remark A.542 Where each hypothesis is spent

open in the book · appendices/A-long-proofs.tex:26389 · p. 3056

remark -- no derivation owed by its kind

Rests on

depends_on example 13.68 Why each hypothesis is there ¶
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶
  - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.47 Differentiable structure ¶
      - depends_on definition 13.44 Atlas ¶
  - depends_on definition 13.49 Smooth map between manifolds ¶
    - depends_on definition 13.44 Atlas ¶ ↺
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶ ↺
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on theorem 13.67 Quotient manifold theorem ¶
  - depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
  - depends_on definition 13.54 Embedded submanifold ¶
    - depends_on definition 13.53 Immersion, submersion, embedding ¶
      - depends_on definition 13.52 Differential; pushforward ¶
      - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
  - depends_on theorem 13.63 Constant rank theorem ¶
    - depends_on corollary A.292 Inverse function theorem ¶
      - depends_on proposition 7.104 Chain rule in several variables ¶
      - depends_on theorem A.286 Implicit function theorem ¶
      - proves proof app:A-long-proofs@proof-186 ¶
    - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on theorem A.286 Implicit function theorem ¶ ↺
    - proves proof ch:11-manifolds-tensors-curvature@proof-12 ¶
  - proves proof app:A-long-proofs@proof-329 ¶
  - proves proof ch:11-manifolds-tensors-curvature@prooflink-2 ¶
depends_on proposition A.535 Existence of a slice ¶
- depends_on corollary A.292 Inverse function theorem ¶ ↺
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
- depends_on lemma A.533 The orbit map has constant rank $d$ ¶
  - depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
  - depends_on definition 13.53 Immersion, submersion, embedding ¶ ↺
  - depends_on theorem 13.63 Constant rank theorem ¶ ↺
  - proves proof app:A-long-proofs@proof-321 ¶
- proves proof app:A-long-proofs@proof-323 ¶
depends_on proposition A.532 Separation and countability of the quotient ¶
- depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
- depends_on lemma A.530 Two elementary facts about compactness ¶
  - depends_on definition A.529 Hausdorff; second countable; locally compact ¶
    - depends_on definition 6.9 Compact set ¶ ↺
    - depends_on definition 6.2 Open set ¶ ↺
    - depends_on definition 6.1 Topological space ¶
      - depends_on definition 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶ ↺
  - depends_on definition 6.9 Compact set ¶ ↺
  - depends_on proposition 6.10 Continuous images of compact sets ¶
    - depends_on definition 6.9 Compact set ¶ ↺
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶ ↺
      - depends_on definition 6.1 Topological space ¶ ↺
    - proves proof ch:04-topology@proof-1 ¶
  - proves proof app:A-long-proofs@proof-318 ¶
- depends_on lemma A.531 The projection is open ¶
  - depends_on definition 13.66 Smooth action; free; proper; orbit ¶ ↺
  - depends_on definition 6.6 Continuous map ¶ ↺
  - depends_on definition 6.2 Open set ¶ ↺
  - proves proof app:A-long-proofs@proof-319 ¶
- proves proof app:A-long-proofs@proof-320 ¶

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type	direction	node	provenance	where
`depends_on`	→	Why each hypothesis is there	declared	appendices/A-long-proofs.tex:26414
`depends_on`	→	Existence of a slice	declared	appendices/A-long-proofs.tex:26414
`depends_on`	→	Separation and countability of the quotient	declared	appendices/A-long-proofs.tex:26414