:: SIMPLEX1 semantic presentation

REAL is set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal non empty-membered Element of bool REAL
bool REAL is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of NAT
- 1 is non empty V21() V29() ext-real non positive negative set
K87(1) is non empty V21() V29() V30() ext-real non positive negative finite set
COMPLEX is set
RAT is set
INT is set
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:NAT,NAT:] is Relation-like non empty non trivial non finite non empty-membered set
[:[:NAT,NAT:],NAT:] is Relation-like non empty non trivial non finite non empty-membered set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite V47() subset-closed non with_non-empty_elements non empty-membered V290() d.union-closed set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal non empty-membered set
bool omega is non empty non trivial non finite V47() subset-closed non with_non-empty_elements non empty-membered V290() d.union-closed set
bool NAT is non empty non trivial non finite V47() subset-closed non with_non-empty_elements non empty-membered V290() d.union-closed set
{} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
the Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
2 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of NAT
K469() is set
{{}} is functional non empty trivial finite finite-membered 1 -element V47() subset-closed non with_non-empty_elements V290() d.union-closed set
K272({{}}) is M10({{}})
[:K272({{}}),{{}}:] is Relation-like set
bool [:K272({{}}),{{}}:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
K36(K272({{}}),{{}}) is set
K620() is TopStruct
the carrier of K620() is set
0 is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of NAT
- 1 is non empty V21() V29() V30() ext-real non positive negative finite Element of REAL
+infty is non empty V29() ext-real positive non negative set
-infty is non empty V29() ext-real non positive negative set
union {} is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k is set
card k is epsilon-transitive epsilon-connected ordinal cardinal set
V is Relation-like set
card V is epsilon-transitive epsilon-connected ordinal cardinal set
dom V is set
(dom V) \ k is Element of bool (dom V)
bool (dom V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V | ((dom V) \ k) is Relation-like set
card (V | ((dom V) \ k)) is epsilon-transitive epsilon-connected ordinal cardinal set
Aff is epsilon-transitive epsilon-connected ordinal cardinal set
Aff *` (card k) is epsilon-transitive epsilon-connected ordinal cardinal set
(card (V | ((dom V) \ k))) +` (Aff *` (card k)) is epsilon-transitive epsilon-connected ordinal cardinal set
XX is Relation-like Function-like set
dom XX is set
x is set
XX . x is set
Im (V,x) is set
card (Im (V,x)) is epsilon-transitive epsilon-connected ordinal cardinal set
C is Relation-like Function-like set
dom C is set
rng C is set
x is Relation-like Function-like set
dom x is set
C is set
x . C is set
XX . C is set
F is Relation-like Function-like set
dom F is set
rng F is set
C is Relation-like Function-like Function-yielding V197() set
V | k is Relation-like set
F is Relation-like Function-like set
dom F is set
rng F is set
[:k,Aff:] is Relation-like set
X is set
A1 is set
F . A1 is set
A1 `1 is set
C . (A1 `1) is Relation-like Function-like set
A1 `2 is set
(C . (A1 `1)) . (A1 `2) is set
[(A1 `1),((C . (A1 `1)) . (A1 `2))] is V15() set
{(A1 `1),((C . (A1 `1)) . (A1 `2))} is non empty finite set
{(A1 `1)} is non empty trivial finite 1 -element set
{{(A1 `1),((C . (A1 `1)) . (A1 `2))},{(A1 `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(A1 `1),(A1 `2)] is V15() set
{(A1 `1),(A1 `2)} is non empty finite set
{{(A1 `1),(A1 `2)},{(A1 `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
V .: {(A1 `1)} is set
XX . (A1 `1) is set
S is Relation-like Function-like set
dom S is set
rng S is set
Im (V,(A1 `1)) is set
S . (A1 `2) is set
X is set
A1 is set
S is set
[A1,S] is V15() set
{A1,S} is non empty finite set
{A1} is non empty trivial finite 1 -element set
{{A1,S},{A1}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
C . A1 is Relation-like Function-like set
XX . A1 is set
CA is Relation-like Function-like set
dom CA is set
rng CA is set
F is set
CA . F is set
[A1,F] is V15() set
{A1,F} is non empty finite set
{{A1,F},{A1}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[A1,F] `1 is set
C . ([A1,F] `1) is Relation-like Function-like set
[A1,F] `2 is set
(C . ([A1,F] `1)) . ([A1,F] `2) is set
[([A1,F] `1),((C . ([A1,F] `1)) . ([A1,F] `2))] is V15() set
{([A1,F] `1),((C . ([A1,F] `1)) . ([A1,F] `2))} is non empty finite set
{([A1,F] `1)} is non empty trivial finite 1 -element set
{{([A1,F] `1),((C . ([A1,F] `1)) . ([A1,F] `2))},{([A1,F] `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
Im (V,A1) is set
F . [A1,F] is set
X is set
A1 is set
F . X is set
F . A1 is set
X `1 is set
C . (X `1) is Relation-like Function-like set
X `2 is set
(C . (X `1)) . (X `2) is set
[(X `1),((C . (X `1)) . (X `2))] is V15() set
{(X `1),((C . (X `1)) . (X `2))} is non empty finite set
{(X `1)} is non empty trivial finite 1 -element set
{{(X `1),((C . (X `1)) . (X `2))},{(X `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
A1 `1 is set
C . (A1 `1) is Relation-like Function-like set
A1 `2 is set
(C . (A1 `1)) . (A1 `2) is set
[(A1 `1),((C . (A1 `1)) . (A1 `2))] is V15() set
{(A1 `1),((C . (A1 `1)) . (A1 `2))} is non empty finite set
{(A1 `1)} is non empty trivial finite 1 -element set
{{(A1 `1),((C . (A1 `1)) . (A1 `2))},{(A1 `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(X `1),(X `2)] is V15() set
{(X `1),(X `2)} is non empty finite set
{{(X `1),(X `2)},{(X `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
Im (V,(X `1)) is set
[(A1 `1),(A1 `2)] is V15() set
{(A1 `1),(A1 `2)} is non empty finite set
{{(A1 `1),(A1 `2)},{(A1 `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
Im (V,(A1 `1)) is set
XX . (A1 `1) is set
S is Relation-like Function-like set
dom S is set
rng S is set
XX . (X `1) is set
CA is Relation-like Function-like set
dom CA is set
rng CA is set
CA . (X `2) is set
S . (A1 `2) is set
card (V | k) is epsilon-transitive epsilon-connected ordinal cardinal set
card [:k,Aff:] is epsilon-transitive epsilon-connected ordinal cardinal set
[:(card k),Aff:] is Relation-like set
card [:(card k),Aff:] is epsilon-transitive epsilon-connected ordinal cardinal set
X is set
A1 is set
S is set
[A1,S] is V15() set
{A1,S} is non empty finite set
{A1} is non empty trivial finite 1 -element set
{{A1,S},{A1}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
((dom V) \ k) \/ k is set
(dom V) \/ k is set
(V | k) \/ (V | ((dom V) \ k)) is Relation-like set
V | (dom V) is Relation-like set
XX is set
Im (V,XX) is set
x is set
[XX,x] is V15() set
{XX,x} is non empty finite set
{XX} is non empty trivial finite 1 -element set
{{XX,x},{XX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
card {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of omega
x is set
Im (V,x) is set
card (Im (V,x)) is epsilon-transitive epsilon-connected ordinal cardinal set
C is set
[x,C] is V15() set
{x,C} is non empty finite set
{x} is non empty trivial finite 1 -element set
{{x,C},{x}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
k is set
card k is epsilon-transitive epsilon-connected ordinal cardinal set
V is non empty finite set
card V is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
(card V) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
[:k,V:] is Relation-like set
bool [:k,V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(card V) - 1 is V21() V29() V30() ext-real finite Element of REAL
XX is Relation-like k -defined V -valued Function-like quasi_total Element of bool [:k,V:]
rng XX is finite Element of bool V
bool V is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is non empty finite set
[:Aff,V:] is Relation-like non empty finite set
bool [:Aff,V:] is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom XX is Element of bool k
bool k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is Relation-like Aff -defined V -valued Function-like quasi_total finite Element of bool [:Aff,V:]
C is set
{C} is non empty trivial finite 1 -element set
x " {C} is finite Element of bool Aff
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card (x " {C}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
C is set
{C} is non empty trivial finite 1 -element set
x " {C} is finite Element of bool Aff
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card (x " {C}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
C is set
{C} is non empty trivial finite 1 -element set
x " {C} is finite Element of bool Aff
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card (x " {C}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(dom XX) \ (x " {C}) is Element of bool k
x | ((dom XX) \ (x " {C})) is Relation-like Aff -defined (dom XX) \ (x " {C}) -defined Aff -defined V -valued Function-like finite Element of bool [:Aff,V:]
XX " {C} is Element of bool k
card (XX " {C}) is epsilon-transitive epsilon-connected ordinal cardinal set
1 + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
dom (x | ((dom XX) \ (x " {C}))) is finite Element of bool Aff
card (dom (x | ((dom XX) \ (x " {C})))) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card Aff is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
(card Aff) - (card (x " {C})) is V21() V29() ext-real set
- (card (x " {C})) is V21() V29() ext-real non positive set
K87((card (x " {C}))) is V21() V29() V30() ext-real non positive finite set
(card Aff) + (- (card (x " {C}))) is V21() V29() ext-real set
K85((card Aff),(- (card (x " {C})))) is V21() V29() ext-real set
K89((card Aff),(card (x " {C}))) is V21() V29() V30() ext-real finite set
V \ {C} is finite Element of bool V
rng (x | ((dom XX) \ (x " {C}))) is finite Element of bool V
[:(dom (x | ((dom XX) \ (x " {C})))),(V \ {C}):] is Relation-like finite set
bool [:(dom (x | ((dom XX) \ (x " {C})))),(V \ {C}):] is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
card (V \ {C}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card V) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
1 - (card (x " {C})) is V21() V29() V30() ext-real finite Element of REAL
(card V) + (1 - (card (x " {C}))) is V21() V29() V30() ext-real finite Element of REAL
1 - (- 1) is non empty V21() V29() V30() ext-real positive non negative finite Element of REAL
CA is set
{CA} is non empty trivial finite 1 -element set
XX " {CA} is Element of bool k
card (XX " {CA}) is epsilon-transitive epsilon-connected ordinal cardinal set
S is Relation-like dom (x | ((dom XX) \ (x " {C}))) -defined V \ {C} -valued Function-like quasi_total finite Element of bool [:(dom (x | ((dom XX) \ (x " {C})))),(V \ {C}):]
rng S is finite Element of bool (V \ {C})
bool (V \ {C}) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
S " {CA} is finite Element of bool (dom (x | ((dom XX) \ (x " {C}))))
bool (dom (x | ((dom XX) \ (x " {C})))) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is set
{F} is non empty trivial finite 1 -element set
k is 1-sorted
the carrier of k is set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
k is 1-sorted
the carrier of k is set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool (bool the carrier of V)
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
bool ([#] V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool ([#] V))
bool ([#] V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool ([#] V)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
k is 1-sorted
the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is subset-closed SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
{(k,V,Aff)} is non empty trivial finite 1 -element Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {(k,V,Aff)} is strict non void subset-closed non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {(k,V,Aff)} is non empty V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {(k,V,Aff)}) #) is strict TopStruct
the topology of V is Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
{Aff} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
[#] (Complex_of {(k,V,Aff)}) is non proper Element of bool the carrier of (Complex_of {(k,V,Aff)})
the carrier of (Complex_of {(k,V,Aff)}) is set
bool the carrier of (Complex_of {(k,V,Aff)}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the_family_of V is V47() subset-closed Element of bool (bool the carrier of V)
the topology of (Complex_of {(k,V,Aff)}) is Element of bool (bool the carrier of (Complex_of {(k,V,Aff)}))
bool (bool the carrier of (Complex_of {(k,V,Aff)})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
k is non empty RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
{ (conv (k,V,b₁)) where b₁ is Element of bool the carrier of V : not b₁ is dependent } is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is set
XX is Element of bool the carrier of V
(k,V,XX) is Element of bool the carrier of k
conv (k,V,XX) is convex Element of bool the carrier of k
F is Element of bool (bool the carrier of k)
union F is Element of bool the carrier of k
XX is Element of bool the carrier of k
x is set
C is set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
X is Element of bool the carrier of V
(k,V,X) is Element of bool the carrier of k
conv (k,V,X) is convex Element of bool the carrier of k
C is Element of bool the carrier of V
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
Aff is Element of bool the carrier of k
F is Element of bool the carrier of k
XX is set
x is Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
x is Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is SimplicialComplexStr of the carrier of k
the topology of Aff is Element of bool (bool the carrier of Aff)
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,Aff) is Element of bool the carrier of k
F is set
XX is Element of bool the carrier of V
(k,V,XX) is Element of bool the carrier of k
conv (k,V,XX) is convex Element of bool the carrier of k
x is Element of bool the carrier of Aff
(k,Aff,x) is Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
F is set
k is set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is subset-closed SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of Aff
(V,Aff,F) is Element of bool the carrier of V
conv (V,Aff,F) is convex Element of bool the carrier of V
{ (Int b₁) where b₁ is Element of bool the carrier of V : b₁ c= (V,Aff,F) } is set
union { (Int b₁) where b₁ is Element of bool the carrier of V : b₁ c= (V,Aff,F) } is set
XX is set
x is Element of bool the carrier of V
Int x is Element of bool the carrier of V
C is Element of bool the carrier of Aff
the topology of Aff is Element of bool (bool the carrier of Aff)
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of Aff
(V,Aff,F) is Element of bool the carrier of V
Int (V,Aff,F) is Element of bool the carrier of V
F is Element of bool the carrier of Aff
(V,Aff,F) is Element of bool the carrier of V
Int (V,Aff,F) is Element of bool the carrier of V
conv (V,Aff,F) is convex Element of bool the carrier of V
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is non empty set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
XX is set
Aff is set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is Element of bool the carrier of k
{V} is non empty trivial finite 1 -element Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {V} is strict non void subset-closed non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {V} is non empty V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {V}) #) is strict TopStruct
(k,(Complex_of {V})) is Element of bool the carrier of k
conv V is convex Element of bool the carrier of k
the carrier of (Complex_of {V}) is set
bool the carrier of (Complex_of {V}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (Complex_of {V}) is Element of bool (bool the carrier of (Complex_of {V}))
bool (bool the carrier of (Complex_of {V})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool V)
bool V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is set
x is Element of bool the carrier of (Complex_of {V})
(k,(Complex_of {V}),x) is Element of bool the carrier of k
conv (k,(Complex_of {V}),x) is convex Element of bool the carrier of k
F is Element of bool the carrier of (Complex_of {V})
(k,(Complex_of {V}),F) is Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is Element of bool (bool the carrier of k)
Aff is Element of bool (bool the carrier of k)
V \/ Aff is Element of bool (bool the carrier of k)
Complex_of (V \/ Aff) is strict subset-closed total SimplicialComplexStr of the carrier of k
subset-closed_closure_of (V \/ Aff) is V47() subset-closed Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of (V \/ Aff)) #) is strict TopStruct
(k,(Complex_of (V \/ Aff))) is Element of bool the carrier of k
Complex_of V is strict subset-closed total SimplicialComplexStr of the carrier of k
subset-closed_closure_of V is V47() subset-closed Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of V) #) is strict TopStruct
(k,(Complex_of V)) is Element of bool the carrier of k
Complex_of Aff is strict subset-closed total SimplicialComplexStr of the carrier of k
subset-closed_closure_of Aff is V47() subset-closed Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of Aff) #) is strict TopStruct
(k,(Complex_of Aff)) is Element of bool the carrier of k
(k,(Complex_of V)) \/ (k,(Complex_of Aff)) is Element of bool the carrier of k
the topology of (Complex_of V) is Element of bool (bool the carrier of (Complex_of V))
the carrier of (Complex_of V) is set
bool the carrier of (Complex_of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of V)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (Complex_of Aff) is Element of bool (bool the carrier of (Complex_of Aff))
the carrier of (Complex_of Aff) is set
bool the carrier of (Complex_of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of Aff)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (Complex_of V) \/ the topology of (Complex_of Aff) is set
the topology of (Complex_of (V \/ Aff)) is Element of bool (bool the carrier of (Complex_of (V \/ Aff)))
the carrier of (Complex_of (V \/ Aff)) is set
bool the carrier of (Complex_of (V \/ Aff)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of (V \/ Aff))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
C is set
F is Element of bool the carrier of (Complex_of (V \/ Aff))
(k,(Complex_of (V \/ Aff)),F) is Element of bool the carrier of k
conv (k,(Complex_of (V \/ Aff)),F) is convex Element of bool the carrier of k
X is Element of bool the carrier of (Complex_of V)
(k,(Complex_of V),X) is Element of bool the carrier of k
X is Element of bool the carrier of (Complex_of Aff)
(k,(Complex_of Aff),X) is Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is (k,V)
(k,Aff) is Element of bool the carrier of k
F is set
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool the carrier of Aff
(k,Aff,XX) is Element of bool the carrier of k
conv (k,Aff,XX) is convex Element of bool the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is non void non empty-membered SimplicialComplexStr of the carrier of k
Aff is (k,V)
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,Aff) is Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
the topology of V is Element of bool (bool the carrier of V)
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of the topology of V is strict subset-closed total SimplicialComplexStr of the carrier of V
subset-closed_closure_of the topology of V is V47() subset-closed Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of the topology of V) #) is strict TopStruct
[#] (Complex_of the topology of V) is non proper Element of bool the carrier of (Complex_of the topology of V)
the carrier of (Complex_of the topology of V) is set
bool the carrier of (Complex_of the topology of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is SimplicialComplexStr of the carrier of k
(k,XX) is Element of bool the carrier of k
x is set
C is Element of bool the carrier of V
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
the carrier of XX is set
bool the carrier of XX is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of XX is Element of bool (bool the carrier of XX)
bool (bool the carrier of XX) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of XX
(k,XX,F) is Element of bool the carrier of k
the carrier of XX is set
bool the carrier of XX is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is Element of bool the carrier of XX
(k,XX,x) is Element of bool the carrier of k
conv (k,XX,x) is convex Element of bool the carrier of k
