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Theorem List for Metamath Proof Explorer - 34101-34200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnn0ltp1ne 34101 Nonnegative integer ordering relation. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐴 ∈ β„•0 ∧ 𝐡 ∈ β„•0) β†’ ((𝐴 + 1) < 𝐡 ↔ (𝐴 < 𝐡 ∧ 𝐡 β‰  (𝐴 + 1))))
 
Theorem0nn0m1nnn0 34102 A number is zero if and only if it's a nonnegative integer that becomes negative after subtracting 1. (Contributed by BTernaryTau, 30-Sep-2023.)
(𝑁 = 0 ↔ (𝑁 ∈ β„•0 ∧ Β¬ (𝑁 βˆ’ 1) ∈ β„•0))
 
Theoremf1resfz0f1d 34103 If a function with a sequence of nonnegative integers (starting at 0) as its domain is one-to-one when 0 is removed, and if the range of that restriction does not contain the function's value at the removed integer, then the function is itself one-to-one. (Contributed by BTernaryTau, 4-Oct-2023.)
(πœ‘ β†’ 𝐾 ∈ β„•0)    &   (πœ‘ β†’ 𝐹:(0...𝐾)βŸΆπ‘‰)    &   (πœ‘ β†’ (𝐹 β†Ύ (1...𝐾)):(1...𝐾)–1-1→𝑉)    &   (πœ‘ β†’ ((𝐹 β€œ {0}) ∩ (𝐹 β€œ (1...𝐾))) = βˆ…)    β‡’   (πœ‘ β†’ 𝐹:(0...𝐾)–1-1→𝑉)
 
Theoremfisshasheq 34104 A finite set is equal to its subset if they are the same size. (Contributed by BTernaryTau, 3-Oct-2023.)
((𝐡 ∈ Fin ∧ 𝐴 βŠ† 𝐡 ∧ (β™―β€˜π΄) = (β™―β€˜π΅)) β†’ 𝐴 = 𝐡)
 
Theoremhashf1dmcdm 34105 The size of the domain of a one-to-one set function is less than or equal to the size of its codomain, if it exists. (Contributed by BTernaryTau, 1-Oct-2023.)
((𝐹 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š ∧ 𝐹:𝐴–1-1→𝐡) β†’ (β™―β€˜π΄) ≀ (β™―β€˜π΅))
 
Theoremrevpfxsfxrev 34106 The reverse of a prefix of a word is equal to the same-length suffix of the reverse of that word. (Contributed by BTernaryTau, 2-Dec-2023.)
((π‘Š ∈ Word 𝑉 ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (reverseβ€˜(π‘Š prefix 𝐿)) = ((reverseβ€˜π‘Š) substr ⟨((β™―β€˜π‘Š) βˆ’ 𝐿), (β™―β€˜π‘Š)⟩))
 
Theoremswrdrevpfx 34107 A subword expressed in terms of reverses and prefixes. (Contributed by BTernaryTau, 3-Dec-2023.)
((π‘Š ∈ Word 𝑉 ∧ 𝐹 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜π‘Š))) β†’ (π‘Š substr ⟨𝐹, 𝐿⟩) = (reverseβ€˜((reverseβ€˜(π‘Š prefix 𝐿)) prefix (𝐿 βˆ’ 𝐹))))
 
21.5.3  Graph theory
 
Theoremlfuhgr 34108* A hypergraph is loop-free if and only if every edge connects at least two vertices. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ UHGraph β†’ (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)2 ≀ (β™―β€˜π‘₯)))
 
Theoremlfuhgr2 34109* A hypergraph is loop-free if and only if every edge is not a loop. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ UHGraph β†’ (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ βˆ€π‘₯ ∈ (Edgβ€˜πΊ)(β™―β€˜π‘₯) β‰  1))
 
Theoremlfuhgr3 34110* A hypergraph is loop-free if and only if none of its edges connect to only one vertex. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   (𝐺 ∈ UHGraph β†’ (𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)} ↔ Β¬ βˆƒπ‘Ž{π‘Ž} ∈ (Edgβ€˜πΊ)))
 
Theoremcplgredgex 34111* Any two (distinct) vertices in a complete graph are connected to each other by at least one edge. (Contributed by BTernaryTau, 2-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (𝐺 ∈ ComplGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ (𝑉 βˆ– {𝐴})) β†’ βˆƒπ‘’ ∈ 𝐸 {𝐴, 𝐡} βŠ† 𝑒))
 
Theoremcusgredgex 34112 Any two (distinct) vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 3-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (𝐺 ∈ ComplUSGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ (𝑉 βˆ– {𝐴})) β†’ {𝐴, 𝐡} ∈ 𝐸))
 
Theoremcusgredgex2 34113 Any two distinct vertices in a complete simple graph are connected to each other by an edge. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΈ = (Edgβ€˜πΊ)    β‡’   (𝐺 ∈ ComplUSGraph β†’ ((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐴 β‰  𝐡) β†’ {𝐴, 𝐡} ∈ 𝐸))
 
Theorempfxwlk 34114 A prefix of a walk is a walk. (Contributed by BTernaryTau, 2-Dec-2023.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 prefix 𝐿)(Walksβ€˜πΊ)(𝑃 prefix (𝐿 + 1)))
 
