theorem 7.24 Extreme value theorem

open in the book · parts/02-mathematical-methods/05-real-analysis.tex:391 · p. 207

Rests on

depends_on axiom 7.1 Completeness of $\R$ ¶
depends_on proposition 7.22 Sequential characterization ¶
- depends_on definition 7.20 Continuity at a point ¶
  - depends_on definition 7.16 Limit ¶
    - depends_on definition 7.2 Absolute value ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.7 eq:ana-limit-left ¶
  - depends_on equation 7.5 eq:ana-limit-right ¶
- depends_on definition 7.4 Convergence ¶
  - depends_on definition 7.2 Absolute value ¶ ↺
- proves proof ch:05-real-analysis@proof-7 ¶
depends_on theorem 7.7 Bolzano–Weierstrass ¶
- depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
  - depends_on definition A.36 Cut ¶
  - depends_on theorem A.40 Least-upper-bound property ¶
    - depends_on definition A.36 Cut ¶ ↺
    - depends_on definition A.38 Order on $\R$ ¶
      - depends_on definition A.36 Cut ¶ ↺
    - proves proof app:A-long-proofs@proof-29 ¶
  - proves proof app:A-long-proofs@proof-32 ¶
- depends_on corollary A.46 Monotone convergence ¶
  - depends_on theorem A.40 Least-upper-bound property ¶ ↺
  - proves proof app:A-long-proofs@proof-33 ¶
- proves proof ch:05-real-analysis@proof-4 ¶
proves proof ch:05-real-analysis@proof-9 ¶

Supports

depends_on lemma A.221 Counting identity ¶
depends_on lemma 6.19 Continuous argument along a path ¶
- depends_on definition 6.20 Winding number ¶
  - depends_on example 6.21 ex:top-winding-values ¶
  - depends_on lemma 6.22 Nearby loops wind alike ¶
    - depends_on proposition 6.23 The punctured plane is not simply connected ¶
- depends_on lemma 6.22 Nearby loops wind alike ¶ ↺
depends_on theorem 7.40 Continuous functions are integrable ¶
- depends_on corollary A.502 All iterated orders agree ¶
  - depends_on lemma A.507 Coordinate permutations ¶
- depends_on definition 16.11 Admissible class; functional ¶
  - depends_on definition 16.59 Weierstrass excess function ¶
  - depends_on definition 16.12 The two norms; weak and strong neighbourhoods ¶
    - depends_on definition 16.13 Weak and strong extrema ¶
      - depends_on example 16.14 A weak minimum that is not strong ¶
      - depends_on proposition 16.16 Stationarity is necessary ¶
      - depends_on theorem 16.60 Weierstrass' necessary condition ¶
  - depends_on definition 16.15 Variation; the first variation ¶
    - depends_on definition 16.83 One-parameter transformation group; variational symmetry ¶
    - depends_on lemma 16.18 Fundamental lemma ¶
      - depends_on theorem A.688 The Kirchhoff plate equation ¶
      - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶
      - depends_on theorem 16.38 Euler–Poisson equation ¶
      - depends_on theorem 16.41 Multiplier rule for pointwise constraints ¶
      - depends_on theorem 16.31 Natural boundary condition ¶
      - depends_on theorem 25.19 Hamilton's equations from the first-order action ¶
      - depends_on theorem 25.22 Faddeev–Jackiw equations and brackets ¶
    - depends_on proposition 16.16 Stationarity is necessary ¶ ↺
    - depends_on theorem 16.43 Euler's rule for integral constraints ¶
      - depends_on example 16.45 Two solved isoperimetric problems ¶
      - depends_on proposition 16.4 The hanging chain is a catenary ¶
      - depends_on proposition 16.6 The isoperimetric extremal is a circle ¶
  - depends_on proposition 16.68 Weierstrass' counterexample ¶
- depends_on definition 9.137 Elliptic integrals of the three kinds ¶
  - depends_on definition 9.140 Amplitude and the Jacobi elliptic functions ¶
    - depends_on proposition 9.143 A cubic quadrature between two turning points ¶
    - depends_on proposition 9.141 First properties ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