the topology of XX is Element of bool (bool the carrier of XX)
bool (bool the carrier of XX) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
C is set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is subset-closed SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
Sub_of_Fin the topology of V is finite-membered Element of bool the topology of V
bool the topology of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool (bool the carrier of V)
Complex_of F is strict subset-closed total SimplicialComplexStr of the carrier of V
subset-closed_closure_of F is V47() subset-closed Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of F) #) is strict TopStruct
[#] (Complex_of F) is non proper Element of bool the carrier of (Complex_of F)
the carrier of (Complex_of F) is set
bool the carrier of (Complex_of F) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the_family_of V is V47() subset-closed Element of bool (bool the carrier of V)
x is SimplicialComplexStr of the carrier of k
the topology of x is Element of bool (bool the carrier of x)
the carrier of x is set
bool the carrier of x is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of x) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,x) is Element of bool the carrier of k
C is set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
X is non empty Element of bool the carrier of k
ConvexComb k is set
{ (Sum b₁) where b₁ is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() convex Linear_Combination of X : b₁ in ConvexComb k } is set
A1 is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() convex Linear_Combination of X
Sum A1 is Element of the carrier of k
Carrier A1 is finite Element of bool the carrier of k
{ b₁ where b₁ is Element of the carrier of k : not A1 . b₁ = 0 } is set
S is non empty Element of bool the carrier of k
CA is Element of bool the carrier of x
F is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of S
{ (Sum b₁) where b₁ is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() convex Linear_Combination of S : b₁ in ConvexComb k } is set
conv S is non empty convex Element of bool the carrier of k
(k,x,CA) is Element of bool the carrier of k
C is Element of bool the carrier of x
(k,x,C) is Element of bool the carrier of k
conv (k,x,C) is convex Element of bool the carrier of k
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
Aff is (k,V)
F is (k,Aff)
(k,F) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,Aff) is Element of bool the carrier of k
(k,V) is Element of bool the carrier of k
the carrier of F is set
bool the carrier of F is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool the carrier of F
(k,F,XX) is Element of bool the carrier of k
conv (k,F,XX) is convex Element of bool the carrier of k
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is Element of bool the carrier of Aff
(k,Aff,x) is Element of bool the carrier of k
conv (k,Aff,x) is convex Element of bool the carrier of k
C is Element of bool the carrier of V
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the carrier of V is set
the topology of V is Element of bool (bool the carrier of V)
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of the topology of V is strict subset-closed total SimplicialComplexStr of the carrier of V
subset-closed_closure_of the topology of V is V47() subset-closed Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of the topology of V) #) is strict TopStruct
Aff is (k,V)
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of Aff is Element of bool (bool the carrier of Aff)
Sub_of_Fin the topology of Aff is finite-membered Element of bool the topology of Aff
bool the topology of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool (bool the carrier of Aff)
Complex_of F is strict subset-closed total SimplicialComplexStr of the carrier of Aff
subset-closed_closure_of F is V47() subset-closed Element of bool (bool the carrier of Aff)
TopStruct(# the carrier of Aff,(subset-closed_closure_of F) #) is strict TopStruct
XX is (k,V)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
subdivision (Aff,V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of (subdivision (Aff,V)) is set
bool the carrier of (subdivision (Aff,V)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom Aff is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
XX + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
x is set
C is finite non dependent Element of bool the carrier of V
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
card C is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
{C} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
union {C} is finite Element of bool the carrier of V
X is Element of bool the carrier of V
F is non empty Element of bool the carrier of k
Aff . F is set
Aff . C is set
Aff .: {C} is finite Element of bool the carrier of k
Im (Aff,C) is set
X is Element of the carrier of k
{X} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
conv F is non empty convex Element of bool the carrier of k
[#] (subdivision (Aff,V)) is non proper Element of bool the carrier of (subdivision (Aff,V))
S is set
CA is set
S is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
Aff .: S is finite Element of bool the carrier of k
union S is finite Element of bool the carrier of V
A1 is Element of bool the carrier of (subdivision (Aff,V))
(k,(subdivision (Aff,V)),A1) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),A1) is convex Element of bool the carrier of k
conv {X} is non empty convex Element of bool the carrier of k
S is Element of the carrier of k
CA is Element of the carrier of k
{S} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
F \ {S} is Element of bool the carrier of k
conv (F \ {S}) is convex Element of bool the carrier of k
F is V21() V29() ext-real Element of REAL
F * X is Element of the carrier of k
1 - F is V21() V29() ext-real Element of REAL
(1 - F) * CA is Element of the carrier of k
(F * X) + ((1 - F) * CA) is Element of the carrier of k
C \ {S} is finite non dependent Element of bool the carrier of V
(k,V,(C \ {S})) is Element of bool the carrier of k
card (C \ {S}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
XX is Element of bool the carrier of (subdivision (Aff,V))
XX is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
Aff .: XX is finite Element of bool the carrier of k
(k,(subdivision (Aff,V)),XX) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),XX) is convex Element of bool the carrier of k
union XX is finite Element of bool the carrier of V
XX \/ {C} is non empty finite finite-membered Element of bool (bool the carrier of V)
XXA is set
m1 is set
XXA is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
Aff .: XXA is finite Element of bool the carrier of k
A1 is Element of bool the carrier of (subdivision (Aff,V))
XX \/ A1 is Element of bool the carrier of (subdivision (Aff,V))
m1 is Element of bool the carrier of (subdivision (Aff,V))
(k,(subdivision (Aff,V)),m1) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),m1) is convex Element of bool the carrier of k
union XXA is finite Element of bool the carrier of V
(union XX) \/ C is finite Element of bool the carrier of V
XX is set
x is finite non dependent Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
card x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k,(subdivision (Aff,V))) is Element of bool the carrier of k
XX is set
x is Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
C is finite non dependent Element of bool the carrier of V
card C is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is set
X is finite non dependent Element of bool the carrier of V
(k,V,X) is Element of bool the carrier of k
conv (k,V,X) is convex Element of bool the carrier of k
card X is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
X is Element of bool the carrier of (subdivision (Aff,V))
F is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
Aff .: F is finite Element of bool the carrier of k
(k,(subdivision (Aff,V)),X) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),X) is convex Element of bool the carrier of k
union F is finite Element of bool the carrier of V
XX is Element of bool the carrier of (subdivision (Aff,V))
(k,(subdivision (Aff,V)),XX) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),XX) is convex Element of bool the carrier of k
x is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
Aff .: x is finite Element of bool the carrier of k
{} V is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() non dependent subset-closed Function-yielding V197() Element of bool the carrier of V
(k,V,({} V)) is Element of bool the carrier of k
conv (k,V,({} V)) is convex Element of bool the carrier of k
union x is finite Element of bool the carrier of V
C is finite Element of bool the carrier of V
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
F is set
X is set
Aff . X is set
A1 is finite non dependent Element of bool the carrier of V
Aff . A1 is set
(k,V,A1) is Element of bool the carrier of k
conv (k,V,A1) is convex Element of bool the carrier of k
XX is Element of bool the carrier of (subdivision (Aff,V))
(k,(subdivision (Aff,V)),XX) is Element of bool the carrier of k
conv (k,(subdivision (Aff,V)),XX) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
Aff is subset-closed finite-membered (k,V)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
F is finite non dependent Element of bool the carrier of V
(center_of_mass k) . F is set
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
subdivision (k,(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
subdivision (F,(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
subdivision ((F + 1),(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
XX is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
(V,XX) is Element of bool the carrier of V
[#] XX is non proper Element of bool the carrier of XX
the carrier of XX is set
bool the carrier of XX is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision (1,(center_of_mass V),(subdivision (F,(center_of_mass V),Aff))) is SimplicialComplexStr of the carrier of V
subdivision ((center_of_mass V),XX) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,XX) is non void subset-closed finite-membered non with_non-empty_elements (V,XX)
subdivision ({},(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
({},k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ({},(center_of_mass k),V) is SimplicialComplexStr of the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(1,k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision (1,(center_of_mass k),V) is SimplicialComplexStr of the carrier of k
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
[#] (k,V,Aff) is non proper Element of bool the carrier of (k,V,Aff)
the carrier of (k,V,Aff) is set
bool the carrier of (k,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision (k,(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
(V,(k,V,Aff)) is Element of bool the carrier of V
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
((k + 1),V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
(V,(k,V,Aff)) is non void subset-closed finite-membered non with_non-empty_elements (V,(k,V,Aff))
(V,(k,V,Aff)) is Element of bool the carrier of V
[#] (k,V,Aff) is non proper Element of bool the carrier of (k,V,Aff)
the carrier of (k,V,Aff) is set
bool the carrier of (k,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
1 + k is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
subdivision ((1 + k),(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
subdivision (k,(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
subdivision (1,(center_of_mass V),(subdivision (k,(center_of_mass V),Aff))) is SimplicialComplexStr of the carrier of V
subdivision (1,(center_of_mass V),(k,V,Aff)) is SimplicialComplexStr of the carrier of V
(1,V,(k,V,Aff)) is non void subset-closed finite-membered non with_non-empty_elements (V,(k,V,Aff))
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree V is ext-real set
the topology of V is Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
TopStruct(# the carrier of V, the topology of V #) is strict TopStruct
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
[#] (k,V) is non proper Element of bool the carrier of (k,V)
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is ext-real set
{} + Aff is ext-real set
{} + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(degree V) + Aff is ext-real set
the topology of (k,V) is Element of bool (bool the carrier of (k,V))
bool (bool the carrier of (k,V)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is set
C is finite non dependent Element of bool the carrier of (k,V)
X is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: X is finite Element of bool the carrier of k
X /\ (dom (center_of_mass k)) is finite Element of bool (BOOL the carrier of k)
(center_of_mass k) .: (X /\ (dom (center_of_mass k))) is finite Element of bool the carrier of k
F is Element of bool the carrier of V
union X is finite Element of bool the carrier of V
A1 is finite non dependent Element of bool the carrier of V
S is set
(k,V,A1) is Element of bool the carrier of k
card A1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree V) + 1 is ext-real set
S is set
{S} is non empty trivial finite 1 -element set
{A1} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
F is set
Im ((center_of_mass k),A1) is set
(center_of_mass k) . A1 is set
{((center_of_mass k) . A1)} is non empty trivial finite 1 -element set
CA is Element of the carrier of k
{CA} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
Sum {CA} is Element of the carrier of k
1 / 1 is non empty V21() V29() ext-real positive non negative Element of REAL
(1 / 1) * (Sum {CA}) is Element of the carrier of k
{((1 / 1) * (Sum {CA}))} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
{(Sum {CA})} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
x is set
C is finite non dependent Element of bool the carrier of V
F is Element of bool the carrier of (k,V)
{C} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
X is Element of bool the carrier of V
A1 is set
X is finite finite-membered simplex-like Element of bool (bool the carrier of V)
S is set
card C is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree V) + 1 is ext-real set
(k,V,C) is Element of bool the carrier of k
A1 is set
{A1} is non empty trivial finite 1 -element set
(center_of_mass k) .: X is finite Element of bool the carrier of k
Im ((center_of_mass k),C) is set
(center_of_mass k) . C is set
{((center_of_mass k) . C)} is non empty trivial finite 1 -element set
S is Element of the carrier of k
{S} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
Sum {S} is Element of the carrier of k
1 / 1 is non empty V21() V29() ext-real positive non negative Element of REAL
(1 / 1) * (Sum {S}) is Element of the carrier of k
{((1 / 1) * (Sum {S}))} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
{(Sum {S})} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
F is Element of bool the carrier of (k,V)
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree Aff is ext-real set
the topology of Aff is Element of bool (bool the carrier of Aff)
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
TopStruct(# the carrier of Aff, the topology of Aff #) is strict TopStruct
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom (center_of_mass V) is Element of bool (BOOL the carrier of V)
bool (BOOL the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(F,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree (F,V,Aff) is ext-real set
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((F + 1),V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree ((F + 1),V,Aff) is ext-real set
subdivision ((center_of_mass V),Aff) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
degree (subdivision ((center_of_mass V),Aff)) is ext-real set
(V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
(V,(F,V,Aff)) is Element of bool the carrier of V
[#] (F,V,Aff) is non proper Element of bool the carrier of (F,V,Aff)
the carrier of (F,V,Aff) is set
bool the carrier of (F,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,(F,V,Aff)) is non void subset-closed finite-membered non with_non-empty_elements (V,(F,V,Aff))
({},V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree ({},V,Aff) is ext-real set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
F is non void subset-closed finite-membered non with_non-empty_elements SubSimplicialComplex of Aff
(V,F) is Element of bool the carrier of V
[#] F is non proper Element of bool the carrier of F
the carrier of F is set
bool the carrier of F is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,F) is non void subset-closed finite-membered non with_non-empty_elements (V,F)
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision (k,(center_of_mass V),F) is SimplicialComplexStr of the carrier of V
subdivision (k,(center_of_mass V),Aff) is SimplicialComplexStr of the carrier of V
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
card k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card (card k) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Vertices Aff is Element of bool the carrier of Aff
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
Vertices (k,V,Aff) is Element of bool the carrier of (k,V,Aff)
the carrier of (k,V,Aff) is set
bool the carrier of (k,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Skeleton_of (Aff,{}) is non void subset-closed finite-membered finite-degree non with_non-empty_elements SubSimplicialComplex of Aff
the topology of Aff is Element of bool (bool the carrier of Aff)
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
{} + 1 is non empty V21() V29() ext-real positive non negative set
K85({},1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal set
the_subsets_with_limited_card (({} + 1), the topology of Aff, the carrier of Aff) is Element of bool (bool the carrier of Aff)
Complex_of (the_subsets_with_limited_card (({} + 1), the topology of Aff, the carrier of Aff)) is strict subset-closed total SimplicialComplexStr of the carrier of Aff
subset-closed_closure_of (the_subsets_with_limited_card (({} + 1), the topology of Aff, the carrier of Aff)) is V47() subset-closed Element of bool (bool the carrier of Aff)
TopStruct(# the carrier of Aff,(subset-closed_closure_of (the_subsets_with_limited_card (({} + 1), the topology of Aff, the carrier of Aff))) #) is strict TopStruct
the topology of (Skeleton_of (Aff,{})) is Element of bool (bool the carrier of (Skeleton_of (Aff,{})))
the carrier of (Skeleton_of (Aff,{})) is set
bool the carrier of (Skeleton_of (Aff,{})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Skeleton_of (Aff,{}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,(Skeleton_of (Aff,{}))) is Element of bool the carrier of V
[#] (Skeleton_of (Aff,{})) is non proper Element of bool the carrier of (Skeleton_of (Aff,{}))
degree (Skeleton_of (Aff,{})) is V21() V29() V30() ext-real finite set
(k,V,(Skeleton_of (Aff,{}))) is non void subset-closed finite-membered non with_non-empty_elements (V, Skeleton_of (Aff,{}))
Vertices (Skeleton_of (Aff,{})) is Element of bool the carrier of (Skeleton_of (Aff,{}))
XX is set
x is Element of the carrier of Aff
C is Element of bool the carrier of Aff
{x} is non empty trivial finite 1 -element set
F is Element of bool the carrier of Aff
card F is epsilon-transitive epsilon-connected ordinal cardinal set
card 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
the_subsets_with_limited_card (1, the topology of Aff, the carrier of Aff) is finite-membered Element of bool (bool the carrier of Aff)
X is finite non dependent Element of bool the carrier of (Skeleton_of (Aff,{}))
A1 is Element of the carrier of (Skeleton_of (Aff,{}))
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
Aff is non void subset-closed finite-membered non with_non-empty_elements total SimplicialComplexStr of the carrier of V
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non empty non proper Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,Aff) is Element of bool the carrier of V
[#] (k,V,Aff) is non proper Element of bool the carrier of (k,V,Aff)
the carrier of (k,V,Aff) is set
bool the carrier of (k,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
Aff is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of V
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements total (V,Aff)
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non empty non proper Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,Aff) is Element of bool the carrier of V