Theoremrevwlk 34115 The reverse of a walk is a walk. (Contributed by BTernaryTau, 30-Nov-2023.)
(𝐹(Walksβ€˜πΊ)𝑃 β†’ (reverseβ€˜πΉ)(Walksβ€˜πΊ)(reverseβ€˜π‘ƒ))
 
Theoremrevwlkb 34116 Two words represent a walk if and only if their reverses also represent a walk. (Contributed by BTernaryTau, 4-Dec-2023.)
((𝐹 ∈ Word π‘Š ∧ 𝑃 ∈ Word π‘ˆ) β†’ (𝐹(Walksβ€˜πΊ)𝑃 ↔ (reverseβ€˜πΉ)(Walksβ€˜πΊ)(reverseβ€˜π‘ƒ)))
 
Theoremswrdwlk 34117 Two matching subwords of a walk also represent a walk. (Contributed by BTernaryTau, 7-Dec-2023.)
((𝐹(Walksβ€˜πΊ)𝑃 ∧ 𝐡 ∈ (0...𝐿) ∧ 𝐿 ∈ (0...(β™―β€˜πΉ))) β†’ (𝐹 substr ⟨𝐡, 𝐿⟩)(Walksβ€˜πΊ)(𝑃 substr ⟨𝐡, (𝐿 + 1)⟩))
 
Theorempthhashvtx 34118 A graph containing a path has at least as many vertices as there are edges in the path. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (𝐹(Pathsβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) ≀ (β™―β€˜π‘‰))
 
Theorempthisspthorcycl 34119 A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.)
(𝐹(Pathsβ€˜πΊ)𝑃 β†’ (𝐹(SPathsβ€˜πΊ)𝑃 ∨ 𝐹(Cyclesβ€˜πΊ)𝑃))
 
Theoremspthcycl 34120 A walk is a trivial path if and only if it is both a simple path and a cycle. (Contributed by BTernaryTau, 8-Oct-2023.)
((𝐹(Pathsβ€˜πΊ)𝑃 ∧ 𝐹 = βˆ…) ↔ (𝐹(SPathsβ€˜πΊ)𝑃 ∧ 𝐹(Cyclesβ€˜πΊ)𝑃))
 
Theoremusgrgt2cycl 34121 A non-trivial cycle in a simple graph has a length greater than 2. (Contributed by BTernaryTau, 24-Sep-2023.)
((𝐺 ∈ USGraph ∧ 𝐹(Cyclesβ€˜πΊ)𝑃 ∧ 𝐹 β‰  βˆ…) β†’ 2 < (β™―β€˜πΉ))
 
Theoremusgrcyclgt2v 34122 A simple graph with a non-trivial cycle must have at least 3 vertices. (Contributed by BTernaryTau, 5-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ USGraph ∧ 𝐹(Cyclesβ€˜πΊ)𝑃 ∧ 𝐹 β‰  βˆ…) β†’ 2 < (β™―β€˜π‘‰))
 
Theoremsubgrwlk 34123 If a walk exists in a subgraph of a graph 𝐺, then that walk also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 β†’ (𝐹(Walksβ€˜π‘†)𝑃 β†’ 𝐹(Walksβ€˜πΊ)𝑃))
 
Theoremsubgrtrl 34124 If a trail exists in a subgraph of a graph 𝐺, then that trail also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 β†’ (𝐹(Trailsβ€˜π‘†)𝑃 β†’ 𝐹(Trailsβ€˜πΊ)𝑃))
 
Theoremsubgrpth 34125 If a path exists in a subgraph of a graph 𝐺, then that path also exists in 𝐺. (Contributed by BTernaryTau, 22-Oct-2023.)
(𝑆 SubGraph 𝐺 β†’ (𝐹(Pathsβ€˜π‘†)𝑃 β†’ 𝐹(Pathsβ€˜πΊ)𝑃))
 
Theoremsubgrcycl 34126 If a cycle exists in a subgraph of a graph 𝐺, then that cycle also exists in 𝐺. (Contributed by BTernaryTau, 23-Oct-2023.)
(𝑆 SubGraph 𝐺 β†’ (𝐹(Cyclesβ€˜π‘†)𝑃 β†’ 𝐹(Cyclesβ€˜πΊ)𝑃))
 
Theoremcusgr3cyclex 34127* Every complete simple graph with more than two vertices has a 3-cycle. (Contributed by BTernaryTau, 4-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ ComplUSGraph ∧ 2 < (β™―β€˜π‘‰)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 3))
 
Theoremloop1cycl 34128* A hypergraph has a cycle of length one if and only if it has a loop. (Contributed by BTernaryTau, 13-Oct-2023.)
(𝐺 ∈ UHGraph β†’ (βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 1 ∧ (π‘β€˜0) = 𝐴) ↔ {𝐴} ∈ (Edgβ€˜πΊ)))
 