      - depends_on proposition 9.142 The differential equation of the sine amplitude ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶
  - depends_on proposition 9.139 Series for the complete integral of the first kind ¶
    - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
- depends_on lemma A.501 Iterated integration over a box ¶
  - depends_on corollary A.502 All iterated orders agree ¶ ↺
  - depends_on lemma A.509 Primitive maps have the substitution property ¶
    - depends_on proposition A.511 The substitution property is universal ¶
      - depends_on remark A.516 The hypotheses of the global form ¶
      - depends_on theorem A.513 Change of variables ¶
- depends_on lemma 16.18 Fundamental lemma ¶ ↺
- depends_on remark 7.128 What the derivations below take as given ¶
  - depends_on lemma A.312 The computation in one chart ¶
  - depends_on theorem 7.129 Change of variables in a multiple integral ¶
    - depends_on example 7.130 The two Jacobians this treatise uses ¶
      - depends_on lemma A.518 The excised ball ¶
    - depends_on lemma A.518 The excised ball ¶ ↺
    - depends_on proposition A.519 The convolution exists ¶
      - depends_on proposition A.524 Rate of decay ¶
  - depends_on theorem 7.133 Gauss ¶
    - depends_on lemma A.708 The disturbance flux vanishes ¶
      - depends_on corollary A.711 The disturbance is a dipole at leading order ¶
    - depends_on lemma A.712 The force is a far-field integral ¶
    - depends_on lemma A.117 Laplacian in orthogonal coordinates ¶
      - depends_on theorem A.120 Stäckel and Robertson conditions ¶
    - depends_on lemma 44.10 Divergence theorem on $(M,g)$ ¶
      - depends_on proposition 44.11 The boundary term, and the Gibbons–Hawking–York action ¶
    - depends_on lemma 31.11 Transport theorem for a material volume ¶
      - depends_on theorem 31.12 Conservation of mass ¶
      - depends_on theorem 31.22 Euler's equations of motion ¶
    - depends_on lemma 106.86 A shift of a divergent integral leaves a surface term ¶
      - depends_on proposition 106.88 The obstruction ¶
    - depends_on lemma 10.44 Green's identities ¶
      - depends_on proposition 10.71 Green's representation formula ¶
      - depends_on proposition 10.67 The Newtonian potential is the fundamental solution ¶
      - depends_on theorem 10.45 Symmetry of the Green's function ¶
    - depends_on lemma 10.58 Darboux's equation for spherical means ¶
      - depends_on theorem 10.59 Kirchhoff's formula ¶
      - depends_on theorem 10.69 Mean-value property ¶
    - depends_on proposition 23.54 The geometrical amplitude diverges ¶
      - depends_on proposition 23.55 The fold, and the quarter-power law ¶
      - depends_on remark 23.59 What is observed ¶
    - depends_on proposition 10.57 Energy in a backward cone ¶
      - depends_on proposition 28.63 Energy density, flux, and the conservation law ¶
    - depends_on theorem 16.36 Euler–Lagrange equations for several independent variables ¶ ↺
    - depends_on theorem 32.6 Evolution of phase volume ¶
      - depends_on definition 32.7 Conservative and dissipative flows ¶
      - depends_on definition 32.62 The Lorenz system ¶
      - depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      - depends_on proposition 32.76 The exponents sum to the mean divergence ¶
    - … 1 more
  - depends_on theorem 7.131 Green ¶
    - depends_on proposition 32.33 Bendixson's negative criterion ¶
    - depends_on proposition 9.36 Bendixson–Dulac negative criterion ¶
    - depends_on theorem 7.132 Stokes ¶
      - depends_on definition 31.63 Circulation ¶
      - depends_on proposition 7.136 Properties of conservative fields ¶
    - depends_on theorem 8.12 Cauchy ¶
      - depends_on corollary 8.15 Deformation of contours ¶
      - depends_on corollary 17.71 Residue evaluation and causality ¶
      - depends_on lemma 100.27 Wick rotation ¶