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(F,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements total (V,Aff)
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((F + 1),V,Aff) is non void subset-closed finite-membered non with_non-empty_elements total (V,Aff)
[#] (F,V,Aff) is non proper Element of bool the carrier of (F,V,Aff)
the carrier of (F,V,Aff) is set
bool the carrier of (F,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,(F,V,Aff)) is Element of bool the carrier of V
(V,(F,V,Aff)) is non void subset-closed finite-membered non with_non-empty_elements (V,(F,V,Aff))
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ((center_of_mass V),(F,V,Aff)) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
({},V,Aff) is non void subset-closed finite-membered non with_non-empty_elements total (V,Aff)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
k is non empty RLSStruct
the carrier of k is non empty set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is affinely-independent Element of bool (bool the carrier of k)
Complex_of V is strict subset-closed total SimplicialComplexStr of the carrier of k
subset-closed_closure_of V is V47() subset-closed Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of V) #) is strict TopStruct
the carrier of (Complex_of V) is set
bool the carrier of (Complex_of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of (Complex_of V)
(k,(Complex_of V),Aff) is Element of bool the carrier of k
F is set
k is non empty RLSStruct
the carrier of k is non empty set
V is SimplicialComplexStr of the carrier of k
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
F is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
conv (k,V,Aff) is convex Element of bool the carrier of k
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
(conv (k,V,Aff)) /\ (conv (k,V,F)) is Element of bool the carrier of k
Aff /\ F is Element of bool the carrier of V
(k,V,(Aff /\ F)) is Element of bool the carrier of k
conv (k,V,(Aff /\ F)) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is affinely-independent Element of bool the carrier of k
{V} is non empty trivial finite 1 -element affinely-independent Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {V} is strict non void subset-closed non with_non-empty_elements total total (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {V} is non empty V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {V}) #) is strict TopStruct
the carrier of (Complex_of {V}) is set
bool the carrier of (Complex_of {V}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of (Complex_of {V})
XX is Element of bool the carrier of (Complex_of {V})
(k,(Complex_of {V}),F) is Element of bool the carrier of k
conv (k,(Complex_of {V}),F) is convex Element of bool the carrier of k
(k,(Complex_of {V}),XX) is Element of bool the carrier of k
conv (k,(Complex_of {V}),XX) is convex Element of bool the carrier of k
(conv (k,(Complex_of {V}),F)) /\ (conv (k,(Complex_of {V}),XX)) is Element of bool the carrier of k
F /\ XX is Element of bool the carrier of (Complex_of {V})
(k,(Complex_of {V}),(F /\ XX)) is Element of bool the carrier of k
conv (k,(Complex_of {V}),(F /\ XX)) is convex Element of bool the carrier of k
the topology of (Complex_of {V}) is Element of bool (bool the carrier of (Complex_of {V}))
bool (bool the carrier of (Complex_of {V})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool V)
bool V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Affin (k,(Complex_of {V}),XX) is Element of bool the carrier of k
Affin (k,(Complex_of {V}),F) is Element of bool the carrier of k
Affin V is Element of bool the carrier of k
x is set
V \ (k,(Complex_of {V}),(F /\ XX)) is Element of bool the carrier of k
V /\ (k,(Complex_of {V}),(F /\ XX)) is Element of bool the carrier of k
V \ (V \ (k,(Complex_of {V}),(F /\ XX))) is Element of bool the carrier of k
x |-- (k,(Complex_of {V}),F) is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (k,(Complex_of {V}),F)
x |-- V is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of V
x |-- (k,(Complex_of {V}),XX) is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (k,(Complex_of {V}),XX)
Carrier (x |-- (k,(Complex_of {V}),F)) is finite Element of bool the carrier of k
{ b₁ where b₁ is Element of the carrier of k : not (x |-- (k,(Complex_of {V}),F)) . b₁ = 0 } is set
Carrier (x |-- (k,(Complex_of {V}),XX)) is finite Element of bool the carrier of k
{ b₁ where b₁ is Element of the carrier of k : not (x |-- (k,(Complex_of {V}),XX)) . b₁ = 0 } is set
F is set
(x |-- V) . F is V21() V29() ext-real set
F is Element of the carrier of k
(x |-- (k,(Complex_of {V}),F)) . F is V21() V29() ext-real Element of REAL
x |-- (k,(Complex_of {V}),(F /\ XX)) is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (k,(Complex_of {V}),(F /\ XX))
(x |-- (k,(Complex_of {V}),(F /\ XX))) . F is V21() V29() ext-real Element of REAL
Affin (k,(Complex_of {V}),(F /\ XX)) is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the Element of the carrier of k is Element of the carrier of k
{ the Element of the carrier of k} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
the Element of the carrier of k is Element of the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
{ the Element of the carrier of k} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
{{ the Element of the carrier of k}} is non empty trivial finite finite-membered 1 -element affinely-independent with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {{ the Element of the carrier of k}} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {{ the Element of the carrier of k}} is non empty finite-membered V47() subset-closed non with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {{ the Element of the carrier of k}}) #) is strict TopStruct
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is (k) SimplicialComplexStr of the carrier of k
Aff is subset-closed finite-membered SubSimplicialComplex of V
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of Aff
(k,Aff,F) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of Aff is Element of bool (bool the carrier of Aff)
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool the carrier of V
(k,V,XX) is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is (k) SimplicialComplexStr of the carrier of k
Aff is subset-closed finite-membered SubSimplicialComplex of V
the topology of Aff is Element of bool (bool the carrier of Aff)
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of Aff
XX is Element of bool the carrier of Aff
(k,Aff,F) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
conv (k,Aff,F) is convex Element of bool the carrier of k
(k,Aff,XX) is Element of bool the carrier of k
conv (k,Aff,XX) is convex Element of bool the carrier of k
(conv (k,Aff,F)) /\ (conv (k,Aff,XX)) is Element of bool the carrier of k
F /\ XX is Element of bool the carrier of Aff
(k,Aff,(F /\ XX)) is Element of bool the carrier of k
conv (k,Aff,(F /\ XX)) is convex Element of bool the carrier of k
x is Element of bool the carrier of V
C is Element of bool the carrier of V
(k,V,x) is Element of bool the carrier of k
conv (k,V,x) is convex Element of bool the carrier of k
(k,V,C) is Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
(conv (k,V,x)) /\ (conv (k,V,C)) is Element of bool the carrier of k
x /\ C is Element of bool the carrier of V
(k,V,(x /\ C)) is Element of bool the carrier of k
conv (k,V,(x /\ C)) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is subset-closed SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
F is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Int (k,V,Aff) is Element of bool the carrier of k
(k,V,F) is Element of bool the carrier of k
Int (k,V,F) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
(conv (k,V,Aff)) /\ (conv (k,V,F)) is Element of bool the carrier of k
Aff /\ F is Element of bool the carrier of V
(k,V,(Aff /\ F)) is Element of bool the carrier of k
conv (k,V,(Aff /\ F)) is convex Element of bool the carrier of k
XX is set
Aff is Element of bool the carrier of V
F is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
(conv (k,V,Aff)) /\ (conv (k,V,F)) is Element of bool the carrier of k
Aff /\ F is Element of bool the carrier of V
(k,V,(Aff /\ F)) is Element of bool the carrier of k
conv (k,V,(Aff /\ F)) is convex Element of bool the carrier of k
XX is set
the_family_of V is V47() subset-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
{ (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,Aff) } is set
union { (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,Aff) } is set
x is set
C is Element of bool the carrier of k
Int C is Element of bool the carrier of k
{ (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
union { (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
F is set
X is Element of bool the carrier of k
Int X is Element of bool the carrier of k
A1 is Element of bool the carrier of V
S is Element of bool the carrier of V
(k,V,A1) is Element of bool the carrier of k
Int (k,V,A1) is Element of bool the carrier of k
(k,V,S) is Element of bool the carrier of k
Int (k,V,S) is Element of bool the carrier of k
conv C is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite non dependent Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is subset-closed finite-membered (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Int (k,V,Aff) is Element of bool the carrier of k
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
XX is set
{ (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
union { (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
x is set
C is Element of bool the carrier of k
Int C is Element of bool the carrier of k
the topology of V is Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
Int (k,V,F) is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is subset-closed finite-membered (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is Element of bool the carrier of V
(k,V,Aff) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Int (k,V,Aff) is Element of bool the carrier of k
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
the topology of V is Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is set
{ (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
union { (Int b₁) where b₁ is Element of bool the carrier of k : b₁ c= (k,V,F) } is set
x is set
C is Element of bool the carrier of k
Int C is Element of bool the carrier of k
F is Element of bool the carrier of V
(k,V,F) is Element of bool the carrier of k
Int (k,V,F) is Element of bool the carrier of k
conv (k,V,Aff) is convex Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements (k) SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool the carrier of (k,V)
(k,(k,V),XX) is Element of bool the carrier of k
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: x is finite Element of bool the carrier of k
union x is finite Element of bool the carrier of V
bool (union x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union x))
bool (union x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union x)) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,(union x)) is Element of bool the carrier of k
bool (k,V,(union x)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (k,V,(union x)))
bool (k,V,(union x)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (k,V,(union x))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
C is c=-linear finite Element of bool (bool the carrier of k)
union C is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements total (k) SimplicialComplexStr of the carrier of k
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(Aff,k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k,V)
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] k is non empty non proper Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(F,k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k,V)
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((F + 1),k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k,V)
(k,(F,k,V)) is non void subset-closed finite-membered non with_non-empty_elements (k,(F,k,V))
({},k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k,V)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements (k) (k) SimplicialComplexStr of the carrier of k
the topology of V is Element of bool (bool the carrier of V)
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass k) | the topology of V is Relation-like BOOL the carrier of k -defined the topology of V -defined BOOL the carrier of k -defined the carrier of k -valued Function-like Element of bool [:(BOOL the carrier of k), the carrier of k:]
XX is set
dom ((center_of_mass k) | the topology of V) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is set
((center_of_mass k) | the topology of V) . XX is set
((center_of_mass k) | the topology of V) . x is set
(center_of_mass k) . XX is set
(center_of_mass k) . x is set
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
(dom (center_of_mass k)) /\ the topology of V is Element of bool (bool the carrier of V)
F is finite non dependent Element of bool the carrier of V
(center_of_mass k) . F is set
(k,V,F) is affinely-independent Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
X is finite non dependent Element of bool the carrier of V
(center_of_mass k) . X is set
(k,V,X) is affinely-independent Element of bool the carrier of k
conv (k,V,X) is convex Element of bool the carrier of k
(conv (k,V,F)) /\ (conv (k,V,X)) is Element of bool the carrier of k
F /\ X is finite non dependent Element of bool the carrier of V
(k,V,(F /\ X)) is affinely-independent Element of bool the carrier of k
conv (k,V,(F /\ X)) is convex Element of bool the carrier of k
Affin (k,V,(F /\ X)) is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements (k) (k) SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
XX + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
x is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: x is finite Element of bool the carrier of k
C is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: C is finite Element of bool the carrier of k
union x is finite Element of bool the carrier of V
card (union x) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union C is finite Element of bool the carrier of V
card (union C) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is finite non dependent Element of bool the carrier of (k,V)
X is finite non dependent Element of bool the carrier of (k,V)
(k,(k,V),F) is Element of bool the carrier of k
Int (k,(k,V),F) is Element of bool the carrier of k
(k,(k,V),X) is Element of bool the carrier of k
Int (k,(k,V),X) is Element of bool the carrier of k
bool (union C) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union C))
bool (union C) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union C)) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,(union C)) is Element of bool the carrier of k
bool (k,V,(union C)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (k,V,(union C)))
bool (k,V,(union C)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (k,V,(union C))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (union x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union x))
bool (union x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union x)) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V,(union x)) is Element of bool the carrier of k
bool (k,V,(union x)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (k,V,(union x)))
bool (k,V,(union x)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (k,V,(union x))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Int ((center_of_mass k) .: x) is Element of bool the carrier of k
Int (k,V,(union x)) is Element of bool the carrier of k
Int ((center_of_mass k) .: C) is Element of bool the carrier of k
Int (k,V,(union C)) is Element of bool the carrier of k
(center_of_mass k) . (union x) is set
S is Element of the carrier of k
{S} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
conv {S} is non empty convex Element of bool the carrier of k
CA is set
F is Element of the carrier of k
{(union x)} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
x \ {(union x)} is finite Element of bool (bool the carrier of V)
C \ {(union x)} is finite Element of bool (bool the carrier of V)
the topology of V is Element of bool (bool the carrier of V)
XX is set
XXA is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
XX is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
XX is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
F |-- (k,(k,V),F) is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (k,(k,V),F)
F |-- (k,(k,V),X) is Relation-like the carrier of k -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (k,(k,V),X)
(center_of_mass k) | the topology of V is Relation-like BOOL the carrier of k -defined the topology of V -defined BOOL the carrier of k -defined BOOL the carrier of k -defined the topology of V -defined the carrier of k -valued the carrier of k -valued Function-like one-to-one Element of bool [:(BOOL the carrier of k), the carrier of k:]
union XX is finite Element of bool the carrier of V
XXA is Element of bool (bool the carrier of k)
union XXA is Element of bool the carrier of k
YA is set
YA is Element of the carrier of k
{YA} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
(union x) \ {YA} is finite Element of bool the carrier of V
conv (union XXA) is convex Element of bool the carrier of k
(k,V,((union x) \ {YA})) is Element of bool the carrier of k
conv (k,V,((union x) \ {YA})) is convex Element of bool the carrier of k
conv (k,(k,V),F) is convex Element of bool the carrier of k
(F |-- (k,(k,V),F)) . S is V21() V29() ext-real Element of REAL
Affin (k,(k,V),F) is Element of bool the carrier of k
Sum (F |-- (k,(k,V),F)) is Element of the carrier of k
Carrier (F |-- (k,(k,V),F)) is finite Element of bool the carrier of k
{ b₁ where b₁ is Element of the carrier of k : not (F |-- (k,(k,V),F)) . b₁ = 0 } is set
(k,(k,V),F) \ {S} is Element of bool the carrier of k
conv ((k,(k,V),F) \ {S}) is convex Element of bool the carrier of k
((F |-- (k,(k,V),F)) . S) * S is Element of the carrier of k
1 - ((F |-- (k,(k,V),F)) . S) is V21() V29() ext-real Element of REAL
1 / ((F |-- (k,(k,V),F)) . S) is V21() V29() ext-real Element of REAL
(1 / ((F |-- (k,(k,V),F)) . S)) * F is Element of the carrier of k
1 - (1 / ((F |-- (k,(k,V),F)) . S)) is V21() V29() ext-real Element of REAL
YA is Element of the carrier of k
(1 - ((F |-- (k,(k,V),F)) . S)) * YA is Element of the carrier of k
(((F |-- (k,(k,V),F)) . S) * S) + ((1 - ((F |-- (k,(k,V),F)) . S)) * YA) is Element of the carrier of k
(1 - (1 / ((F |-- (k,(k,V),F)) . S))) * YA is Element of the carrier of k
((1 / ((F |-- (k,(k,V),F)) . S)) * F) + ((1 - (1 / ((F |-- (k,(k,V),F)) . S))) * YA) is Element of the carrier of k
Int ((k,(k,V),F) \ {S}) is Element of bool the carrier of k
Im ((center_of_mass k),(union x)) is set
(center_of_mass k) .: {(union x)} is finite Element of bool the carrier of k
F \ {S} is finite non dependent Element of bool the carrier of (k,V)
((center_of_mass k) | the topology of V) .: x is finite Element of bool the carrier of k
(((center_of_mass k) | the topology of V) .: x) \ ((center_of_mass k) .: {(union x)}) is finite Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: {(union x)} is finite Element of bool the carrier of k
(((center_of_mass k) | the topology of V) .: x) \ (((center_of_mass k) | the topology of V) .: {(union x)}) is finite Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: (x \ {(union x)}) is finite Element of bool the carrier of k
(center_of_mass k) .: (x \ {(union x)}) is finite Element of bool the carrier of k
XX is Element of bool (bool the carrier of k)
union XX is Element of bool the carrier of k
conv (union XX) is convex Element of bool the carrier of k
card (union XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union XX is finite Element of bool the carrier of V
conv (k,(k,V),X) is convex Element of bool the carrier of k
(F |-- (k,(k,V),X)) . S is V21() V29() ext-real Element of REAL
Affin (k,(k,V),X) is Element of bool the carrier of k
Sum (F |-- (k,(k,V),X)) is Element of the carrier of k
Carrier (F |-- (k,(k,V),X)) is finite Element of bool the carrier of k
{ b₁ where b₁ is Element of the carrier of k : not (F |-- (k,(k,V),X)) . b₁ = 0 } is set
(k,(k,V),X) \ {S} is Element of bool the carrier of k
conv ((k,(k,V),X) \ {S}) is convex Element of bool the carrier of k
((F |-- (k,(k,V),X)) . S) * S is Element of the carrier of k
1 - ((F |-- (k,(k,V),X)) . S) is V21() V29() ext-real Element of REAL
1 / ((F |-- (k,(k,V),X)) . S) is V21() V29() ext-real Element of REAL
(1 / ((F |-- (k,(k,V),X)) . S)) * F is Element of the carrier of k