Theorem2cycld 34129 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = βŸ¨β€œπ΄π΅πΆβ€βŸ©    &   πΉ = βŸ¨β€œπ½πΎβ€βŸ©    &   (πœ‘ β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉 ∧ 𝐢 ∈ 𝑉))    &   (πœ‘ β†’ (𝐴 β‰  𝐡 ∧ 𝐡 β‰  𝐢))    &   (πœ‘ β†’ ({𝐴, 𝐡} βŠ† (πΌβ€˜π½) ∧ {𝐡, 𝐢} βŠ† (πΌβ€˜πΎ)))    &   π‘‰ = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ 𝐴 = 𝐢)    β‡’   (πœ‘ β†’ 𝐹(Cyclesβ€˜πΊ)𝑃)
 
Theorem2cycl2d 34130 Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023.)
𝑃 = βŸ¨β€œπ΄π΅π΄β€βŸ©    &   πΉ = βŸ¨β€œπ½πΎβ€βŸ©    &   (πœ‘ β†’ (𝐴 ∈ 𝑉 ∧ 𝐡 ∈ 𝑉))    &   (πœ‘ β†’ 𝐴 β‰  𝐡)    &   (πœ‘ β†’ ({𝐴, 𝐡} βŠ† (πΌβ€˜π½) ∧ {𝐴, 𝐡} βŠ† (πΌβ€˜πΎ)))    &   π‘‰ = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    β‡’   (πœ‘ β†’ 𝐹(Cyclesβ€˜πΊ)𝑃)
 
Theoremumgr2cycllem 34131* Lemma for umgr2cycl 34132. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐹 = βŸ¨β€œπ½πΎβ€βŸ©    &   πΌ = (iEdgβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ UMGraph)    &   (πœ‘ β†’ 𝐽 ∈ dom 𝐼)    &   (πœ‘ β†’ 𝐽 β‰  𝐾)    &   (πœ‘ β†’ (πΌβ€˜π½) = (πΌβ€˜πΎ))    β‡’   (πœ‘ β†’ βˆƒπ‘ 𝐹(Cyclesβ€˜πΊ)𝑝)
 
Theoremumgr2cycl 34132* A multigraph with two distinct edges that connect the same vertices has a 2-cycle. (Contributed by BTernaryTau, 17-Oct-2023.)
𝐼 = (iEdgβ€˜πΊ)    β‡’   ((𝐺 ∈ UMGraph ∧ βˆƒπ‘— ∈ dom πΌβˆƒπ‘˜ ∈ dom 𝐼((πΌβ€˜π‘—) = (πΌβ€˜π‘˜) ∧ 𝑗 β‰  π‘˜)) β†’ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ (β™―β€˜π‘“) = 2))
 
21.5.3.1  Acyclic graphs
 
Syntaxcacycgr 34133 Extend class notation with acyclic graphs.
class AcyclicGraph
 
Definitiondf-acycgr 34134* Define the class of all acyclic graphs. A graph is called acyclic if it has no (non-trivial) cycles. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜π‘”)𝑝 ∧ 𝑓 β‰  βˆ…)}
 
Theoremdfacycgr1 34135* An alternate definition of the class of all acyclic graphs that requires all cycles to be trivial. (Contributed by BTernaryTau, 11-Oct-2023.)
AcyclicGraph = {𝑔 ∣ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜π‘”)𝑝 β†’ 𝑓 = βˆ…)}
 
Theoremisacycgr 34136* The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…)))
 
Theoremisacycgr1 34137* The property of being an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
(𝐺 ∈ π‘Š β†’ (𝐺 ∈ AcyclicGraph ↔ βˆ€π‘“βˆ€π‘(𝑓(Cyclesβ€˜πΊ)𝑝 β†’ 𝑓 = βˆ…)))
 
Theoremacycgrcycl 34138 Any cycle in an acyclic graph is trivial (i.e. has one vertex and no edges). (Contributed by BTernaryTau, 12-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Cyclesβ€˜πΊ)𝑃) β†’ 𝐹 = βˆ…)
 
Theoremacycgr0v 34139 A null graph (with no vertices) is an acyclic graph. (Contributed by BTernaryTau, 11-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ π‘Š ∧ 𝑉 = βˆ…) β†’ 𝐺 ∈ AcyclicGraph)
 
Theoremacycgr1v 34140 A multigraph with one vertex is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ UMGraph ∧ (β™―β€˜π‘‰) = 1) β†’ 𝐺 ∈ AcyclicGraph)
 
Theoremacycgr2v 34141 A simple graph with two vertices is an acyclic graph. (Contributed by BTernaryTau, 12-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   ((𝐺 ∈ USGraph ∧ (β™―β€˜π‘‰) = 2) β†’ 𝐺 ∈ AcyclicGraph)
 
Theoremprclisacycgr 34142* A proper class (representing a null graph, see vtxvalprc 28305) has the property of an acyclic graph (see also acycgr0v 34139). (Contributed by BTernaryTau, 11-Oct-2023.) (New usage is discouraged.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (Β¬ 𝐺 ∈ V β†’ Β¬ βˆƒπ‘“βˆƒπ‘(𝑓(Cyclesβ€˜πΊ)𝑝 ∧ 𝑓 β‰  βˆ…))
 
Theoremacycgrislfgr 34143* An acyclic hypergraph is a loop-free hypergraph. (Contributed by BTernaryTau, 15-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    &   πΌ = (iEdgβ€˜πΊ)    β‡’   ((𝐺 ∈ AcyclicGraph ∧ 𝐺 ∈ UHGraph) β†’ 𝐼:dom 𝐼⟢{π‘₯ ∈ 𝒫 𝑉 ∣ 2 ≀ (β™―β€˜π‘₯)})
 
