶
      • depends_on lemma 7.71 Derivatives; the Pythagorean identity ¶
        • depends_on equation 7.53 eq:ana-cos-sin-series ¶ ↺
        • depends_on equation 5.3 eq:lin-binomial ¶
        • depends_on proposition 7.49 Cauchy product ¶
        • depends_on theorem 7.51 Termwise differentiation ¶ ↺
        • proves proof ch:05-real-analysis@proof-46 ¶
      • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
        • depends_on definition 7.74 $\pi$ ¶ ↺
        • depends_on equation 7.58 eq:ana-half-angle ¶
        • depends_on lemma 7.65 The integer part ¶
        • depends_on lemma 7.75 The first quadrant ¶
        • depends_on lemma 7.72 Addition theorems ¶
        • proves proof ch:05-real-analysis@proof-51 ¶
      • proves proof ch:05-real-analysis@proof-52 ¶
    • depends_on lemma 7.72 Addition theorems ¶ ↺
    • depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺
    • proves proof ch:05-real-analysis@proof-53 ¶
  • proves proof ch:05-real-analysis@proof-54 ¶
• depends_on lemma 7.72 Addition theorems ¶ ↺
• depends_on proposition 7.77 Special values, periodicity, and the kernel ¶ ↺

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