1 - (1 / ((F |-- (k,(k,V),X)) . S)) is V21() V29() ext-real Element of REAL
Xm1 is Element of the carrier of k
(1 - ((F |-- (k,(k,V),X)) . S)) * Xm1 is Element of the carrier of k
(((F |-- (k,(k,V),X)) . S) * S) + ((1 - ((F |-- (k,(k,V),X)) . S)) * Xm1) is Element of the carrier of k
(1 - (1 / ((F |-- (k,(k,V),X)) . S))) * Xm1 is Element of the carrier of k
((1 / ((F |-- (k,(k,V),X)) . S)) * F) + ((1 - (1 / ((F |-- (k,(k,V),X)) . S))) * Xm1) is Element of the carrier of k
Int ((k,(k,V),X) \ {S}) is Element of bool the carrier of k
conv (k,V,(union x)) is convex Element of bool the carrier of k
Xm is set
R1 is Element of bool the carrier of V
Xm is set
R1 is Element of the carrier of k
{R1} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
(union x) \ {R1} is finite Element of bool the carrier of V
(k,V,((union x) \ {R1})) is Element of bool the carrier of k
conv (k,V,((union x) \ {R1})) is convex Element of bool the carrier of k
card (union XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
X \ {S} is finite non dependent Element of bool the carrier of (k,V)
((center_of_mass k) | the topology of V) .: C is finite Element of bool the carrier of k
(((center_of_mass k) | the topology of V) .: C) \ ((center_of_mass k) .: {(union x)}) is finite Element of bool the carrier of k
(((center_of_mass k) | the topology of V) .: C) \ (((center_of_mass k) | the topology of V) .: {(union x)}) is finite Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: (C \ {(union x)}) is finite Element of bool the carrier of k
(center_of_mass k) .: (C \ {(union x)}) is finite Element of bool the carrier of k
(k,(k,V),(F \ {S})) is Element of bool the carrier of k
(k,(k,V),(X \ {S})) is Element of bool the carrier of k
(X \ {S}) \/ {S} is non empty finite set
XX is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: XX is finite Element of bool the carrier of k
x is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: x is finite Element of bool the carrier of k
union XX is finite Element of bool the carrier of V
card (union XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union x is finite Element of bool the carrier of V
card (union x) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
C is finite non dependent Element of bool the carrier of (k,V)
F is finite non dependent Element of bool the carrier of (k,V)
(k,(k,V),C) is Element of bool the carrier of k
Int (k,(k,V),C) is Element of bool the carrier of k
(k,(k,V),F) is Element of bool the carrier of k
Int (k,(k,V),F) is Element of bool the carrier of k
X is set
A1 is set
(center_of_mass k) . A1 is set
XX is Element of bool the carrier of (k,V)
x is Element of bool the carrier of (k,V)
(k,(k,V),XX) is Element of bool the carrier of k
Int (k,(k,V),XX) is Element of bool the carrier of k
(k,(k,V),x) is Element of bool the carrier of k
Int (k,(k,V),x) is Element of bool the carrier of k
C is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: C is finite Element of bool the carrier of k
F is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: F is finite Element of bool the carrier of k
union C is finite Element of bool the carrier of V
card (union C) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union F is finite Element of bool the carrier of V
card (union F) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements total (k) (k) SimplicialComplexStr of the carrier of k
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k) (k,V)
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(Aff,k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k) (k,V)
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] k is non empty non proper Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(F,k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k) (k,V)
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((F + 1),k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k) (k,V)
(k,(F,k,V)) is non void subset-closed finite-membered non with_non-empty_elements (k) (k,(F,k,V))
({},k,V) is non void subset-closed finite-membered non with_non-empty_elements total (k) (k,V)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree V is ext-real set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
degree (k,V) is ext-real set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
Aff + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
F is finite non dependent Element of bool the carrier of V
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k,V,F) is Element of bool the carrier of k
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
BOOL F is set
bool F is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool F is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool F)
bool (bool F) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool F) \ {{}} is finite finite-membered Element of bool (bool F)
(center_of_mass k) .: (BOOL F) is Element of bool the carrier of k
(center_of_mass k) | (BOOL F) is Relation-like BOOL the carrier of k -defined BOOL F -defined BOOL the carrier of k -defined the carrier of k -valued Function-like Element of bool [:(BOOL the carrier of k), the carrier of k:]
{(k,V,F)} is non empty trivial finite 1 -element Element of bool (bool the carrier of k)
Complex_of {(k,V,F)} is strict non void subset-closed non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {(k,V,F)} is non empty V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {(k,V,F)}) #) is strict TopStruct
the topology of (Complex_of {(k,V,F)}) is Element of bool (bool the carrier of (Complex_of {(k,V,F)}))
the carrier of (Complex_of {(k,V,F)}) is set
bool the carrier of (Complex_of {(k,V,F)}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {(k,V,F)})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
conv (k,V,F) is convex Element of bool the carrier of k
x is set
C is set
(center_of_mass k) . C is set
F is non empty Element of bool the carrier of k
conv F is non empty convex Element of bool the carrier of k
bool (k,V,F) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (k,V,F))
bool (k,V,F) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (k,V,F)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass k) | (bool F) is Relation-like BOOL the carrier of k -defined bool F -defined BOOL the carrier of k -defined the carrier of k -valued Function-like finite Element of bool [:(BOOL the carrier of k), the carrier of k:]
((center_of_mass k) | (bool F)) | (BOOL F) is Relation-like BOOL the carrier of k -defined BOOL F -defined BOOL the carrier of k -defined the carrier of k -valued Function-like finite Element of bool [:(BOOL the carrier of k), the carrier of k:]
XX is affinely-independent Element of bool (bool the carrier of k)
Complex_of XX is strict subset-closed total total (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of XX is V47() subset-closed Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of XX) #) is strict TopStruct
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
V is non void subset-closed finite-membered non with_non-empty_elements (k) SimplicialComplexStr of the carrier of k
(k,V) is Element of bool the carrier of k
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree V is ext-real set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
degree (k,V) is ext-real set
Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
Aff + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
the finite non dependent Simplex of Aff,V is finite non dependent Simplex of Aff,V
card the finite non dependent Simplex of Aff,V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is ext-real set
F + 1 is ext-real set
(k,V, the finite non dependent Simplex of Aff,V) is affinely-independent Element of bool the carrier of k
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
Aff is non void subset-closed finite-membered non with_non-empty_elements (V) SimplicialComplexStr of the carrier of V
(V,Aff) is Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] Aff is non proper Element of bool the carrier of Aff
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree Aff is ext-real set
(k,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree (k,V,Aff) is ext-real set
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(F,V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree (F,V,Aff) is ext-real set
F + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((F + 1),V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree ((F + 1),V,Aff) is ext-real set
[#] (F,V,Aff) is non proper Element of bool the carrier of (F,V,Aff)
the carrier of (F,V,Aff) is set
bool the carrier of (F,V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,(F,V,Aff)) is non void subset-closed finite-membered non with_non-empty_elements (V,(F,V,Aff))
(V,(F,V,Aff)) is Element of bool the carrier of V
({},V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V,Aff)
degree ({},V,Aff) is ext-real set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements (k) (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
XX is finite-membered simplex-like Element of bool (bool the carrier of V)
card XX is epsilon-transitive epsilon-connected ordinal cardinal set
(center_of_mass k) .: XX is Element of bool the carrier of k
card ((center_of_mass k) .: XX) is epsilon-transitive epsilon-connected ordinal cardinal set
[#] V is non proper Element of bool the carrier of V
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass k) | XX is Relation-like BOOL the carrier of k -defined XX -defined BOOL the carrier of k -defined the carrier of k -valued Function-like Element of bool [:(BOOL the carrier of k), the carrier of k:]
dom ((center_of_mass k) | XX) is finite-membered Element of bool XX
bool XX is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
x is set
(center_of_mass k) | the topology of V is Relation-like BOOL the carrier of k -defined the topology of V -defined BOOL the carrier of k -defined BOOL the carrier of k -defined the topology of V -defined the carrier of k -valued the carrier of k -valued Function-like one-to-one Element of bool [:(BOOL the carrier of k), the carrier of k:]
((center_of_mass k) | the topology of V) | XX is Relation-like BOOL the carrier of k -defined XX -defined BOOL the carrier of k -defined the carrier of k -valued Function-like Element of bool [:(BOOL the carrier of k), the carrier of k:]
rng ((center_of_mass k) | XX) is Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements (k) (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is Element of bool the carrier of k
[#] V is non proper Element of bool the carrier of V
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k,V)
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is finite-membered simplex-like Element of bool (bool the carrier of V)
x is finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: x is Element of bool the carrier of k
(center_of_mass k) .: XX is Element of bool the carrier of k
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
C is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: C is finite Element of bool the carrier of k
x \ {{}} is Element of bool (bool the carrier of V)
the topology of V is Element of bool (bool the carrier of V)
(center_of_mass k) | the topology of V is Relation-like BOOL the carrier of k -defined the topology of V -defined BOOL the carrier of k -defined BOOL the carrier of k -defined the topology of V -defined the carrier of k -valued the carrier of k -valued Function-like one-to-one Element of bool [:(BOOL the carrier of k), the carrier of k:]
dom ((center_of_mass k) | the topology of V) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
(dom (center_of_mass k)) /\ the topology of V is Element of bool (bool the carrier of V)
F is finite-membered simplex-like Element of bool (bool the carrier of V)
(k,V,x) is Element of bool (bool the carrier of k)
(k,V,x) \ {{}} is Element of bool (bool the carrier of k)
C /\ (dom (center_of_mass k)) is finite Element of bool (BOOL the carrier of k)
S is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: (C /\ (dom (center_of_mass k))) is finite Element of bool the carrier of k
(center_of_mass k) .: S is finite Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: S is finite Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: x is Element of bool the carrier of k
x /\ (dom (center_of_mass k)) is Element of bool (BOOL the carrier of k)
(k,V,x) /\ (bool the carrier of k) is Element of bool (bool the carrier of k)
((k,V,x) /\ (bool the carrier of k)) \ {{}} is Element of bool (bool the carrier of k)
(center_of_mass k) .: F is Element of bool the carrier of k
((center_of_mass k) | the topology of V) .: F is Element of bool the carrier of k
(k,V,XX) is Element of bool (bool the carrier of k)
((center_of_mass k) | the topology of V) .: XX is Element of bool the carrier of k
CA is set
F is set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite affinely-independent Element of bool the carrier of V
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
{Aff} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
Complex_of {Aff} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {Aff} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {Aff}) #) is strict TopStruct
(V,(Complex_of {Aff})) is non void subset-closed finite-membered non with_non-empty_elements (V) (V) (V, Complex_of {Aff})
F is finite Element of bool the carrier of V
XX is finite Element of bool (bool the carrier of V)
union XX is Element of bool the carrier of V
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) + k is V21() V29() ext-real non negative set
K85((card XX),k) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
((card XX) + k) + 1 is non empty V21() V29() ext-real positive non negative Element of REAL
k + (card XX) is V21() V29() ext-real non negative set
K85(k,(card XX)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(center_of_mass V) .: XX is finite Element of bool the carrier of V
the topology of (Complex_of {Aff}) is Element of bool (bool the carrier of (Complex_of {Aff}))
the carrier of (Complex_of {Aff}) is set
bool the carrier of (Complex_of {Aff}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass V) | the topology of (Complex_of {Aff}) is Relation-like BOOL the carrier of V -defined the topology of (Complex_of {Aff}) -defined BOOL the carrier of V -defined BOOL the carrier of V -defined the topology of (Complex_of {Aff}) -defined the carrier of V -valued the carrier of V -valued Function-like one-to-one Element of bool [:(BOOL the carrier of V), the carrier of V:]
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool Aff)
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool Aff) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree (Complex_of {Aff}) is V21() V29() V30() ext-real finite set
X is ext-real set
X - 1 is ext-real set
X + (- 1) is ext-real set
(card Aff) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
(V,(Complex_of {Aff})) is Element of bool the carrier of V
[#] (Complex_of {Aff}) is non proper Element of bool the carrier of (Complex_of {Aff})
subdivision ((center_of_mass V),(Complex_of {Aff})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of V
(card Aff) - 1 is V21() V29() V30() ext-real finite Element of REAL
degree (V,(Complex_of {Aff})) is ext-real set
F is set
((center_of_mass V) | the topology of (Complex_of {Aff})) .: XX is finite Element of bool the carrier of V
dom (center_of_mass V) is Element of bool (BOOL the carrier of V)
bool (BOOL the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom ((center_of_mass V) | the topology of (Complex_of {Aff})) is Element of bool (BOOL the carrier of V)
(dom (center_of_mass V)) /\ the topology of (Complex_of {Aff}) is Element of bool (bool the carrier of (Complex_of {Aff}))
F is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of (Complex_of {Aff}))
(center_of_mass V) .: F is finite Element of bool the carrier of V
F /\ (dom (center_of_mass V)) is finite Element of bool (BOOL the carrier of V)
XX is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of (Complex_of {Aff}))
(center_of_mass V) .: XX is finite Element of bool the carrier of V
XX \ XX is finite Element of bool (bool the carrier of (Complex_of {Aff}))
((center_of_mass V) | the topology of (Complex_of {Aff})) .: XX is finite Element of bool the carrier of V
XX is Element of bool (bool the carrier of V)
XX \/ XX is Element of bool (bool the carrier of V)
XX \/ XX is finite set
(center_of_mass V) .: (F /\ (dom (center_of_mass V))) is finite Element of bool the carrier of V
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: XX is Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: XX) is Element of bool the carrier of V
XX is finite Element of bool (bool the carrier of V)
CA is ext-real set
CA + 1 is ext-real set
union (bool Aff) is finite Element of bool Aff
card (XX \ XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) + (card (XX \ XX)) is V21() V29() ext-real non negative set
K85((card XX),(card (XX \ XX))) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
XXA is finite Element of bool (bool the carrier of V)
XXA \/ XX is finite Element of bool (bool the carrier of V)
card XXA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union XXA is Element of bool the carrier of V
(center_of_mass V) .: XXA is finite Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: XXA) is finite Element of bool the carrier of V
F is finite Element of bool (bool the carrier of V)
F \/ XX is finite Element of bool (bool the carrier of V)
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union F is Element of bool the carrier of V
(center_of_mass V) .: F is finite Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: F) is finite Element of bool the carrier of V
XX is Element of bool (bool the carrier of (Complex_of {Aff}))
card XX is epsilon-transitive epsilon-connected ordinal cardinal set
XX is finite Element of bool (bool the carrier of V)
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) + (card XX) is V21() V29() ext-real non negative set
K85((card XX),(card XX)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
union (F \/ XX) is Element of bool the carrier of V
(union F) \/ (union XX) is Element of bool the carrier of V
bool (union (F \/ XX)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union (F \/ XX)))
bool (union (F \/ XX)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union (F \/ XX))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool the carrier of (Complex_of {Aff})
[#] (V,(Complex_of {Aff})) is non proper Element of bool the carrier of (V,(Complex_of {Aff}))
the carrier of (V,(Complex_of {Aff})) is set
bool the carrier of (V,(Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass V) .: XX is Element of bool the carrier of V
card ((center_of_mass V) .: XX) is epsilon-transitive epsilon-connected ordinal cardinal set
XX is finite non dependent Element of bool the carrier of (V,(Complex_of {Aff}))
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is finite affinely-independent Element of bool the carrier of k
{V} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {V} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {V} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {V}) #) is strict TopStruct
(k,(Complex_of {V})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {V})
Aff is c=-linear finite finite-membered Element of bool (bool the carrier of k)
union Aff is finite Element of bool the carrier of k
(center_of_mass k) .: Aff is finite Element of bool the carrier of k
card (union Aff) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card (union Aff)) - 1 is V21() V29() V30() ext-real finite Element of REAL
the carrier of (Complex_of {V}) is set
bool the carrier of (Complex_of {V}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {V})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (Complex_of {V}) is Element of bool (bool the carrier of (Complex_of {V}))
bool V is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool V)
bool V is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool V) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
F - 1 is V21() V29() V30() ext-real finite Element of REAL
(card V) - 1 is V21() V29() V30() ext-real finite Element of REAL
bool (union Aff) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union Aff))
bool (union Aff) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union Aff)) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is c=-linear finite Element of bool (bool the carrier of (Complex_of {V}))
X is Element of bool the carrier of (Complex_of {V})
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card ((center_of_mass k) .: Aff) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