Theoremupgracycumgr 34144 An acyclic pseudograph is a multigraph. (Contributed by BTernaryTau, 15-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) β†’ 𝐺 ∈ UMGraph)
 
Theoremumgracycusgr 34145 An acyclic multigraph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UMGraph ∧ 𝐺 ∈ AcyclicGraph) β†’ 𝐺 ∈ USGraph)
 
Theoremupgracycusgr 34146 An acyclic pseudograph is a simple graph. (Contributed by BTernaryTau, 17-Oct-2023.)
((𝐺 ∈ UPGraph ∧ 𝐺 ∈ AcyclicGraph) β†’ 𝐺 ∈ USGraph)
 
Theoremcusgracyclt3v 34147 A complete simple graph is acyclic if and only if it has fewer than three vertices. (Contributed by BTernaryTau, 20-Oct-2023.)
𝑉 = (Vtxβ€˜πΊ)    β‡’   (𝐺 ∈ ComplUSGraph β†’ (𝐺 ∈ AcyclicGraph ↔ (β™―β€˜π‘‰) < 3))
 
Theorempthacycspth 34148 A path in an acyclic graph is a simple path. (Contributed by BTernaryTau, 21-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝐹(Pathsβ€˜πΊ)𝑃) β†’ 𝐹(SPathsβ€˜πΊ)𝑃)
 
Theoremacycgrsubgr 34149 The subgraph of an acyclic graph is also acyclic. (Contributed by BTernaryTau, 23-Oct-2023.)
((𝐺 ∈ AcyclicGraph ∧ 𝑆 SubGraph 𝐺) β†’ 𝑆 ∈ AcyclicGraph)
 
21.6  Mathbox for Mario Carneiro
 
21.6.1  Predicate calculus with all distinct variables
 
Axiomax-7d 34150* Distinct variable version of ax-11 2155. (Contributed by Mario Carneiro, 14-Aug-2015.)
(βˆ€π‘₯βˆ€π‘¦πœ‘ β†’ βˆ€π‘¦βˆ€π‘₯πœ‘)
 
Axiomax-8d 34151* Distinct variable version of ax-7 2012. (Contributed by Mario Carneiro, 14-Aug-2015.)
(π‘₯ = 𝑦 β†’ (π‘₯ = 𝑧 β†’ 𝑦 = 𝑧))
 
Axiomax-9d1 34152 Distinct variable version of ax-6 1972, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
Β¬ βˆ€π‘₯ Β¬ π‘₯ = π‘₯
 
Axiomax-9d2 34153* Distinct variable version of ax-6 1972, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
Β¬ βˆ€π‘₯ Β¬ π‘₯ = 𝑦
 
Axiomax-10d 34154* Distinct variable version of axc11n 2426. (Contributed by Mario Carneiro, 14-Aug-2015.)
(βˆ€π‘₯ π‘₯ = 𝑦 β†’ βˆ€π‘¦ 𝑦 = π‘₯)
 
Axiomax-11d 34155* Distinct variable version of ax-12 2172. (Contributed by Mario Carneiro, 14-Aug-2015.)
(π‘₯ = 𝑦 β†’ (βˆ€π‘¦πœ‘ β†’ βˆ€π‘₯(π‘₯ = 𝑦 β†’ πœ‘)))
 
21.6.2  Miscellaneous stuff
 
Theoremquartfull 34156 The quartic equation, written out in full. This actually makes a fairly good Metamath stress test. Note that the length of this formula could be shortened significantly if the intermediate expressions were expanded and simplified, but it's not like this theorem will be used anyway. (Contributed by Mario Carneiro, 6-May-2015.)
(πœ‘ β†’ 𝐴 ∈ β„‚)    &   (πœ‘ β†’ 𝐡 ∈ β„‚)    &   (πœ‘ β†’ 𝐢 ∈ β„‚)    &   (πœ‘ β†’ 𝐷 ∈ β„‚)    &   (πœ‘ β†’ 𝑋 ∈ β„‚)    &   (πœ‘ β†’ (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)) β‰  0)    &   (πœ‘ β†’ -((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3) β‰  0)    β‡’   (πœ‘ β†’ ((((𝑋↑4) + (𝐴 Β· (𝑋↑3))) + ((𝐡 Β· (𝑋↑2)) + ((𝐢 Β· 𝑋) + 𝐷))) = 0 ↔ ((𝑋 = ((-(𝐴 / 4) βˆ’ ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (βˆšβ€˜((-(((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) βˆ’ ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) / 2)) + ((((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8)) / 4) / ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) βˆ’ ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) βˆ’ (βˆšβ€˜((-(((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) βˆ’ ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) / 2)) + ((((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8)) / 4) / ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))) ∨ (𝑋 = ((-(𝐴 / 4) + ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) + (βˆšβ€˜((-(((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) βˆ’ ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) / 2)) βˆ’ ((((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8)) / 4) / ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2))))) ∨ 𝑋 = ((-(𝐴 / 4) + ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)) βˆ’ (βˆšβ€˜((-(((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)↑2) βˆ’ ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) / 2)) βˆ’ ((((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8)) / 4) / ((βˆšβ€˜-((((2 Β· (𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))) + (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3))) + ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))) / (((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))) + (βˆšβ€˜((((-(2 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑3)) βˆ’ (27 Β· (((𝐢 βˆ’ ((𝐴 Β· 𝐡) / 2)) + ((𝐴↑3) / 8))↑2))) + (72 Β· ((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2))) Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4)))))))↑2) βˆ’ (4 Β· ((((𝐡 βˆ’ ((3 / 8) Β· (𝐴↑2)))↑2) + (12 Β· ((𝐷 βˆ’ ((𝐢 Β· 𝐴) / 4)) + ((((𝐴↑2) Β· 𝐡) / 16) βˆ’ ((3 / 256) Β· (𝐴↑4))))))↑3))))) / 2)↑𝑐(1 / 3)))) / 3)) / 2)))))))))
 