      - depends_on proposition 106.14 The Feynman contour permits the rotation ¶
      - depends_on theorem 17.78 Causality implies dispersion relations ¶
      - depends_on theorem 17.94 Mellin inversion ¶
    - depends_on theorem 13.41 Holonomy equals the enclosed curvature; local Gauss–Bonnet ¶
      - depends_on example 13.42 The sphere, the solid angle, and the pole ¶
    - depends_on theorem 10.96 Rankine–Hugoniot condition ¶
      - depends_on definition 10.99 Entropy condition ¶
      - depends_on example 10.97 Burgers shock ¶
      - depends_on example 10.98 Weak solutions are not unique ¶
      - depends_on proposition 10.102 Jump form of the entropy inequality ¶
  - depends_on theorem A.303 Change of variables for multiple integrals; quoted ¶
    - depends_on lemma A.305 The chart integral is well defined ¶
      - depends_on definition A.308 Integral over the manifold ¶
- depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
  - depends_on corollary 7.110 Variable limits of integration ¶
  - depends_on definition A.450 The space $W^{1,r}(a,b)$ ¶
    - depends_on definition A.462 The weighted space and the energy space ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶
    - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶
      - depends_on theorem A.455 Compact embedding of $W^{1,r}(a,b)$ into the continuous functions ¶
    - depends_on lemma A.452 Young and Hölder ¶
      - depends_on lemma A.489 Courant–Lebesgue ¶
      - depends_on lemma A.463 Poincaré inequality; $B_{K}$ is an inner product ¶ ↺
      - depends_on lemma A.453 Uniform bound and uniform Hölder continuity ¶ ↺
  - depends_on lemma A.520 A $C^{1}$ limit ¶
    - depends_on proposition A.521 Every derivative passes onto $g$ ¶
      - depends_on proposition A.524 Rate of decay ¶ ↺
      - depends_on proposition A.522 The Newtonian potential inverts the Laplacian ¶
  - depends_on lemma A.440 Grönwall's inequality ¶
    - depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
      - depends_on corollary A.445 Displacing the initial point ¶
  - depends_on lemma 16.19 du Bois-Reymond ¶
    - depends_on lemma 16.20 Mixed form ¶
      - depends_on theorem 16.35 System of Euler–Lagrange equations ¶
      - depends_on theorem 16.22 Euler–Lagrange ¶
      - depends_on theorem 16.43 Euler's rule for integral constraints ¶ ↺
  - depends_on lemma 9.10 Grönwall's inequality ¶
    - depends_on proposition 9.11 Continuous dependence on the initial data ¶
    - depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
  - depends_on proposition 11.22 Properties of the distribution function ¶
    - depends_on proposition 11.27 Probability integral transform ¶
  - depends_on proposition 9.18 The linear equation: integrating factor ¶
    - depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      - depends_on proposition 9.88 The model problem, and the error of the composite ¶
  - depends_on proposition 9.19 Separation of variables ¶
  - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
    - depends_on corollary 7.44 Substitution and integration by parts ¶
      - depends_on definition 8.8 Contour integral ¶
      - depends_on lemma 16.20 Mixed form ¶ ↺
      - depends_on lemma 9.136 Wallis integrals ¶
      - depends_on proposition 7.145 Delta as a limit; elementary properties ¶
      - depends_on proposition 16.50 The second variation ¶
      - depends_on proposition 5.95 The focal polar equation of a conic ¶
      - depends_on proposition 9.143 A cubic quadrature between two turning points ¶ ↺
      - depends_on proposition 9.144 The pendulum equation at finite amplitude ¶ ↺
      - depends_on theorem 16.38 Euler–Poisson equation ¶ ↺
    - depends_on corollary 16.23 du Bois-Reymond form ¶
      - depends_on corollary 16.33 Weierstrass–Erdmann corner conditions ¶