degree (Complex_of {V}) is V21() V29() V30() ext-real finite set
A1 is ext-real set
A1 - 1 is ext-real set
A1 + (- 1) is ext-real set
(card V) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
(k,(Complex_of {V})) is Element of bool the carrier of k
[#] (Complex_of {V}) is non proper Element of bool the carrier of (Complex_of {V})
subdivision ((center_of_mass k),(Complex_of {V})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
[#] (k,(Complex_of {V})) is non proper Element of bool the carrier of (k,(Complex_of {V}))
the carrier of (k,(Complex_of {V})) is set
bool the carrier of (k,(Complex_of {V})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree (k,(Complex_of {V})) is ext-real set
XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card Aff) - {} is V21() V29() ext-real non negative set
- {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
K87({}) is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
(card Aff) + (- {}) is V21() V29() ext-real non negative set
K85((card Aff),(- {})) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
K89((card Aff),{}) is V21() V29() V30() ext-real non negative finite set
(card Aff) - XX is V21() V29() ext-real set
- XX is V21() V29() ext-real non positive set
K87(XX) is V21() V29() V30() ext-real non positive finite set
(card Aff) + (- XX) is V21() V29() ext-real set
K85((card Aff),(- XX)) is V21() V29() ext-real set
K89((card Aff),XX) is V21() V29() V30() ext-real finite set
CA is finite non dependent Simplex of (card (union Aff)) - 1,(k,(Complex_of {V}))
card CA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is ext-real set
F + 1 is ext-real set
((card (union Aff)) - 1) + 1 is V21() V29() V30() ext-real finite Element of REAL
XX is finite finite-membered Element of bool (bool (union Aff))
XX is finite finite-membered Element of bool (bool (union Aff))
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
XXA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card Aff) - XXA is V21() V29() ext-real set
- XXA is V21() V29() ext-real non positive set
K87(XXA) is V21() V29() V30() ext-real non positive finite set
(card Aff) + (- XXA) is V21() V29() ext-real set
K85((card Aff),(- XXA)) is V21() V29() ext-real set
K89((card Aff),XXA) is V21() V29() V30() ext-real finite set
XXA + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(card Aff) - (XXA + 1) is V21() V29() ext-real set
- (XXA + 1) is non empty V21() V29() ext-real non positive negative set
K87((XXA + 1)) is non empty V21() V29() V30() ext-real non positive negative finite set
(card Aff) + (- (XXA + 1)) is V21() V29() ext-real set
K85((card Aff),(- (XXA + 1))) is V21() V29() ext-real set
K89((card Aff),(XXA + 1)) is V21() V29() V30() ext-real finite set
m1 is finite set
card m1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
m1 is finite set
card m1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(XXA + 1) - XXA is V21() V29() V30() ext-real finite Element of REAL
M is set
{M} is non empty trivial finite 1 -element set
m1 \ {M} is finite Element of bool m1
bool m1 is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(card m1) - 1 is V21() V29() V30() ext-real finite Element of REAL
card (m1 \ {M}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
cA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
cA + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(card Aff) - ((card Aff) - XX) is V21() V29() ext-real set
- ((card Aff) - XX) is V21() V29() ext-real set
K87(((card Aff) - XX)) is V21() V29() ext-real set
(card Aff) + (- ((card Aff) - XX)) is V21() V29() ext-real set
K85((card Aff),(- ((card Aff) - XX))) is V21() V29() ext-real set
K89((card Aff),((card Aff) - XX)) is V21() V29() ext-real set
XXA is finite set
card XXA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
CA is ext-real set
CA + 1 is ext-real set
S is Element of bool the carrier of (k,(Complex_of {V}))
card S is epsilon-transitive epsilon-connected ordinal cardinal set
F is set
card F is epsilon-transitive epsilon-connected ordinal cardinal set
Seg (card (union Aff)) is Element of bool NAT
XX is set
XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
[:(Seg (card (union Aff))),Aff:] is Relation-like set
bool [:(Seg (card (union Aff))),Aff:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Relation-like Seg (card (union Aff)) -defined Aff -valued Function-like quasi_total Element of bool [:(Seg (card (union Aff))),Aff:]
XX is set
dom XX is Element of bool (Seg (card (union Aff)))
bool (Seg (card (union Aff))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is set
XX . XX is set
XX . XX is set
card (XX . XX) is epsilon-transitive epsilon-connected ordinal cardinal set
rng XX is c=-linear finite finite-membered Element of bool Aff
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card (Seg (card (union Aff))) is epsilon-transitive epsilon-connected ordinal cardinal set
S is Element of bool the carrier of (k,(Complex_of {V}))
card S is epsilon-transitive epsilon-connected ordinal cardinal set
CA is ext-real set
CA + 1 is ext-real set
((card (union Aff)) - 1) + 1 is V21() V29() V30() ext-real finite Element of REAL
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is finite finite-membered Element of bool (bool the carrier of k)
card V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union V is finite Element of bool the carrier of k
card (union V) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass k) .: V is finite Element of bool the carrier of k
XX is non empty finite Element of bool the carrier of k
(union V) \/ XX is non empty finite Element of bool the carrier of k
{((union V) \/ XX)} is non empty trivial finite finite-membered 1 -element with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
(center_of_mass k) .: {((union V) \/ XX)} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {((union V) \/ XX)}) is finite Element of bool the carrier of k
Complex_of {((union V) \/ XX)} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {((union V) \/ XX)} is non empty finite-membered V47() subset-closed non with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {((union V) \/ XX)}) #) is strict TopStruct
(k,(Complex_of {((union V) \/ XX)})) is non void subset-closed finite-membered non with_non-empty_elements (k, Complex_of {((union V) \/ XX)})
x is finite affinely-independent Element of bool the carrier of k
{x} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {x} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {x} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {x}) #) is strict TopStruct
the carrier of (Complex_of {x}) is set
bool the carrier of (Complex_of {x}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {x})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V \/ {x} is non empty finite finite-membered Element of bool (bool the carrier of k)
the topology of (Complex_of {x}) is Element of bool (bool the carrier of (Complex_of {x}))
bool x is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool x)
bool x is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Element of bool (bool the carrier of (Complex_of {x}))
X is set
X is Element of bool the carrier of (Complex_of {x})
(k,(Complex_of {x})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {x})
(k,(Complex_of {x})) is Element of bool the carrier of k
[#] (Complex_of {x}) is non proper Element of bool the carrier of (Complex_of {x})
subdivision ((center_of_mass k),(Complex_of {x})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
[#] (k,(Complex_of {x})) is non proper Element of bool the carrier of (k,(Complex_of {x}))
the carrier of (k,(Complex_of {x})) is set
bool the carrier of (k,(Complex_of {x})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass k) .: F is Element of bool the carrier of k
S is set
CA is set
A1 is Element of bool the carrier of (k,(Complex_of {x}))
card x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
degree (Complex_of {x}) is V21() V29() V30() ext-real finite set
CA is ext-real set
CA - 1 is ext-real set
CA + (- 1) is ext-real set
(card x) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
card F is epsilon-transitive epsilon-connected ordinal cardinal set
(card V) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(card (union V)) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(card x) - 1 is V21() V29() V30() ext-real finite Element of REAL
S is finite non dependent Element of bool the carrier of (k,(Complex_of {x}))
card S is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is ext-real set
F + 1 is ext-real set
degree (k,(Complex_of {x})) is ext-real set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is finite Element of bool (bool the carrier of k)
card V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union V is Element of bool the carrier of k
card (union V) is epsilon-transitive epsilon-connected ordinal cardinal set
(center_of_mass k) .: V is finite Element of bool the carrier of k
XX is finite Element of bool the carrier of k
(union V) \/ XX is Element of bool the carrier of k
x is finite Element of bool the carrier of k
{((union V) \/ XX)} is non empty trivial finite 1 -element Element of bool (bool the carrier of k)
(center_of_mass k) .: {((union V) \/ XX)} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {((union V) \/ XX)}) is finite Element of bool the carrier of k
{x} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
Complex_of {x} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {x} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {x}) #) is strict TopStruct
(k,(Complex_of {x})) is non void subset-closed finite-membered non with_non-empty_elements (k, Complex_of {x})
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
C is finite Element of bool the carrier of k
(center_of_mass k) . C is set
{((center_of_mass k) . C)} is non empty trivial finite 1 -element set
Im ((center_of_mass k),C) is set
{C} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
(center_of_mass k) .: {C} is finite Element of bool the carrier of k
Complex_of {C} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {C} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {C}) #) is strict TopStruct
A1 is set
subdivision ((center_of_mass k),(Complex_of {C})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
subdivision ((center_of_mass k),(Complex_of {x})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,(Complex_of {C})) is Element of bool the carrier of k
[#] (Complex_of {C}) is non proper Element of bool the carrier of (Complex_of {C})
the carrier of (Complex_of {C}) is set
bool the carrier of (Complex_of {C}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,(Complex_of {C})) is non void subset-closed finite-membered non with_non-empty_elements (k, Complex_of {C})
(k,(Complex_of {x})) is Element of bool the carrier of k
[#] (Complex_of {x}) is non proper Element of bool the carrier of (Complex_of {x})
the carrier of (Complex_of {x}) is set
bool the carrier of (Complex_of {x}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
A1 is set
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {C}) is finite Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is c=-linear finite finite-membered Element of bool (bool the carrier of k)
card V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union V is finite Element of bool the carrier of k
card (union V) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass k) .: V is finite Element of bool the carrier of k
XX is Element of the carrier of k
{XX} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
(union V) \/ {XX} is non empty finite Element of bool the carrier of k
{((union V) \/ {XX})} is non empty trivial finite finite-membered 1 -element with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
Complex_of {((union V) \/ {XX})} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {((union V) \/ {XX})} is non empty finite-membered V47() subset-closed non with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {((union V) \/ {XX})}) #) is strict TopStruct
(k,(Complex_of {((union V) \/ {XX})})) is non void subset-closed finite-membered non with_non-empty_elements (k, Complex_of {((union V) \/ {XX})})
{ b₁ where b₁ is finite non dependent Simplex of card V,(k,(Complex_of {((union V) \/ {XX})})) : (center_of_mass k) .: V c= b₁ } is set
(center_of_mass k) .: {((union V) \/ {XX})} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {((union V) \/ {XX})}) is finite Element of bool the carrier of k
{(((center_of_mass k) .: V) \/ ((center_of_mass k) .: {((union V) \/ {XX})}))} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
x is finite affinely-independent Element of bool the carrier of k
{x} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {x} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {x} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {x}) #) is strict TopStruct
(k,(Complex_of {x})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {x})
S is set
F is finite non dependent Simplex of card V,(k,(Complex_of {x}))
CA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card V) + CA is V21() V29() ext-real non negative set
K85((card V),CA) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
((card V) + CA) + 1 is non empty V21() V29() ext-real positive non negative Element of REAL
card x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
CA + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(k,(k,(Complex_of {x})),F) is affinely-independent Element of bool the carrier of k
XX is finite Element of bool (bool the carrier of k)
XX \/ V is finite Element of bool (bool the carrier of k)
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union XX is Element of bool the carrier of k
(center_of_mass k) .: XX is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: XX) is finite Element of bool the carrier of k
union (XX \/ V) is Element of bool the carrier of k
(union XX) \/ (union V) is Element of bool the carrier of k
XX is set
XX is finite finite-membered Element of bool (bool the carrier of k)
union XX is finite Element of bool the carrier of k
card (union XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card V) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
XX is set
{XX} is non empty trivial finite 1 -element set
S is set
(center_of_mass k) .: {x} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {x}) is finite Element of bool the carrier of k
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is c=-linear finite finite-membered Element of bool (bool the carrier of k)
card V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card V) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
union V is finite Element of bool the carrier of k
card (union V) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
{(union V)} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
Complex_of {(union V)} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of k
subset-closed_closure_of {(union V)} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {(union V)}) #) is strict TopStruct
(k,(Complex_of {(union V)})) is non void subset-closed finite-membered non with_non-empty_elements (k, Complex_of {(union V)})
(center_of_mass k) .: V is finite Element of bool the carrier of k
{ b₁ where b₁ is finite non dependent Simplex of card V,(k,(Complex_of {(union V)})) : (center_of_mass k) .: V c= b₁ } is set
card { b₁ where b₁ is finite non dependent Simplex of card V,(k,(Complex_of {(union V)})) : (center_of_mass k) .: V c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
F is finite affinely-independent Element of bool the carrier of k
bool F is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool F) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is finite finite-membered Element of bool (bool F)
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
x is finite finite-membered Element of bool (bool F)
card x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
x \ XX is finite finite-membered Element of bool (bool F)
card (x \ XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
((card V) + 1) - (card V) is V21() V29() V30() ext-real finite Element of REAL
C is set
{C} is non empty trivial finite 1 -element set
{F} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {F} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {F} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {F}) #) is strict TopStruct
the topology of (Complex_of {F}) is Element of bool (bool the carrier of (Complex_of {F}))
the carrier of (Complex_of {F}) is set
bool the carrier of (Complex_of {F}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {F})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool F is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool F)
(k,(Complex_of {F})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {F})
(card F) - 1 is V21() V29() V30() ext-real finite Element of REAL
(k,(Complex_of {F})) is Element of bool the carrier of k
[#] (Complex_of {F}) is non proper Element of bool the carrier of (Complex_of {F})
subdivision ((center_of_mass k),(Complex_of {F})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
[#] (k,(Complex_of {F})) is non proper Element of bool the carrier of (k,(Complex_of {F}))
the carrier of (k,(Complex_of {F})) is set
bool the carrier of (k,(Complex_of {F})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
S is Element of bool (bool the carrier of (Complex_of {F}))
(center_of_mass k) .: S is Element of bool the carrier of k
XX is Element of bool the carrier of (Complex_of {F})
XX is Element of bool the carrier of (k,(Complex_of {F}))
card XX is epsilon-transitive epsilon-connected ordinal cardinal set
F is ext-real set
F + 1 is ext-real set
degree (Complex_of {F}) is V21() V29() V30() ext-real finite set
F is ext-real set
F - 1 is ext-real set
F + (- 1) is ext-real set
(card F) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
degree (k,(Complex_of {F})) is ext-real set
XX is finite Element of bool the carrier of k
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) - 1 is V21() V29() V30() ext-real finite Element of REAL
XXA is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
card XXA is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
card XXA is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of omega
m1 is set
M is set
{M} is non empty trivial finite 1 -element set
XXA is set
{XXA} is non empty trivial finite 1 -element set
XX \ {XXA} is finite Element of bool the carrier of k
m1 is finite Element of bool the carrier of k
card m1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
m1 \/ {XXA} is non empty finite set
XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
XX + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
card m1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
M is set
XXA is set
card XXA is epsilon-transitive epsilon-connected ordinal cardinal set
XXA is set
card XXA is epsilon-transitive epsilon-connected ordinal cardinal set
union x is finite Element of bool F
union S is Element of bool the carrier of (Complex_of {F})
(card XX) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
m1 is set
card m1 is epsilon-transitive epsilon-connected ordinal cardinal set
cA is finite Element of bool the carrier of k
M is finite Element of bool the carrier of k
(card XX) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
S \ XX is Element of bool (bool the carrier of (Complex_of {F}))
cA \ M is finite Element of bool the carrier of k
card (cA \ M) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card cA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card M is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card cA) - (card M) is V21() V29() ext-real set
- (card M) is V21() V29() ext-real non positive set
K87((card M)) is V21() V29() V30() ext-real non positive finite set
(card cA) + (- (card M)) is V21() V29() ext-real set
K85((card cA),(- (card M))) is V21() V29() ext-real set
K89((card cA),(card M)) is V21() V29() V30() ext-real finite set
YA is set
YA is set
{YA,YA} is non empty finite set
YA is Element of the carrier of k
{YA} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
M \/ {YA} is non empty finite Element of bool the carrier of k
Xm1 is Element of the carrier of k
{Xm1} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of k
M \/ {Xm1} is non empty finite Element of bool the carrier of k
{(M \/ {YA})} is non empty trivial finite finite-membered 1 -element with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
V \/ {(M \/ {YA})} is non empty finite finite-membered Element of bool (bool the carrier of k)
{(M \/ {Xm1})} is non empty trivial finite finite-membered 1 -element with_non-empty_elements non empty-membered Element of bool (bool the carrier of k)