21.6.3  Derangements and the Subfactorial
 
Theoremderanglem 34157* Lemma for derangements. (Contributed by Mario Carneiro, 19-Jan-2015.)
(𝐴 ∈ Fin β†’ {𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ πœ‘)} ∈ Fin)
 
Theoremderangval 34158* Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   (𝐴 ∈ Fin β†’ (π·β€˜π΄) = (β™―β€˜{𝑓 ∣ (𝑓:𝐴–1-1-onto→𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘“β€˜π‘¦) β‰  𝑦)}))
 
Theoremderangf 34159* The derangement number is a function from finite sets to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   π·:FinβŸΆβ„•0
 
Theoremderang0 34160* The derangement number of the empty set. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   (π·β€˜βˆ…) = 1
 
Theoremderangsn 34161* The derangement number of a singleton. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   (𝐴 ∈ 𝑉 β†’ (π·β€˜{𝐴}) = 0)
 
Theoremderangenlem 34162* One half of derangen 34163. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (π·β€˜π΄) ≀ (π·β€˜π΅))
 
Theoremderangen 34163* The derangement number is a cardinal invariant, i.e. it only depends on the size of a set and not on its contents. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   ((𝐴 β‰ˆ 𝐡 ∧ 𝐡 ∈ Fin) β†’ (π·β€˜π΄) = (π·β€˜π΅))
 
Theoremsubfacval 34164* The subfactorial is defined as the number of derangements (see derangval 34158) of the set (1...𝑁). (Contributed by Mario Carneiro, 21-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„•0 β†’ (π‘†β€˜π‘) = (π·β€˜(1...𝑁)))
 
Theoremderangen2 34165* Write the derangement number in terms of the subfactorial. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝐴 ∈ Fin β†’ (π·β€˜π΄) = (π‘†β€˜(β™―β€˜π΄)))
 
Theoremsubfacf 34166* The subfactorial is a function from nonnegative integers to nonnegative integers. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   π‘†:β„•0βŸΆβ„•0
 
Theoremsubfaclefac 34167* The subfactorial is less than the factorial. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„•0 β†’ (π‘†β€˜π‘) ≀ (!β€˜π‘))
 
Theoremsubfac0 34168* The subfactorial at zero. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (π‘†β€˜0) = 1
 
Theoremsubfac1 34169* The subfactorial at one. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (π‘†β€˜1) = 0
 
Theoremsubfacp1lem1 34170* Lemma for subfacp1 34177. The set 𝐾 together with {1, 𝑀} partitions the set 1...(𝑁 + 1). (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    β‡’   (πœ‘ β†’ ((𝐾 ∩ {1, 𝑀}) = βˆ… ∧ (𝐾 βˆͺ {1, 𝑀}) = (1...(𝑁 + 1)) ∧ (β™―β€˜πΎ) = (𝑁 βˆ’ 1)))
 
Theoremsubfacp1lem2a 34171* Lemma for subfacp1 34177. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 βˆͺ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    &   πΉ = (𝐺 βˆͺ {⟨1, π‘€βŸ©, βŸ¨π‘€, 1⟩})    &   (πœ‘ β†’ 𝐺:𝐾–1-1-onto→𝐾)    β‡’   (πœ‘ β†’ (𝐹:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ (πΉβ€˜1) = 𝑀 ∧ (πΉβ€˜π‘€) = 1))
 
Theoremsubfacp1lem2b 34172* Lemma for subfacp1 34177. Properties of a bijection on 𝐾 augmented with the two-element flip to get a bijection on 𝐾 βˆͺ {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    &   πΉ = (𝐺 βˆͺ {⟨1, π‘€βŸ©, βŸ¨π‘€, 1⟩})    &   (πœ‘ β†’ 𝐺:𝐾–1-1-onto→𝐾)    β‡’   ((πœ‘ ∧ 𝑋 ∈ 𝐾) β†’ (πΉβ€˜π‘‹) = (πΊβ€˜π‘‹))
 