    - depends_on lemma 7.116 Functions vanishing on a regular zero set ¶
    - depends_on lemma A.72 Iterated integral inequality ¶
      - depends_on theorem A.74 Flow of a time-dependent vector field ¶
    - depends_on lemma A.520 A $C^{1}$ limit ¶ ↺
    - depends_on lemma A.172 Dirichlet ¶
      - depends_on lemma A.193 A cosine integral ¶
      - depends_on theorem A.177 Fourier inversion for a distribution function ¶
    - depends_on lemma A.221 Counting identity ¶ ↺
    - depends_on lemma A.312 The computation in one chart ¶ ↺
    - depends_on proposition 7.85 Irrationality of $\pi$ ¶
      - depends_on remark 7.94 Three numbers, two kinds ¶
      - depends_on remark 7.87 Priority, and what irrationality does not give ¶
    - depends_on proposition 8.10 Fundamental theorem for contours ¶
      - depends_on theorem 8.13 Goursat ¶
      - depends_on theorem 8.18 Liouville ¶
    - depends_on proposition 17.41 The identities physics uses ¶
      - depends_on proposition 17.90 Hankel transform as the axially symmetric Fourier transform ¶
      - depends_on proposition 17.52 Linear translation-invariant systems ¶
    - depends_on proposition 17.73 Initial- and final-value theorems ¶
    - … 4 more
  - depends_on theorem 10.52 Duhamel's principle ¶
    - depends_on corollary 10.53 Duhamel for the wave equation ¶
depends_on theorem 7.42 Fundamental theorem of calculus, I ¶ ↺
depends_on theorem 7.34 Rolle ¶
- depends_on theorem 7.37 Cauchy mean value theorem and l'Hôpital's rule ¶
  - depends_on lemma A.306 The standard smooth bump ¶
    - depends_on lemma A.307 Partition of unity on a compact manifold ¶
      - depends_on definition A.308 Integral over the manifold ¶ ↺
- depends_on theorem 7.35 Mean value theorem ¶
  - depends_on corollary 7.36 cor:ana-mvt-consequences ¶
    - depends_on definition 7.41 Antiderivative ¶
      - depends_on theorem 7.43 Fundamental theorem of calculus, II ¶ ↺
    - depends_on lemma 7.67 The limit over real exponents ¶
      - depends_on proposition 7.68 The natural base ¶
    - depends_on lemma 7.75 The first quadrant ¶
      - depends_on proposition 7.76 $\pi$ as the circle constant ¶
      - depends_on proposition 7.77 Special values, periodicity, and the kernel ¶
      - depends_on proposition 125.2 Crystallographic restriction ¶
    - depends_on lemma 7.78 Chord, arc, tangent ¶
      - depends_on proposition 7.79 The polygon recursion ¶
    - depends_on lemma 9.10 Grönwall's inequality ¶ ↺
    - depends_on proposition 7.108 Euler's theorem on homogeneous functions ¶
    - depends_on proposition 7.61 The logarithm ¶
      - depends_on corollary 7.69 $\ee$ is the unique self-reproducing base ¶
      - depends_on definition 7.62 Real powers ¶
      - depends_on lemma 7.67 The limit over real exponents ¶ ↺
      - depends_on proposition 7.68 The natural base ¶ ↺
      - depends_on proposition 7.63 Laws of real powers ¶
    - depends_on proposition 9.19 Separation of variables ¶ ↺
    - depends_on theorem 7.115 Constant rank ¶
  - depends_on lemma 7.93 A polynomial has at most $n$ roots ¶
  - depends_on lemma A.500 Graphs and $C^{1}$ images have zero content ¶
    - depends_on corollary A.514 Degeneracy on a negligible set ¶
    - depends_on lemma A.512 The boundary strip is thin ¶
      - depends_on theorem A.513 Change of variables ¶ ↺
  - depends_on lemma A.73 Differentiation under the integral sign ¶
    - depends_on lemma A.77 Poincaré lemma, converse form ¶
  - depends_on lemma A.287 The Newton map contracts ¶
    - depends_on lemma A.288 The iteration converges ¶
      - depends_on corollary A.289 The zero set is a graph ¶
      - depends_on lemma A.290 $h$ is Lipschitz ¶
  - depends_on lemma A.290 $h$ is Lipschitz ¶ ↺
  - depends_on lemma A.440 Grönwall's inequality ¶ ↺