V \/ {(M \/ {Xm1})} is non empty finite finite-membered Element of bool (bool the carrier of k)
DY is Element of bool (bool the carrier of (Complex_of {F}))
(center_of_mass k) .: DY is Element of bool the carrier of k
RDY is Element of bool (bool the carrier of (Complex_of {F}))
(center_of_mass k) .: RDY is Element of bool the carrier of k
R is Element of bool the carrier of (k,(Complex_of {F}))
(center_of_mass k) .: {(M \/ {YA})} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {(M \/ {YA})}) is finite Element of bool the carrier of k
card (M \/ {YA}) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
(card M) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
card (M \/ {Xm1}) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
RDX is set
RDX is Element of bool the carrier of (Complex_of {F})
DX is Element of bool the carrier of (k,(Complex_of {F}))
card DX is epsilon-transitive epsilon-connected ordinal cardinal set
card RDY is epsilon-transitive epsilon-connected ordinal cardinal set
RDX is set
(center_of_mass k) | the topology of (Complex_of {F}) is Relation-like BOOL the carrier of k -defined the topology of (Complex_of {F}) -defined BOOL the carrier of k -defined BOOL the carrier of k -defined the topology of (Complex_of {F}) -defined the carrier of k -valued the carrier of k -valued Function-like one-to-one Element of bool [:(BOOL the carrier of k), the carrier of k:]
((center_of_mass k) | the topology of (Complex_of {F})) .: DY is Element of bool the carrier of k
RDX is Element of bool the carrier of (Complex_of {F})
card R is epsilon-transitive epsilon-connected ordinal cardinal set
card DY is epsilon-transitive epsilon-connected ordinal cardinal set
RDX is set
DX is finite set
card DX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
RDX is set
DX is set
(center_of_mass k) .: {(M \/ {Xm1})} is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: {(M \/ {Xm1})}) is finite Element of bool the carrier of k
{R,DX} is non empty finite Element of bool (bool the carrier of (k,(Complex_of {F})))
bool (bool the carrier of (k,(Complex_of {F}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
DX is set
FA is finite non dependent Simplex of card V,(k,(Complex_of {F}))
DX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card V) + DX is V21() V29() ext-real non negative set
K85((card V),DX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
((card V) + DX) + 1 is non empty V21() V29() ext-real positive non negative Element of REAL
DX + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(k,(k,(Complex_of {F})),FA) is affinely-independent Element of bool the carrier of k
FA is finite Element of bool (bool the carrier of k)
FA \/ V is finite Element of bool (bool the carrier of k)
card FA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union FA is Element of bool the carrier of k
(center_of_mass k) .: FA is finite Element of bool the carrier of k
((center_of_mass k) .: V) \/ ((center_of_mass k) .: FA) is finite Element of bool the carrier of k
XXa is set
{XXa} is non empty trivial finite 1 -element set
n is finite Element of bool F
card n is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
cnc is set
card cnc is epsilon-transitive epsilon-connected ordinal cardinal set
{XX} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
V \/ FA is finite Element of bool (bool the carrier of k)
card n is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
cnc is set
card cnc is epsilon-transitive epsilon-connected ordinal cardinal set
{XX} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
V \/ FA is finite Element of bool (bool the carrier of k)
n \/ M is finite set
V \/ FA is finite Element of bool (bool the carrier of k)
n \ M is finite Element of bool F
M \/ cA is finite Element of bool the carrier of k
(cA \ M) \/ M is finite Element of bool the carrier of k
(n \ M) \/ M is finite set
((center_of_mass k) | the topology of (Complex_of {F})) .: RDY is Element of bool the carrier of k
R \ DX is Element of bool the carrier of (k,(Complex_of {F}))
DY \ RDY is Element of bool (bool the carrier of (Complex_of {F}))
((center_of_mass k) | the topology of (Complex_of {F})) .: (DY \ RDY) is Element of bool the carrier of k
dom ((center_of_mass k) | the topology of (Complex_of {F})) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
DX \ R is Element of bool the carrier of (k,(Complex_of {F}))
RDY \ DY is Element of bool (bool the carrier of (Complex_of {F}))
((center_of_mass k) | the topology of (Complex_of {F})) .: (RDY \ DY) is Element of bool the carrier of k
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
(dom (center_of_mass k)) /\ the topology of (Complex_of {F}) is Element of bool (bool the carrier of (Complex_of {F}))
DX is set
DX is set
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements total (k) (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k,V)
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite affinely-independent Element of bool the carrier of k
{Aff} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {Aff} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {Aff} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {Aff}) #) is strict TopStruct
(k,(Complex_of {Aff})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {Aff})
the carrier of (k,(Complex_of {Aff})) is set
bool the carrier of (k,(Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
conv Aff is convex Element of bool the carrier of k
F is Element of bool the carrier of k
conv F is convex Element of bool the carrier of k
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (Complex_of {Aff}) is Element of bool (bool the carrier of (Complex_of {Aff}))
the carrier of (Complex_of {Aff}) is set
bool the carrier of (Complex_of {Aff}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool Aff)
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool Aff) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] V is non proper Element of bool the carrier of V
(k,V) is Element of bool the carrier of k
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
C is finite non dependent Element of bool the carrier of V
(k,V,C) is affinely-independent Element of bool the carrier of k
the topology of V is Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,(Complex_of {Aff})) is Element of bool the carrier of k
[#] (Complex_of {Aff}) is non proper Element of bool the carrier of (Complex_of {Aff})
[#] k is non empty non proper Element of bool the carrier of k
subdivision ((center_of_mass k),(Complex_of {Aff})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
the topology of (k,(Complex_of {Aff})) is Element of bool (bool the carrier of (k,(Complex_of {Aff})))
bool (bool the carrier of (k,(Complex_of {Aff}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of (k,V) is Element of bool (bool the carrier of (k,V))
bool (bool the carrier of (k,V)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is finite non dependent Element of bool the carrier of (k,(Complex_of {Aff}))
(k,(k,(Complex_of {Aff}))) is Element of bool the carrier of k
(k,(k,(Complex_of {Aff})),F) is affinely-independent Element of bool the carrier of k
conv (k,(k,(Complex_of {Aff})),F) is convex Element of bool the carrier of k
X is finite non dependent Element of bool the carrier of (k,V)
(k,(k,V),X) is affinely-independent Element of bool the carrier of k
conv (k,(k,V),X) is convex Element of bool the carrier of k
X is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of V)
(center_of_mass k) .: X is finite Element of bool the carrier of k
[#] (subdivision ((center_of_mass k),(Complex_of {Aff}))) is non proper Element of bool the carrier of (subdivision ((center_of_mass k),(Complex_of {Aff})))
the carrier of (subdivision ((center_of_mass k),(Complex_of {Aff}))) is set
bool the carrier of (subdivision ((center_of_mass k),(Complex_of {Aff}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
A1 is c=-linear finite Element of bool (bool the carrier of (Complex_of {Aff}))
(center_of_mass k) .: A1 is finite Element of bool the carrier of k
dom (center_of_mass k) is Element of bool (BOOL the carrier of k)
bool (BOOL the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
CA is Element of bool the carrier of (Complex_of {Aff})
F is finite non dependent Element of bool the carrier of V
F is finite non dependent Element of bool the carrier of V
(center_of_mass k) . F is set
F is finite non dependent Element of bool the carrier of (k,V)
(k,V,F) is affinely-independent Element of bool the carrier of k
Int (k,V,F) is Element of bool the carrier of k
(k,(k,V),F) is affinely-independent Element of bool the carrier of k
conv (k,(k,V),F) is convex Element of bool the carrier of k
conv (k,V,C) is convex Element of bool the carrier of k
F is finite non dependent Element of bool the carrier of V
S is Element of bool the carrier of (k,(Complex_of {Aff}))
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is non void subset-closed finite-membered non with_non-empty_elements total (V) (V) SimplicialComplexStr of the carrier of V
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree Aff is ext-real set
(V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V) (V) (V,Aff)
F is finite Element of bool the carrier of V
XX is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) + k is V21() V29() ext-real non negative set
K85((card XX),k) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k + (card XX) is V21() V29() ext-real non negative set
K85(k,(card XX)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(center_of_mass V) .: XX is finite Element of bool the carrier of V
F is ext-real set
F + 1 is ext-real set
(F + 1) - 1 is ext-real set
(F + 1) + (- 1) is ext-real set
[#] Aff is non proper Element of bool the carrier of Aff
(V,Aff) is Element of bool the carrier of V
subdivision ((center_of_mass V),Aff) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
degree (V,Aff) is ext-real set
X is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
(center_of_mass V) .: X is finite Element of bool the carrier of V
union X is finite Element of bool the carrier of Aff
(V,Aff,(union X)) is Element of bool the carrier of V
A1 is finite affinely-independent Element of bool the carrier of V
(V,Aff,X) is Element of bool (bool the carrier of V)
union (V,Aff,X) is Element of bool the carrier of V
conv F is convex Element of bool the carrier of V
conv A1 is convex Element of bool the carrier of V
{A1} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
Complex_of {A1} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {A1} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {A1}) #) is strict TopStruct
(V,(Complex_of {A1})) is non void subset-closed finite-membered non with_non-empty_elements (V) (V) (V, Complex_of {A1})
the carrier of (V,(Complex_of {A1})) is set
bool the carrier of (V,(Complex_of {A1})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
S is finite non dependent Element of bool the carrier of (V,(Complex_of {A1}))
card A1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(V,(Complex_of {A1})) is Element of bool the carrier of V
[#] (Complex_of {A1}) is non proper Element of bool the carrier of (Complex_of {A1})
the carrier of (Complex_of {A1}) is set
bool the carrier of (Complex_of {A1}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
degree (Complex_of {A1}) is V21() V29() V30() ext-real finite set
degree (V,(Complex_of {A1})) is ext-real set
F is ext-real set
F - 1 is ext-real set
F + (- 1) is ext-real set
card S is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree (V,(Complex_of {A1}))) + 1 is ext-real set
((card XX) + k) + 1 is non empty V21() V29() ext-real positive non negative Element of REAL
the_family_of Aff is V47() subset-closed Element of bool (bool the carrier of Aff)
the topology of Aff is Element of bool (bool the carrier of Aff)
bool A1 is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool A1)
bool A1 is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool A1) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,Aff,XX) is Element of bool (bool the carrier of V)
union (V,Aff,XX) is Element of bool the carrier of V
XX is finite Element of bool (bool the carrier of V)
XX \/ XX is finite set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union XX is Element of bool the carrier of V
(center_of_mass V) .: XX is finite Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: XX) is finite Element of bool the carrier of V
XX is Element of bool (bool the carrier of Aff)
union XX is Element of bool the carrier of Aff
bool (union XX) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (union XX))
bool (union XX) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (union XX)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
XXA is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
XXA \/ XX is finite finite-membered Element of bool (bool the carrier of Aff)
card XXA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: XXA is finite Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: XXA) is finite Element of bool the carrier of V
X is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
X \/ XX is finite finite-membered Element of bool (bool the carrier of Aff)
card X is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: X is finite Element of bool the carrier of V
((center_of_mass V) .: XX) \/ ((center_of_mass V) .: X) is finite Element of bool the carrier of V
XX \/ X is finite finite-membered Element of bool (bool the carrier of Aff)
[#] (V,Aff) is non proper Element of bool the carrier of (V,Aff)
the carrier of (V,Aff) is set
bool the carrier of (V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass V) .: (XX \/ X) is finite Element of bool the carrier of V
S is Element of bool the carrier of (V,Aff)
card (XX \/ X) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card XX) + (card X) is V21() V29() ext-real non negative set
K85((card XX),(card X)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
CA is finite non dependent Element of bool the carrier of (V,Aff)
card CA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements total (k) (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k,V)
Aff is finite finite-membered simplex-like Element of bool (bool the carrier of V)
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union Aff is finite Element of bool the carrier of V
card (union Aff) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card Aff) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(center_of_mass k) .: Aff is finite Element of bool the carrier of k
F is finite non dependent Element of bool the carrier of V
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k,V,F) is affinely-independent Element of bool the carrier of k
conv (k,V,F) is convex Element of bool the carrier of k
{ b₁ where b₁ is finite non dependent Simplex of card Aff,(k,V) : ( (center_of_mass k) .: Aff c= b₁ & conv (k,(k,V),b₁) c= conv (k,V,F) ) } is set
{F} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
(center_of_mass k) .: {F} is finite Element of bool the carrier of k
((center_of_mass k) .: Aff) \/ ((center_of_mass k) .: {F}) is finite Element of bool the carrier of k
{(((center_of_mass k) .: Aff) \/ ((center_of_mass k) .: {F}))} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
F \ (union Aff) is finite non dependent Element of bool the carrier of V
card (F \ (union Aff)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
((card Aff) + 1) - (card Aff) is V21() V29() V30() ext-real finite Element of REAL
x is set
{x} is non empty trivial finite 1 -element set
C is finite affinely-independent Element of bool the carrier of k
{C} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {C} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {C} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {C}) #) is strict TopStruct
degree (Complex_of {C}) is V21() V29() V30() ext-real finite set
X is ext-real set
X - 1 is ext-real set
X + (- 1) is ext-real set
(card F) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
(k,V,Aff) is Element of bool (bool the carrier of k)
(k,(Complex_of {C})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {C})
{ b₁ where b₁ is finite non dependent Simplex of card Aff,(k,(Complex_of {C})) : (center_of_mass k) .: Aff c= b₁ } is set
[#] V is non proper Element of bool the carrier of V
(k,V) is Element of bool the carrier of k
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
the topology of (Complex_of {C}) is Element of bool (bool the carrier of (Complex_of {C}))
the carrier of (Complex_of {C}) is set
bool the carrier of (Complex_of {C}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {C})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
(k,(Complex_of {C})) is Element of bool the carrier of k
[#] (Complex_of {C}) is non proper Element of bool the carrier of (Complex_of {C})
[#] k is non empty non proper Element of bool the carrier of k
degree (k,(Complex_of {C})) is ext-real set
subdivision ((center_of_mass k),(Complex_of {C})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
degree (k,V) is ext-real set
F is set
A1 is c=-linear finite finite-membered Element of bool (bool the carrier of k)
(center_of_mass k) .: A1 is finite Element of bool the carrier of k
XX is finite non dependent Simplex of card Aff,(k,(Complex_of {C}))
(k,(k,(Complex_of {C})),XX) is affinely-independent Element of bool the carrier of k
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree (k,(Complex_of {C}))) + 1 is ext-real set
XX is finite non dependent Element of bool the carrier of (k,V)
conv (k,(k,(Complex_of {C})),XX) is convex Element of bool the carrier of k
(k,(k,V),XX) is affinely-independent Element of bool the carrier of k
[#] (subdivision ((center_of_mass k),(Complex_of {C}))) is non proper Element of bool the carrier of (subdivision ((center_of_mass k),(Complex_of {C})))
the carrier of (subdivision ((center_of_mass k),(Complex_of {C}))) is set
bool the carrier of (subdivision ((center_of_mass k),(Complex_of {C}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is set
XX is finite non dependent Simplex of card Aff,(k,V)
(k,(k,V),XX) is affinely-independent Element of bool the carrier of k
conv (k,(k,V),XX) is convex Element of bool the carrier of k
the carrier of (k,(Complex_of {C})) is set
bool the carrier of (k,(Complex_of {C})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
XX is ext-real set
XX + 1 is ext-real set
(card XX) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
XXA is ext-real set
XXA - 1 is ext-real set
XXA + (- 1) is ext-real set
XX is Element of bool the carrier of (k,(Complex_of {C}))
F \/ (union Aff) is finite Element of bool the carrier of V
A1 is c=-linear finite finite-membered Element of bool (bool the carrier of k)
union A1 is finite Element of bool the carrier of k
{x} \/ (union A1) is non empty finite set
(center_of_mass k) .: A1 is finite Element of bool the carrier of k
((center_of_mass k) .: A1) \/ ((center_of_mass k) .: {F}) is finite Element of bool the carrier of k
{(((center_of_mass k) .: A1) \/ ((center_of_mass k) .: {F}))} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of k)
k is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of k is non empty set
BOOL the carrier of k is set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool the carrier of k is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of k)
bool (bool the carrier of k) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(bool the carrier of k) \ {{}} is Element of bool (bool the carrier of k)
center_of_mass k is Relation-like BOOL the carrier of k -defined the carrier of k -valued Function-like quasi_total Element of bool [:(BOOL the carrier of k), the carrier of k:]
[:(BOOL the carrier of k), the carrier of k:] is Relation-like set
bool [:(BOOL the carrier of k), the carrier of k:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
V is non void subset-closed finite-membered non with_non-empty_elements total (k) (k) SimplicialComplexStr of the carrier of k
the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(k,V) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k,V)
Aff is finite finite-membered simplex-like Element of bool (bool the carrier of V)
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card Aff) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
union Aff is finite Element of bool the carrier of V
card (union Aff) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass k) .: Aff is finite Element of bool the carrier of k
(k,V,(union Aff)) is Element of bool the carrier of k
conv (k,V,(union Aff)) is convex Element of bool the carrier of k
{ b₁ where b₁ is finite non dependent Simplex of card Aff,(k,V) : ( (center_of_mass k) .: Aff c= b₁ & conv (k,(k,V),b₁) c= conv (k,V,(union Aff)) ) } is set