Theoremsubfacp1lem3 34173* Lemma for subfacp1 34177. In subfacp1lem6 34176 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (π‘“β€˜(π‘“β€˜1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements that satisfy this for fixed 𝑀 = (π‘“β€˜1) is in bijection with 𝑁 βˆ’ 1 derangements, by simply dropping the π‘₯ = 1 and π‘₯ = 𝑀 points from the function to get a derangement on 𝐾 = (1...(𝑁 βˆ’ 1)) βˆ– {1, 𝑀}. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    &   π΅ = {𝑔 ∈ 𝐴 ∣ ((π‘”β€˜1) = 𝑀 ∧ (π‘”β€˜π‘€) = 1)}    &   πΆ = {𝑓 ∣ (𝑓:𝐾–1-1-onto→𝐾 ∧ βˆ€π‘¦ ∈ 𝐾 (π‘“β€˜π‘¦) β‰  𝑦)}    β‡’   (πœ‘ β†’ (β™―β€˜π΅) = (π‘†β€˜(𝑁 βˆ’ 1)))
 
Theoremsubfacp1lem4 34174* Lemma for subfacp1 34177. The function 𝐹, which swaps 1 with 𝑀 and leaves all other elements alone, is a bijection of order 2, i.e. it is its own inverse. (Contributed by Mario Carneiro, 19-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    &   π΅ = {𝑔 ∈ 𝐴 ∣ ((π‘”β€˜1) = 𝑀 ∧ (π‘”β€˜π‘€) β‰  1)}    &   πΉ = (( I β†Ύ 𝐾) βˆͺ {⟨1, π‘€βŸ©, βŸ¨π‘€, 1⟩})    β‡’   (πœ‘ β†’ ◑𝐹 = 𝐹)
 
Theoremsubfacp1lem5 34175* Lemma for subfacp1 34177. In subfacp1lem6 34176 we cut up the set of all derangements on 1...(𝑁 + 1) first according to the value at 1, and then by whether or not (π‘“β€˜(π‘“β€˜1)) = 1. In this lemma, we show that the subset of all 𝑁 + 1 derangements with (π‘“β€˜(π‘“β€˜1)) β‰  1 for fixed 𝑀 = (π‘“β€˜1) is in bijection with derangements of 2...(𝑁 + 1), because pre-composing with the function 𝐹 swaps 1 and 𝑀 and turns the function into a bijection with (π‘“β€˜1) = 1 and (π‘“β€˜π‘₯) β‰  π‘₯ for all other π‘₯, so dropping the point at 1 yields a derangement on the 𝑁 remaining points. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (2...(𝑁 + 1)))    &   π‘€ ∈ V    &   πΎ = ((2...(𝑁 + 1)) βˆ– {𝑀})    &   π΅ = {𝑔 ∈ 𝐴 ∣ ((π‘”β€˜1) = 𝑀 ∧ (π‘”β€˜π‘€) β‰  1)}    &   πΉ = (( I β†Ύ 𝐾) βˆͺ {⟨1, π‘€βŸ©, βŸ¨π‘€, 1⟩})    &   πΆ = {𝑓 ∣ (𝑓:(2...(𝑁 + 1))–1-1-ontoβ†’(2...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (2...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    β‡’   (πœ‘ β†’ (β™―β€˜π΅) = (π‘†β€˜π‘))
 
Theoremsubfacp1lem6 34176* Lemma for subfacp1 34177. By induction, we cut up the set of all derangements on 𝑁 + 1 according to the 𝑁 possible values of (π‘“β€˜1) (since (π‘“β€˜1) β‰  1), and for each set for fixed 𝑀 = (π‘“β€˜1), the subset of derangements with (π‘“β€˜π‘€) = 1 has size 𝑆(𝑁 βˆ’ 1) (by subfacp1lem3 34173), while the subset with (π‘“β€˜π‘€) β‰  1 has size 𝑆(𝑁) (by subfacp1lem5 34175). Adding it all up yields the desired equation 𝑁(𝑆(𝑁) + 𝑆(𝑁 βˆ’ 1)) for the number of derangements on 𝑁 + 1. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    &   π΄ = {𝑓 ∣ (𝑓:(1...(𝑁 + 1))–1-1-ontoβ†’(1...(𝑁 + 1)) ∧ βˆ€π‘¦ ∈ (1...(𝑁 + 1))(π‘“β€˜π‘¦) β‰  𝑦)}    β‡’   (𝑁 ∈ β„• β†’ (π‘†β€˜(𝑁 + 1)) = (𝑁 Β· ((π‘†β€˜π‘) + (π‘†β€˜(𝑁 βˆ’ 1)))))
 
Theoremsubfacp1 34177* A two-term recurrence for the subfactorial. This theorem allows to forget the combinatorial definition of the derangement number in favor of the recursive definition provided by this theorem and subfac0 34168, subfac1 34169. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„• β†’ (π‘†β€˜(𝑁 + 1)) = (𝑁 Β· ((π‘†β€˜π‘) + (π‘†β€˜(𝑁 βˆ’ 1)))))
 
Theoremsubfacval2 34178* A closed-form expression for the subfactorial. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„•0 β†’ (π‘†β€˜π‘) = ((!β€˜π‘) Β· Ξ£π‘˜ ∈ (0...𝑁)((-1β†‘π‘˜) / (!β€˜π‘˜))))
 
Theoremsubfaclim 34179* The subfactorial converges rapidly to 𝑁! / e. This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„• β†’ (absβ€˜(((!β€˜π‘) / e) βˆ’ (π‘†β€˜π‘))) < (1 / 𝑁))
 