  - depends_on lemma A.226 The level factor ¶
  - depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
    - depends_on theorem A.470 $T$ is compact ¶
      - depends_on theorem A.471 Spectral decomposition and completeness in $L^{2}_{r}$ ¶
  - depends_on lemma 11.68 Jensen's inequality for the logarithm ¶
  - depends_on proposition 7.105 Clairaut–Schwarz ¶
    - depends_on definition 10.4 The second-order operator ¶
      - depends_on definition 10.10 Characteristic surface ¶
      - depends_on definition 10.42 Green's function ¶
      - depends_on definition 10.6 Type of a second-order operator ¶
    - depends_on lemma A.75 Differentiating a pullback along a flow ¶
      - depends_on lemma A.77 Poincaré lemma, converse form ¶ ↺
    - depends_on lemma 22.20 The symplectic condition ¶
      - depends_on corollary 22.24 Invariance of the phase-space volume ¶
      - depends_on proposition 22.27 Properties of the Lagrange bracket ¶
      - depends_on proposition 22.30 Properties of the Poisson bracket ¶
      - depends_on remark 22.33 What the duality is good for ¶
      - depends_on remark 22.28 Where the content of the fundamental brackets lies ¶
      - depends_on theorem 22.22 Poincaré ¶
    - depends_on proposition 7.123 Second-order identities of the nabla calculus ¶
      - depends_on remark 7.124 The Laplacian of a vector field is a Cartesian notion ¶
      - depends_on theorem 7.137 Helmholtz decomposition ¶
      - depends_on theorem A.523 Helmholtz decomposition: existence ¶
      - depends_on theorem A.525 Helmholtz decomposition: uniqueness ¶
    - depends_on proposition 30.25 Twenty-one constants ¶
      - depends_on definition 30.37 Voigt notation ¶
      - depends_on proposition 30.38 The Cauchy relations ¶
      - depends_on theorem A.675 Three constants for a cubic crystal, two for an isotropic solid ¶
    - depends_on proposition 30.14 Saint-Venant compatibility is necessary ¶
      - depends_on theorem 30.15 Compatibility is sufficient on a simply connected body ¶
    - depends_on proposition 22.30 Properties of the Poisson bracket ¶ ↺
    - depends_on proposition 13.148 prop:mfd-torsion-tensor ¶
    - depends_on proposition 10.16 Cauchy's characteristic strips ¶
    - depends_on theorem 7.132 Stokes ¶ ↺
    - depends_on theorem 7.106 Taylor's theorem in several variables ¶
      - depends_on remark 7.107 What the Hessian is for ¶
    - depends_on theorem 13.152 Riemann tensor; Ricci identity with torsion ¶
      - depends_on definition 13.153 Contractions ¶
      - depends_on example 21.74 Rindler coordinates ¶
      - depends_on lemma A.640 Variation of the integrated three-curvature ¶
      - depends_on lemma 44.8 Palatini identity ¶
      - depends_on proposition 44.44 Harmonic-gauge reduction ¶
      - depends_on proposition 13.155 Geodesic deviation; Jacobi equation ¶
      - depends_on proposition 13.140 Killing's equation ¶
      - depends_on proposition 13.162 Curvature induced on the quadric ¶
      - depends_on proposition 13.154 Symmetries of the curvature ¶
      - depends_on remark 30.16 Incompatibility is curvature ¶
      - depends_on theorem 13.156 Cartan structure equations ¶
    - … 2 more
  - depends_on theorem 7.100 $C^{1}$ implies differentiable ¶
    - depends_on lemma A.291 $h$ is differentiable, with the stated derivative ¶
    - depends_on remark 7.101 Partial derivatives alone do not suffice ¶
    - depends_on theorem 7.112 Implicit function theorem ¶
      - depends_on corollary 7.113 Inverse function theorem ¶
      - depends_on proposition 7.120 The envelope touches every member it meets ¶
      - depends_on proposition 7.117 Lagrange multipliers in finitely many variables ¶
      - depends_on remark 7.114 Where the proof is, and why ¶
    - depends_on theorem A.286 Implicit function theorem ¶
      - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on corollary A.293 Solving one scalar equation for one coordinate ¶
      - depends_on lemma A.650 Flattening the constraints ¶
      - depends_on remark A.296 Both hypotheses are needed, and the conclusion is local ¶
      - depends_on remark 23.3 What Part II owes this chapter ¶
      - depends_on theorem 23.6 Jacobi ¶
      - depends_on theorem 13.63 Constant rank theorem ¶
      - depends_on theorem 13.62 Regular value theorem in codimension $k$ ¶
  - … 2 more
- depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
  - depends_on definition 32.57 The Hénon–Heiles Hamiltonian ¶
  - depends_on definition 32.12 Linearization ¶
    - depends_on proposition 32.17 Classification of planar fixed points ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶
      - depends_on theorem 32.25 Hopf bifurcation, quoted ¶
    - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
      - depends_on definition 32.22 Bifurcation ¶
      - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
    - depends_on theorem 32.13 Linear stability ¶
      - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
      - depends_on proposition 32.23 Saddle-node bifurcation ¶
      - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
      - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
  - depends_on lemma A.754 Exact depth expansion ¶
    - depends_on proposition A.755 The Boussinesq pair ¶
      - depends_on theorem A.756 The Korteweg–de Vries equation ¶
  - depends_on lemma A.187 Exponentiation ¶
  - depends_on lemma A.138 The flat exponential ¶
    - depends_on definition A.139 The standard radial mollifier ¶
      - depends_on definition A.141 Mollification ¶
      - depends_on lemma A.140 Properties of the mollifier ¶
    - depends_on lemma A.140 Properties of the mollifier ¶ ↺
  - depends_on lemma A.452 Young and Hölder ¶ ↺
  - depends_on lemma A.212 Second-order flatness ¶
    - depends_on lemma A.213 The three minima ¶
      - depends_on theorem A.214 Wilks, $k$ parameters and $r$ constraints ¶
  - depends_on lemma A.209 Uniform third-order remainder ¶
    - depends_on corollary A.210 Uniform quadratic approximation on the $n^{-1/2}$ scale ¶
      - depends_on lemma A.213 The three minima ¶ ↺
    - depends_on lemma A.211 Root-$n$ localisation ¶
      - depends_on lemma A.213 The three minima ¶ ↺
  - depends_on phenomenon 28.13 Universality of small oscillations ¶
  - depends_on proposition 16.50 The second variation ¶ ↺
  - depends_on proposition 17.67 Holomorphy ¶
    - depends_on theorem 17.78 Causality implies dispersion relations ¶ ↺
    - depends_on theorem 17.93 Fundamental strip and holomorphy ¶
      - depends_on theorem 17.94 Mellin inversion ¶ ↺
  - depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
  - … 2 more

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Edges

type	direction	node	provenance	where
`depends_on`	→	Completeness of $\R$	declared	parts/02-mathematical-methods/05-real-analysis.tex:394
`depends_on`	→	Sequential characterization	declared	parts/02-mathematical-methods/05-real-analysis.tex:394
`depends_on`	→	Bolzano–Weierstrass	declared	parts/02-mathematical-methods/05-real-analysis.tex:394
`depends_on`	←	Counting identity	declared	appendices/A-long-proofs.tex:11221
`depends_on`	←	Continuous argument along a path	declared	parts/02-mathematical-methods/04-topology.tex:318
`depends_on`	←	Continuous functions are integrable	declared	parts/02-mathematical-methods/05-real-analysis.tex:788
`depends_on`	←	Fundamental theorem of calculus, I	declared	parts/02-mathematical-methods/05-real-analysis.tex:834
`depends_on`	←	Rolle	declared	parts/02-mathematical-methods/05-real-analysis.tex:661
`proves`	←	ch:05-real-analysis@proof-9	declared	parts/02-mathematical-methods/05-real-analysis.tex:397