card { b₁ where b₁ is finite non dependent Simplex of card Aff,(k,V) : ( (center_of_mass k) .: Aff c= b₁ & conv (k,(k,V),b₁) c= conv (k,V,(union Aff)) ) } is epsilon-transitive epsilon-connected ordinal cardinal set
(k,V,Aff) is Element of bool (bool the carrier of k)
XX is finite non dependent Element of bool the carrier of V
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k,V,XX) is affinely-independent Element of bool the carrier of k
x is c=-linear finite finite-membered Element of bool (bool the carrier of k)
union x is finite Element of bool the carrier of k
F is finite affinely-independent Element of bool the carrier of k
{F} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of k)
Complex_of {F} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (k) (k) SimplicialComplexStr of the carrier of k
subset-closed_closure_of {F} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of k)
TopStruct(# the carrier of k,(subset-closed_closure_of {F}) #) is strict TopStruct
degree (Complex_of {F}) is V21() V29() V30() ext-real finite set
C is ext-real set
C - 1 is ext-real set
C + (- 1) is ext-real set
(card XX) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
[#] V is non proper Element of bool the carrier of V
(k,V) is Element of bool the carrier of k
subdivision ((center_of_mass k),V) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of k
(k,(Complex_of {F})) is non void subset-closed finite-membered non with_non-empty_elements (k) (k) (k, Complex_of {F})
{ b₁ where b₁ is finite non dependent Simplex of card Aff,(k,(Complex_of {F})) : (center_of_mass k) .: Aff c= b₁ } is set
the topology of (Complex_of {F}) is Element of bool (bool the carrier of (Complex_of {F}))
the carrier of (Complex_of {F}) is set
bool the carrier of (Complex_of {F}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {F})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the topology of V is Element of bool (bool the carrier of V)
(k,(Complex_of {F})) is Element of bool the carrier of k
[#] (Complex_of {F}) is non proper Element of bool the carrier of (Complex_of {F})
[#] k is non empty non proper Element of bool the carrier of k
degree (k,(Complex_of {F})) is ext-real set
subdivision ((center_of_mass k),(Complex_of {F})) is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements SimplicialComplexStr of the carrier of k
degree (k,V) is ext-real set
CA is set
(center_of_mass k) .: x is finite Element of bool the carrier of k
F is finite non dependent Simplex of card Aff,(k,(Complex_of {F}))
(k,(k,(Complex_of {F})),F) is affinely-independent Element of bool the carrier of k
the carrier of (k,V) is set
bool the carrier of (k,V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree (k,(Complex_of {F}))) + 1 is ext-real set
XX is finite non dependent Element of bool the carrier of (k,V)
conv (k,(k,(Complex_of {F})),F) is convex Element of bool the carrier of k
conv (k,V,XX) is convex Element of bool the carrier of k
(k,(k,V),XX) is affinely-independent Element of bool the carrier of k
[#] (subdivision ((center_of_mass k),(Complex_of {F}))) is non proper Element of bool the carrier of (subdivision ((center_of_mass k),(Complex_of {F})))
the carrier of (subdivision ((center_of_mass k),(Complex_of {F}))) is set
bool the carrier of (subdivision ((center_of_mass k),(Complex_of {F}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
CA is set
conv (k,V,XX) is convex Element of bool the carrier of k
XX is finite non dependent Simplex of card Aff,(k,V)
(k,(k,V),XX) is affinely-independent Element of bool the carrier of k
conv (k,(k,V),XX) is convex Element of bool the carrier of k
the carrier of (k,(Complex_of {F})) is set
bool the carrier of (k,(Complex_of {F})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is ext-real set
F + 1 is ext-real set
(card XX) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
XX is ext-real set
XX - 1 is ext-real set
XX + (- 1) is ext-real set
XX is Element of bool the carrier of (k,(Complex_of {F}))
card { b₁ where b₁ is finite non dependent Simplex of card Aff,(k,(Complex_of {F})) : (center_of_mass k) .: Aff c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
k + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
k - 1 is V21() V29() V30() ext-real finite Element of REAL
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is non void subset-closed finite-membered non with_non-empty_elements total (V) (V) SimplicialComplexStr of the carrier of V
degree Aff is ext-real set
(V,Aff) is non void subset-closed finite-membered non with_non-empty_elements (V) (V) (V,Aff)
F is finite Element of bool the carrier of V
{F} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {F} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total SimplicialComplexStr of the carrier of V
subset-closed_closure_of {F} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {F}) #) is strict TopStruct
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
Int F is Element of bool the carrier of V
(V,(Complex_of {F})) is Element of bool the carrier of V
conv F is convex Element of bool the carrier of V
(V,Aff) is Element of bool the carrier of V
the carrier of Aff is set
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(degree Aff) + 1 is ext-real set
XX is Element of bool the carrier of Aff
card XX is epsilon-transitive epsilon-connected ordinal cardinal set
(V,Aff,XX) is Element of bool the carrier of V
conv (V,Aff,XX) is convex Element of bool the carrier of V
Affin (V,Aff,XX) is Element of bool the carrier of V
Affin F is Element of bool the carrier of V
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
the carrier of Aff is set
the topology of Aff is Element of bool (bool the carrier of Aff)
bool the carrier of Aff is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
TopStruct(# the carrier of Aff, the topology of Aff #) is strict TopStruct
X is finite non dependent Simplex of k - 1,(V,Aff)
{ b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : X c= b₁ } is set
(V,(V,Aff),X) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),X) is convex Element of bool the carrier of V
A1 is set
card A1 is epsilon-transitive epsilon-connected ordinal cardinal set
[#] Aff is non proper Element of bool the carrier of Aff
degree (V,Aff) is ext-real set
k + (- 1) is V21() V29() V30() ext-real finite Element of REAL
card X is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is ext-real set
F + 1 is ext-real set
(k - 1) + 1 is V21() V29() V30() ext-real finite Element of REAL
subdivision ((center_of_mass V),Aff) is strict non void subset-closed finite-membered non with_non-empty_elements SimplicialComplexStr of the carrier of V
S is finite non dependent Element of bool the carrier of Aff
CA is finite non dependent Simplex of k - 1,Aff
{ b₁ where b₁ is finite non dependent Simplex of k,Aff : CA c= b₁ } is set
(V,Aff,CA) is affinely-independent Element of bool the carrier of V
card { b₁ where b₁ is finite non dependent Simplex of k,Aff : CA c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
XX is set
XX is finite non dependent Simplex of k,Aff
the topology of (V,Aff) is Element of bool (bool the carrier of (V,Aff))
the carrier of (V,Aff) is set
bool the carrier of (V,Aff) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (V,Aff)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree Aff) + 1 is ext-real set
XX is finite non dependent Element of bool the carrier of (V,Aff)
XX is set
XX is finite non dependent Simplex of k,(V,Aff)
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree (V,Aff)) + 1 is ext-real set
XX is finite non dependent Element of bool the carrier of Aff
S is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
(center_of_mass V) .: S is finite Element of bool the carrier of V
S \ {{}} is finite Element of bool (bool the carrier of Aff)
CA is c=-linear finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
union CA is finite Element of bool the carrier of Aff
(V,Aff,(union CA)) is Element of bool the carrier of V
bool (V,Aff,(union CA)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool (V,Aff,(union CA)))
bool (V,Aff,(union CA)) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool (V,Aff,(union CA))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
dom (center_of_mass V) is Element of bool (BOOL the carrier of V)
bool (BOOL the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
S /\ (dom (center_of_mass V)) is finite Element of bool (BOOL the carrier of V)
S /\ (bool the carrier of V) is finite Element of bool (bool the carrier of V)
(S /\ (bool the carrier of V)) \ {{}} is finite Element of bool (bool the carrier of V)
CA /\ (bool the carrier of V) is finite Element of bool (bool the carrier of V)
(center_of_mass V) .: CA is finite Element of bool the carrier of V
card CA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F is finite non dependent Element of bool the carrier of Aff
card F is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(degree Aff) + 1 is ext-real set
(V,Aff,F) is affinely-independent Element of bool the carrier of V
conv (V,Aff,F) is convex Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of k,Aff : F c= b₁ } is set
card { b₁ where b₁ is finite non dependent Simplex of k,Aff : F c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
XX is set
XX is set
{XX,XX} is non empty finite set
XXA is finite non dependent Simplex of k,Aff
m1 is finite non dependent Simplex of k,Aff
card m1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(V,Aff,m1) is affinely-independent Element of bool the carrier of V
conv (V,Aff,m1) is convex Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : ( X c= b₁ & conv (V,(V,Aff),b₁) c= conv (V,Aff,m1) ) } is set
{m1} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of Aff)
(center_of_mass V) .: {m1} is finite Element of bool the carrier of V
X \/ ((center_of_mass V) .: {m1}) is finite set
{(X \/ ((center_of_mass V) .: {m1}))} is non empty trivial finite finite-membered 1 -element set
{XXA} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of Aff)
(center_of_mass V) .: {XXA} is finite Element of bool the carrier of V
X \/ ((center_of_mass V) .: {XXA}) is finite set
M is Element of bool the carrier of Aff
CA \/ {m1} is non empty finite finite-membered Element of bool (bool the carrier of Aff)
(center_of_mass V) .: (CA \/ {m1}) is finite Element of bool the carrier of V
CA \/ {XXA} is non empty finite finite-membered Element of bool (bool the carrier of Aff)
(center_of_mass V) .: (CA \/ {XXA}) is finite Element of bool the carrier of V
M is Element of bool the carrier of Aff
M is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),M) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),M) is convex Element of bool the carrier of V
x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card CA) + x is V21() V29() ext-real non negative set
K85((card CA),x) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
{(X \/ ((center_of_mass V) .: {m1})),(X \/ ((center_of_mass V) .: {XXA}))} is non empty finite finite-membered set
M is set
((degree Aff) + 1) - 1 is ext-real set
((degree Aff) + 1) + (- 1) is ext-real set
cA is finite non dependent Simplex of k,(V,Aff)
x + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(V,(V,Aff),cA) is affinely-independent Element of bool the carrier of V
YA is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
YA \/ CA is finite finite-membered Element of bool (bool the carrier of Aff)
card YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: YA is finite Element of bool the carrier of V
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: YA) is finite Element of bool the carrier of V
YA is set
{YA} is non empty trivial finite 1 -element set
card (YA \/ CA) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union (YA \/ CA) is finite Element of bool the carrier of Aff
Xm1 is finite non dependent Element of bool the carrier of Aff
card Xm1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: {m1}) is finite Element of bool the carrier of V
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: {XXA}) is finite Element of bool the carrier of V
card XXA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(V,Aff,XXA) is affinely-independent Element of bool the carrier of V
conv (V,Aff,XXA) is convex Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : ( X c= b₁ & conv (V,(V,Aff),b₁) c= conv (V,Aff,XXA) ) } is set
{(X \/ ((center_of_mass V) .: {XXA}))} is non empty trivial finite finite-membered 1 -element set
M is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),M) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),M) is convex Element of bool the carrier of V
M is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),M) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),M) is convex Element of bool the carrier of V
XX is Element of bool the carrier of V
conv XX is convex Element of bool the carrier of V
XX is finite Element of bool the carrier of V
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
Affin (V,(V,Aff),X) is Element of bool the carrier of V
Affin XX is Element of bool the carrier of V
F \ XX is finite Element of bool the carrier of V
card (F \ XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k + 1) - k is V21() V29() V30() ext-real finite Element of REAL
XXA is set
{XXA} is non empty trivial finite 1 -element set
(center_of_mass V) . F is set
((center_of_mass V) . F) |-- (V,Aff,F) is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (V,Aff,F)
Affin (V,Aff,F) is Element of bool the carrier of V
sum (((center_of_mass V) . F) |-- (V,Aff,F)) is V21() V29() ext-real Element of REAL
Sum (((center_of_mass V) . F) |-- (V,Aff,F)) is Element of the carrier of V
(Sum (((center_of_mass V) . F) |-- (V,Aff,F))) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
M is Element of the carrier of V
((Sum (((center_of_mass V) . F) |-- (V,Aff,F))) |-- F) . M is V21() V29() ext-real Element of REAL
Carrier (((center_of_mass V) . F) |-- (V,Aff,F)) is finite Element of bool the carrier of V
{ b₁ where b₁ is Element of the carrier of V : not (((center_of_mass V) . F) |-- (V,Aff,F)) . b₁ = 0 } is set
cA is Relation-like NAT -defined REAL -valued Function-like FinSequence-like V50() V51() V52() FinSequence of REAL
Sum cA is V21() V29() ext-real Element of REAL
YA is Relation-like NAT -defined the carrier of V -valued Function-like FinSequence-like FinSequence of the carrier of V
len YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
len cA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
rng YA is Element of bool the carrier of V
dom cA is Element of bool NAT
dom YA is Element of bool NAT
conv XX is convex Element of bool the carrier of V
YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
cA . YA is V21() V29() ext-real set
YA . YA is set
(((center_of_mass V) . F) |-- (V,Aff,F)) . (YA . YA) is V21() V29() ext-real set
(YA . YA) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
((YA . YA) |-- F) . M is V21() V29() ext-real Element of REAL
((((center_of_mass V) . F) |-- (V,Aff,F)) . (YA . YA)) * (((YA . YA) |-- F) . M) is V21() V29() ext-real Element of REAL
1 / (card F) is V21() V29() ext-real non negative Element of REAL
YA is set
YA |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
(YA |-- F) . M is V21() V29() ext-real Element of REAL
{M} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of V
F \ {M} is finite Element of bool the carrier of V
YA is set
conv (F \ {M}) is convex Element of bool the carrier of V
YA is set
(YA |-- F) . YA is V21() V29() ext-real set
(((center_of_mass V) . F) |-- (V,Aff,F)) . YA is V21() V29() ext-real set
YA is set
YA . YA is set
Xm1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
cA . Xm1 is V21() V29() ext-real set
((((center_of_mass V) . F) |-- (V,Aff,F)) . YA) * ((YA |-- F) . M) is V21() V29() ext-real Element of REAL
((center_of_mass V) . F) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
(((center_of_mass V) . F) |-- F) . M is V21() V29() ext-real Element of REAL
((center_of_mass V) . F) |-- XX is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of XX
Carrier (((center_of_mass V) . F) |-- XX) is finite Element of bool the carrier of V
{ b₁ where b₁ is Element of the carrier of V : not (((center_of_mass V) . F) |-- XX) . b₁ = 0 } is set
YA is set
{YA} is non empty trivial finite 1 -element set
YA is finite non dependent Simplex of k,Aff
{YA} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of Aff)
(center_of_mass V) .: {YA} is finite Element of bool the carrier of V
X \/ ((center_of_mass V) .: {YA}) is finite set
{(X \/ ((center_of_mass V) .: {YA}))} is non empty trivial finite finite-membered 1 -element set
Xm1 is set
Xm is finite non dependent Simplex of k,(V,Aff)
x + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(V,(V,Aff),Xm) is affinely-independent Element of bool the carrier of V
R1 is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
R1 \/ CA is finite finite-membered Element of bool (bool the carrier of Aff)
card R1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: R1 is finite Element of bool the carrier of V
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: R1) is finite Element of bool the carrier of V
DY is set
{DY} is non empty trivial finite 1 -element set
card (R1 \/ CA) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
union (R1 \/ CA) is finite Element of bool the carrier of Aff
R is finite non dependent Element of bool the carrier of Aff
card R is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: {YA}) is finite Element of bool the carrier of V
card YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(V,Aff,YA) is affinely-independent Element of bool the carrier of V
conv (V,Aff,YA) is convex Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : ( X c= b₁ & conv (V,(V,Aff),b₁) c= conv (V,Aff,YA) ) } is set
Xm1 is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),Xm1) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),Xm1) is convex Element of bool the carrier of V
(center_of_mass V) . F is set
((center_of_mass V) . F) |-- (V,Aff,F) is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of (V,Aff,F)
XX is Element of bool the carrier of V
conv XX is convex Element of bool the carrier of V
Affin (V,(V,Aff),X) is Element of bool the carrier of V
Affin XX is Element of bool the carrier of V
XX is finite Element of bool the carrier of V
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
F \ XX is finite Element of bool the carrier of V
card (F \ XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(k + 1) - k is V21() V29() V30() ext-real finite Element of REAL
XXA is set
{XXA} is non empty trivial finite 1 -element set
Affin (V,Aff,F) is Element of bool the carrier of V
sum (((center_of_mass V) . F) |-- (V,Aff,F)) is V21() V29() ext-real Element of REAL
Sum (((center_of_mass V) . F) |-- (V,Aff,F)) is Element of the carrier of V
(Sum (((center_of_mass V) . F) |-- (V,Aff,F))) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
m1 is Element of the carrier of V
((Sum (((center_of_mass V) . F) |-- (V,Aff,F))) |-- F) . m1 is V21() V29() ext-real Element of REAL
Carrier (((center_of_mass V) . F) |-- (V,Aff,F)) is finite Element of bool the carrier of V
{ b₁ where b₁ is Element of the carrier of V : not (((center_of_mass V) . F) |-- (V,Aff,F)) . b₁ = 0 } is set
M is Relation-like NAT -defined REAL -valued Function-like FinSequence-like V50() V51() V52() FinSequence of REAL
Sum M is V21() V29() ext-real Element of REAL
cA is Relation-like NAT -defined the carrier of V -valued Function-like FinSequence-like FinSequence of the carrier of V
len cA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
len M is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
rng cA is Element of bool the carrier of V
dom M is Element of bool NAT
dom cA is Element of bool NAT
{m1} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of V
F \ {m1} is finite Element of bool the carrier of V
F /\ XX is finite Element of bool the carrier of V
conv XX is convex Element of bool the carrier of V
YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
M . YA is V21() V29() ext-real set
cA . YA is set
(((center_of_mass V) . F) |-- (V,Aff,F)) . (cA . YA) is V21() V29() ext-real set
(cA . YA) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
((cA . YA) |-- F) . m1 is V21() V29() ext-real Element of REAL
((((center_of_mass V) . F) |-- (V,Aff,F)) . (cA . YA)) * (((cA . YA) |-- F) . m1) is V21() V29() ext-real Element of REAL
1 / (card F) is V21() V29() ext-real non negative Element of REAL
YA is set
YA |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
(YA |-- F) . m1 is V21() V29() ext-real Element of REAL
F \ XX is finite Element of bool the carrier of V