Theoremsubfacval3 34180* Another closed form expression for the subfactorial. The expression βŒŠβ€˜(π‘₯ + 1 / 2) is a way of saying "rounded to the nearest integer". (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    &   π‘† = (𝑛 ∈ β„•0 ↦ (π·β€˜(1...𝑛)))    β‡’   (𝑁 ∈ β„• β†’ (π‘†β€˜π‘) = (βŒŠβ€˜(((!β€˜π‘) / e) + (1 / 2))))
 
Theoremderangfmla 34181* The derangements formula, which expresses the number of derangements of a finite nonempty set in terms of the factorial. The expression βŒŠβ€˜(π‘₯ + 1 / 2) is a way of saying "rounded to the nearest integer". This is part of Metamath 100 proof #88. (Contributed by Mario Carneiro, 23-Jan-2015.)
𝐷 = (π‘₯ ∈ Fin ↦ (β™―β€˜{𝑓 ∣ (𝑓:π‘₯–1-1-ontoβ†’π‘₯ ∧ βˆ€π‘¦ ∈ π‘₯ (π‘“β€˜π‘¦) β‰  𝑦)}))    β‡’   ((𝐴 ∈ Fin ∧ 𝐴 β‰  βˆ…) β†’ (π·β€˜π΄) = (βŒŠβ€˜(((!β€˜(β™―β€˜π΄)) / e) + (1 / 2))))
 
21.6.4  The ErdΕ‘s-Szekeres theorem
 
Theoremerdszelem1 34182* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ 𝐴 ∈ 𝑦)}    β‡’   (𝑋 ∈ 𝑆 ↔ (𝑋 βŠ† (1...𝐴) ∧ (𝐹 β†Ύ 𝑋) Isom < , 𝑂 (𝑋, (𝐹 β€œ 𝑋)) ∧ 𝐴 ∈ 𝑋))
 
Theoremerdszelem2 34183* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
𝑆 = {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ 𝐴 ∈ 𝑦)}    β‡’   ((β™― β€œ 𝑆) ∈ Fin ∧ (β™― β€œ 𝑆) βŠ† β„•)
 
Theoremerdszelem3 34184* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    β‡’   (𝐴 ∈ (1...𝑁) β†’ (πΎβ€˜π΄) = sup((β™― β€œ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ 𝐴 ∈ 𝑦)}), ℝ, < ))
 
Theoremerdszelem4 34185* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‚ Or ℝ    β‡’   ((πœ‘ ∧ 𝐴 ∈ (1...𝑁)) β†’ {𝐴} ∈ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ 𝐴 ∈ 𝑦)})
 
Theoremerdszelem5 34186* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‚ Or ℝ    β‡’   ((πœ‘ ∧ 𝐴 ∈ (1...𝑁)) β†’ (πΎβ€˜π΄) ∈ (β™― β€œ {𝑦 ∈ 𝒫 (1...𝐴) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ 𝐴 ∈ 𝑦)}))
 
Theoremerdszelem6 34187* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‚ Or ℝ    β‡’   (πœ‘ β†’ 𝐾:(1...𝑁)βŸΆβ„•)
 
Theoremerdszelem7 34188* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‚ Or ℝ    &   (πœ‘ β†’ 𝐴 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ Β¬ (πΎβ€˜π΄) ∈ (1...(𝑅 βˆ’ 1)))    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ 𝒫 (1...𝑁)(𝑅 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , 𝑂 (𝑠, (𝐹 β€œ 𝑠))))
 
Theoremerdszelem8 34189* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΎ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‚ Or ℝ    &   (πœ‘ β†’ 𝐴 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐡 ∈ (1...𝑁))    &   (πœ‘ β†’ 𝐴 < 𝐡)    β‡’   (πœ‘ β†’ ((πΎβ€˜π΄) = (πΎβ€˜π΅) β†’ Β¬ (πΉβ€˜π΄)𝑂(πΉβ€˜π΅)))
 
Theoremerdszelem9 34190* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΌ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π½ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‡ = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)    β‡’   (πœ‘ β†’ 𝑇:(1...𝑁)–1-1β†’(β„• Γ— β„•))
 
Theoremerdszelem10 34191* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΌ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π½ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‡ = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)    &   (πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < 𝑁)    β‡’   (πœ‘ β†’ βˆƒπ‘š ∈ (1...𝑁)(Β¬ (πΌβ€˜π‘š) ∈ (1...(𝑅 βˆ’ 1)) ∨ Β¬ (π½β€˜π‘š) ∈ (1...(𝑆 βˆ’ 1))))
 
Theoremerdszelem11 34192* Lemma for erdsze 34193. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   πΌ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π½ = (π‘₯ ∈ (1...𝑁) ↦ sup((β™― β€œ {𝑦 ∈ 𝒫 (1...π‘₯) ∣ ((𝐹 β†Ύ 𝑦) Isom < , β—‘ < (𝑦, (𝐹 β€œ 𝑦)) ∧ π‘₯ ∈ 𝑦)}), ℝ, < ))    &   π‘‡ = (𝑛 ∈ (1...𝑁) ↦ ⟨(πΌβ€˜π‘›), (π½β€˜π‘›)⟩)    &   (πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < 𝑁)    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ 𝒫 (1...𝑁)((𝑅 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , < (𝑠, (𝐹 β€œ 𝑠))) ∨ (𝑆 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , β—‘ < (𝑠, (𝐹 β€œ 𝑠)))))
 
Theoremerdsze 34193* The ErdΕ‘s-Szekeres theorem. For any injective sequence 𝐹 on the reals of length at least (𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1) + 1, there is either a subsequence of length at least 𝑅 on which 𝐹 is increasing (i.e. a < , < order isomorphism) or a subsequence of length at least 𝑆 on which 𝐹 is decreasing (i.e. a < , β—‘ < order isomorphism, recalling that β—‘ < is the "greater than" relation). This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝐹:(1...𝑁)–1-1→ℝ)    &   (πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < 𝑁)    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ 𝒫 (1...𝑁)((𝑅 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , < (𝑠, (𝐹 β€œ 𝑠))) ∨ (𝑆 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , β—‘ < (𝑠, (𝐹 β€œ 𝑠)))))
 
Theoremerdsze2lem1 34194* Lemma for erdsze2 34196. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝐹:𝐴–1-1→ℝ)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘ = ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1))    &   (πœ‘ β†’ 𝑁 < (β™―β€˜π΄))    β‡’   (πœ‘ β†’ βˆƒπ‘“(𝑓:(1...(𝑁 + 1))–1-1→𝐴 ∧ 𝑓 Isom < , < ((1...(𝑁 + 1)), ran 𝑓)))
 
Theoremerdsze2lem2 34195* Lemma for erdsze2 34196. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝐹:𝐴–1-1→ℝ)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   π‘ = ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1))    &   (πœ‘ β†’ 𝑁 < (β™―β€˜π΄))    &   (πœ‘ β†’ 𝐺:(1...(𝑁 + 1))–1-1→𝐴)    &   (πœ‘ β†’ 𝐺 Isom < , < ((1...(𝑁 + 1)), ran 𝐺))    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ 𝒫 𝐴((𝑅 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , < (𝑠, (𝐹 β€œ 𝑠))) ∨ (𝑆 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , β—‘ < (𝑠, (𝐹 β€œ 𝑠)))))
 
Theoremerdsze2 34196* Generalize the statement of the ErdΕ‘s-Szekeres theorem erdsze 34193 to "sequences" indexed by an arbitrary subset of ℝ, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.)
(πœ‘ β†’ 𝑅 ∈ β„•)    &   (πœ‘ β†’ 𝑆 ∈ β„•)    &   (πœ‘ β†’ 𝐹:𝐴–1-1→ℝ)    &   (πœ‘ β†’ 𝐴 βŠ† ℝ)    &   (πœ‘ β†’ ((𝑅 βˆ’ 1) Β· (𝑆 βˆ’ 1)) < (β™―β€˜π΄))    β‡’   (πœ‘ β†’ βˆƒπ‘  ∈ 𝒫 𝐴((𝑅 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , < (𝑠, (𝐹 β€œ 𝑠))) ∨ (𝑆 ≀ (β™―β€˜π‘ ) ∧ (𝐹 β†Ύ 𝑠) Isom < , β—‘ < (𝑠, (𝐹 β€œ 𝑠)))))
 
21.6.5  The Kuratowski closure-complement theorem
 
Theoremkur14lem1 34197 Lemma for kur14 34207. (Contributed by Mario Carneiro, 17-Feb-2015.)
𝐴 βŠ† 𝑋    &   (𝑋 βˆ– 𝐴) ∈ 𝑇    &   (πΎβ€˜π΄) ∈ 𝑇    β‡’   (𝑁 = 𝐴 β†’ (𝑁 βŠ† 𝑋 ∧ {(𝑋 βˆ– 𝑁), (πΎβ€˜π‘)} βŠ† 𝑇))
 
Theoremkur14lem2 34198 Lemma for kur14 34207. Write interior in terms of closure and complement: 𝑖𝐴 = π‘π‘˜π‘π΄ where 𝑐 is complement and π‘˜ is closure. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   π‘‹ = βˆͺ 𝐽    &   πΎ = (clsβ€˜π½)    &   πΌ = (intβ€˜π½)    &   π΄ βŠ† 𝑋    β‡’   (πΌβ€˜π΄) = (𝑋 βˆ– (πΎβ€˜(𝑋 βˆ– 𝐴)))
 
Theoremkur14lem3 34199 Lemma for kur14 34207. A closure is a subset of the base set. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   π‘‹ = βˆͺ 𝐽    &   πΎ = (clsβ€˜π½)    &   πΌ = (intβ€˜π½)    &   π΄ βŠ† 𝑋    β‡’   (πΎβ€˜π΄) βŠ† 𝑋
 
Theoremkur14lem4 34200 Lemma for kur14 34207. Complementation is an involution on the set of subsets of a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
𝐽 ∈ Top    &   π‘‹ = βˆͺ 𝐽    &   πΎ = (clsβ€˜π½)    &   πΌ = (intβ€˜π½)    &   π΄ βŠ† 𝑋    β‡’   (𝑋 βˆ– (𝑋 βˆ– 𝐴)) = 𝐴
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47852
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