YA is set
(YA |-- F) . YA is V21() V29() ext-real set
(((center_of_mass V) . F) |-- (V,Aff,F)) . YA is V21() V29() ext-real set
YA is set
cA . YA is set
YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
M . YA is V21() V29() ext-real set
((((center_of_mass V) . F) |-- (V,Aff,F)) . YA) * ((YA |-- F) . m1) is V21() V29() ext-real Element of REAL
((center_of_mass V) . F) |-- F is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of F
(((center_of_mass V) . F) |-- F) . m1 is V21() V29() ext-real Element of REAL
((center_of_mass V) . F) |-- XX is Relation-like the carrier of V -defined REAL -valued Function-like quasi_total V50() V51() V52() Linear_Combination of XX
Carrier (((center_of_mass V) . F) |-- XX) is finite Element of bool the carrier of V
{ b₁ where b₁ is Element of the carrier of V : not (((center_of_mass V) . F) |-- XX) . b₁ = 0 } is set
Affin XX is Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : ( X c= b₁ & conv (V,(V,Aff),b₁) c= conv (V,Aff,F) ) } is set
card { b₁ where b₁ is finite non dependent Simplex of k,(V,Aff) : ( X c= b₁ & conv (V,(V,Aff),b₁) c= conv (V,Aff,F) ) } is epsilon-transitive epsilon-connected ordinal cardinal set
XX is set
XX is set
{XX,XX} is non empty finite set
XXA is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),XXA) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),XXA) is convex Element of bool the carrier of V
m1 is finite non dependent Simplex of k,(V,Aff)
(V,(V,Aff),m1) is affinely-independent Element of bool the carrier of V
conv (V,(V,Aff),m1) is convex Element of bool the carrier of V
M is set
cA is finite non dependent Simplex of k,(V,Aff)
x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(card CA) + x is V21() V29() ext-real non negative set
K85((card CA),x) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
x + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
(V,(V,Aff),cA) is affinely-independent Element of bool the carrier of V
YA is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
YA \/ CA is finite finite-membered Element of bool (bool the carrier of Aff)
card YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(center_of_mass V) .: YA is finite Element of bool the carrier of V
((center_of_mass V) .: CA) \/ ((center_of_mass V) .: YA) is finite Element of bool the carrier of V
YA is finite finite-membered simplex-like Element of bool (bool the carrier of Aff)
(V,Aff,YA) is Element of bool (bool the carrier of V)
(center_of_mass V) .: (V,Aff,YA) is Element of bool the carrier of V
union YA is finite Element of bool the carrier of Aff
YA is finite non dependent Element of bool the carrier of Aff
card YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
conv (V,(V,Aff),cA) is convex Element of bool the carrier of V
k is set
card k is epsilon-transitive epsilon-connected ordinal cardinal set
V is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V - 1 is V21() V29() V30() ext-real finite Element of REAL
V + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
F is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of F is non empty set
bool the carrier of F is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is finite affinely-independent Element of bool the carrier of F
{XX} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of F)
bool (bool the carrier of F) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {XX} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (F) (F) SimplicialComplexStr of the carrier of F
subset-closed_closure_of {XX} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of F)
TopStruct(# the carrier of F,(subset-closed_closure_of {XX}) #) is strict TopStruct
(Aff,F,(Complex_of {XX})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (F) (F) (F, Complex_of {XX})
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
Int XX is Element of bool the carrier of F
x is finite non dependent Simplex of V - 1,(Aff,F,(Complex_of {XX}))
{ b₁ where b₁ is finite non dependent Simplex of V,(Aff,F,(Complex_of {XX})) : x c= b₁ } is set
(F,(Aff,F,(Complex_of {XX})),x) is affinely-independent Element of bool the carrier of F
conv (F,(Aff,F,(Complex_of {XX})),x) is convex Element of bool the carrier of F
F is ext-real set
F - 1 is ext-real set
F + (- 1) is ext-real set
(card XX) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
degree (Complex_of {XX}) is V21() V29() V30() ext-real finite set
[#] (Complex_of {XX}) is non proper Element of bool the carrier of (Complex_of {XX})
the carrier of (Complex_of {XX}) is set
bool the carrier of (Complex_of {XX}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[#] F is non empty non proper Element of bool the carrier of F
(F,(Complex_of {XX})) is Element of bool the carrier of F
({},F,(Complex_of {XX})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (F) (F) (F, Complex_of {XX})
the topology of (Complex_of {XX}) is Element of bool (bool the carrier of (Complex_of {XX}))
bool (bool the carrier of (Complex_of {XX})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool XX is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool XX)
bool XX is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool XX) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
S is finite non dependent Simplex of V - 1,({},F,(Complex_of {XX}))
{ b₁ where b₁ is finite non dependent Simplex of V,({},F,(Complex_of {XX})) : S c= b₁ } is set
(F,({},F,(Complex_of {XX})),S) is affinely-independent Element of bool the carrier of F
conv (F,({},F,(Complex_of {XX})),S) is convex Element of bool the carrier of F
CA is set
card CA is epsilon-transitive epsilon-connected ordinal cardinal set
F is set
XX is finite non dependent Simplex of V, Complex_of {XX}
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
XX is ext-real set
XX + 1 is ext-real set
A1 is finite non dependent Element of bool the carrier of (Complex_of {XX})
V + (- 1) is V21() V29() V30() ext-real finite Element of REAL
card S is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
X is ext-real set
X + 1 is ext-real set
(V - 1) + 1 is V21() V29() V30() ext-real finite Element of REAL
X is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
(X,F,(Complex_of {XX})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (F) (F) (F, Complex_of {XX})
X + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
((X + 1),F,(Complex_of {XX})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (F) (F) (F, Complex_of {XX})
degree (X,F,(Complex_of {XX})) is V21() V29() V30() ext-real finite set
(F,(X,F,(Complex_of {XX}))) is non void subset-closed finite-membered non with_non-empty_elements (F) (F) (F,(X,F,(Complex_of {XX})))
A1 is finite non dependent Simplex of V - 1,((X + 1),F,(Complex_of {XX}))
{ b₁ where b₁ is finite non dependent Simplex of V,((X + 1),F,(Complex_of {XX})) : A1 c= b₁ } is set
(F,((X + 1),F,(Complex_of {XX})),A1) is affinely-independent Element of bool the carrier of F
conv (F,((X + 1),F,(Complex_of {XX})),A1) is convex Element of bool the carrier of F
S is set
card S is epsilon-transitive epsilon-connected ordinal cardinal set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite affinely-independent Element of bool the carrier of V
{Aff} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {Aff} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {Aff} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {Aff}) #) is strict TopStruct
(k,V,(Complex_of {Aff})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (V) (V) (V, Complex_of {Aff})
Vertices (k,V,(Complex_of {Aff})) is Element of bool the carrier of (k,V,(Complex_of {Aff}))
the carrier of (k,V,(Complex_of {Aff})) is set
bool the carrier of (k,V,(Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:(Vertices (k,V,(Complex_of {Aff}))),Aff:] is Relation-like set
bool [:(Vertices (k,V,(Complex_of {Aff}))),Aff:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card Aff) - 1 is V21() V29() V30() ext-real finite Element of REAL
x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
x + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
C is finite affinely-independent Element of bool the carrier of V
card C is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
{C} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
Complex_of {C} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {C} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {C}) #) is strict TopStruct
(k,V,(Complex_of {C})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (V) (V) (V, Complex_of {C})
Vertices (k,V,(Complex_of {C})) is Element of bool the carrier of (k,V,(Complex_of {C}))
the carrier of (k,V,(Complex_of {C})) is set
bool the carrier of (k,V,(Complex_of {C})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:(Vertices (k,V,(Complex_of {C}))),C:] is Relation-like set
bool [:(Vertices (k,V,(Complex_of {C}))),C:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(card C) - 1 is V21() V29() V30() ext-real finite Element of REAL
F is set
X is Element of the carrier of V
{X} is non empty trivial finite 1 -element affinely-independent Element of bool the carrier of V
C \ {X} is finite Element of bool the carrier of V
card (C \ {X}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
A1 is finite affinely-independent Element of bool the carrier of V
{A1} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
Complex_of {A1} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {A1} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {A1}) #) is strict TopStruct
the topology of (Complex_of {A1}) is Element of bool (bool the carrier of (Complex_of {A1}))
the carrier of (Complex_of {A1}) is set
bool the carrier of (Complex_of {A1}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool the carrier of (Complex_of {A1})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool A1 is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool A1)
bool A1 is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool A1) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Vertices (Complex_of {A1}) is Element of bool the carrier of (Complex_of {A1})
union (bool A1) is finite Element of bool A1
[#] (Complex_of {A1}) is non proper Element of bool the carrier of (Complex_of {A1})
[#] V is non empty non proper Element of bool the carrier of V
(V,(Complex_of {A1})) is Element of bool the carrier of V
(k,V,(Complex_of {A1})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (V) (V) (V, Complex_of {A1})
Vertices (k,V,(Complex_of {A1})) is Element of bool the carrier of (k,V,(Complex_of {A1}))
the carrier of (k,V,(Complex_of {A1})) is set
bool the carrier of (k,V,(Complex_of {A1})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Relation-like Vertices (k,V,(Complex_of {C})) -defined C -valued Function-like quasi_total Element of bool [:(Vertices (k,V,(Complex_of {C}))),C:]
{ b₁ where b₁ is finite non dependent Simplex of (card C) - 1,(k,V,(Complex_of {C})) : F .: b₁ = C } is set
card { b₁ where b₁ is finite non dependent Simplex of (card C) - 1,(k,V,(Complex_of {C})) : F .: b₁ = C } is epsilon-transitive epsilon-connected ordinal cardinal set
the topology of (k,V,(Complex_of {C})) is Element of bool (bool the carrier of (k,V,(Complex_of {C})))
bool (bool the carrier of (k,V,(Complex_of {C}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is set
XX is finite non dependent Simplex of (card C) - 1,(k,V,(Complex_of {C}))
F .: XX is finite Element of bool C
bool C is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is Element of bool (bool the carrier of (k,V,(Complex_of {C})))
[#] (Complex_of {C}) is non proper Element of bool the carrier of (Complex_of {C})
the carrier of (Complex_of {C}) is set
bool the carrier of (Complex_of {C}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(V,(Complex_of {C})) is Element of bool the carrier of V
XXA is set
the topology of (Complex_of {C}) is Element of bool (bool the carrier of (Complex_of {C}))
bool (bool the carrier of (Complex_of {C})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool C is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool C)
bool C is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool C) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Vertices (Complex_of {C}) is Element of bool the carrier of (Complex_of {C})
union (bool C) is finite Element of bool C
dom F is Element of bool (Vertices (k,V,(Complex_of {C})))
bool (Vertices (k,V,(Complex_of {C}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is ext-real set
F - 1 is ext-real set
F + (- 1) is ext-real set
degree (Complex_of {C}) is V21() V29() V30() ext-real finite set
2 * {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of REAL
(2 * {}) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
XXA is finite non dependent Element of bool the carrier of (Complex_of {C})
m1 is set
{m1} is non empty trivial finite 1 -element set
conv C is convex Element of bool the carrier of V
F . X is set
XX is finite finite-membered simplex-like Element of bool (bool the carrier of (k,V,(Complex_of {C})))
m1 is set
M is finite non dependent Simplex of {} , Complex_of {C}
F .: M is finite Element of bool C
card M is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
XX is ext-real set
XX + 1 is ext-real set
F .: C is finite Element of bool C
Im (F,X) is set
x - 1 is V21() V29() V30() ext-real finite Element of REAL
Int C is Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } is set
{ b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : A1 = F .: b₁ } is set
YA is set
YA is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: YA is finite Element of bool C
YA is Element of bool (bool the carrier of (k,V,(Complex_of {C})))
{ b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : A1 = F .: b₁ } is set
{ b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : verum } is set
R1 is Relation-like set
dom R1 is set
YA is finite finite-membered simplex-like Element of bool (bool the carrier of (k,V,(Complex_of {C})))
(dom R1) \ YA is Element of bool (dom R1)
bool (dom R1) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
XX is finite finite-membered simplex-like Element of bool (bool the carrier of (k,V,(Complex_of {C})))
RDY is set
R is set
[RDY,R] is V15() set
{RDY,R} is non empty finite set
{RDY} is non empty trivial finite 1 -element set
{{RDY,R},{RDY}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: DX is finite Element of bool C
RDX is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: RDX is finite Element of bool C
card (F .: DX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
R1 | ((dom R1) \ YA) is Relation-like set
dom (R1 | ((dom R1) \ YA)) is set
(dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA) is Element of bool (dom (R1 | ((dom R1) \ YA)))
bool (dom (R1 | ((dom R1) \ YA))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(R1 | ((dom R1) \ YA)) | ((dom (R1 | ((dom R1) \ YA))) \ ((dom R1) \ YA)) is Relation-like set
R is set
DX is set
RDX is set
[DX,RDX] is V15() set
{DX,RDX} is non empty finite set
{DX} is non empty trivial finite 1 -element set
{{DX,RDX},{DX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
card YA is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
2 *` (card YA) is epsilon-transitive epsilon-connected ordinal cardinal set
card 2 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
card (card YA) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card 2) *` (card (card YA)) is epsilon-transitive epsilon-connected ordinal cardinal set
2 * (card YA) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
card (2 * (card YA)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
cA is ext-real set
cA - 1 is ext-real set
cA + (- 1) is ext-real set
(x + 1) + (- 1) is V21() V29() V30() ext-real finite Element of REAL
degree (Complex_of {C}) is V21() V29() V30() ext-real finite set
R is Relation-like set
card R is epsilon-transitive epsilon-connected ordinal cardinal set
card R1 is epsilon-transitive epsilon-connected ordinal cardinal set
DX is set
DX `2 is set
DX `1 is set
[(DX `2),(DX `1)] is V15() set
{(DX `2),(DX `1)} is non empty finite set
{(DX `2)} is non empty trivial finite 1 -element set
{{(DX `2),(DX `1)},{(DX `2)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(DX `1),(DX `2)] is V15() set
{(DX `1),(DX `2)} is non empty finite set
{(DX `1)} is non empty trivial finite 1 -element set
{{(DX `1),(DX `2)},{(DX `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
RDX is set
DX is set
[RDX,DX] is V15() set
{RDX,DX} is non empty finite set
{RDX} is non empty trivial finite 1 -element set
{{RDX,DX},{RDX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[:R,R1:] is Relation-like set
bool [:R,R1:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
DX is Relation-like R -defined R1 -valued Function-like quasi_total Element of bool [:R,R1:]
RDX is set
dom DX is Relation-like Element of bool R
bool R is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
rng DX is Relation-like Element of bool R1
bool R1 is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
RDX is set
RDX `1 is set
RDX `2 is set
[(RDX `1),(RDX `2)] is V15() set
{(RDX `1),(RDX `2)} is non empty finite set
{(RDX `1)} is non empty trivial finite 1 -element set
{{(RDX `1),(RDX `2)},{(RDX `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX is set
DX is set
[DX,DX] is V15() set
{DX,DX} is non empty finite set
{DX} is non empty trivial finite 1 -element set
{{DX,DX},{DX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(RDX `2),(RDX `1)] is V15() set
{(RDX `2),(RDX `1)} is non empty finite set
{(RDX `2)} is non empty trivial finite 1 -element set
{{(RDX `2),(RDX `1)},{(RDX `2)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(RDX `2),(RDX `1)] `2 is set
[(RDX `2),(RDX `1)] `1 is set
[([(RDX `2),(RDX `1)] `2),([(RDX `2),(RDX `1)] `1)] is V15() set
{([(RDX `2),(RDX `1)] `2),([(RDX `2),(RDX `1)] `1)} is non empty finite set
{([(RDX `2),(RDX `1)] `2)} is non empty trivial finite 1 -element set
{{([(RDX `2),(RDX `1)] `2),([(RDX `2),(RDX `1)] `1)},{([(RDX `2),(RDX `1)] `2)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX . [(RDX `2),(RDX `1)] is set
RDX is set
DX is set
DX . RDX is set
DX . DX is set
RDX `2 is set
RDX `1 is set
[(RDX `2),(RDX `1)] is V15() set
{(RDX `2),(RDX `1)} is non empty finite set
{(RDX `2)} is non empty trivial finite 1 -element set
{{(RDX `2),(RDX `1)},{(RDX `2)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX `2 is set
DX `1 is set
[(DX `2),(DX `1)] is V15() set
{(DX `2),(DX `1)} is non empty finite set
{(DX `2)} is non empty trivial finite 1 -element set
{{(DX `2),(DX `1)},{(DX `2)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[(DX `1),(DX `2)] is V15() set
{(DX `1),(DX `2)} is non empty finite set
{(DX `1)} is non empty trivial finite 1 -element set
{{(DX `1),(DX `2)},{(DX `1)}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX is set
FA is set
[DX,FA] is V15() set
{DX,FA} is non empty finite set
{DX} is non empty trivial finite 1 -element set
{{DX,FA},{DX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX is set
FA is set
[DX,FA] is V15() set
{DX,FA} is non empty finite set
{DX} is non empty trivial finite 1 -element set
{{DX,FA},{DX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
(V,(k,V,(Complex_of {A1}))) is Element of bool the carrier of V
conv A1 is convex Element of bool the carrier of V
dom R is set
(dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } is Element of bool (dom R)
bool (dom R) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
RDX is set
DX is set
[RDX,DX] is V15() set
{RDX,DX} is non empty finite set
{RDX} is non empty trivial finite 1 -element set
{{RDX,DX},{RDX}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
DX is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: DX is finite Element of bool C
R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ) is Relation-like set
DX is Element of bool (bool the carrier of (k,V,(Complex_of {C})))
dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )) is set
DX is finite finite-membered simplex-like Element of bool (bool the carrier of (k,V,(Complex_of {C})))
(dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ))) \ DX is Element of bool (dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )))
bool (dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )) | ((dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ))) \ DX) is Relation-like set
FA is set
FA is set
XXa is set
[FA,XXa] is V15() set
{FA,XXa} is non empty finite set
{FA} is non empty trivial finite 1 -element set
{{FA,XXa},{FA}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
m1 is ext-real set
m1 + 1 is ext-real set
(x - 1) + 1 is V21() V29() V30() ext-real finite Element of REAL
F | (Vertices (k,V,(Complex_of {A1}))) is Relation-like Vertices (k,V,(Complex_of {C})) -defined Vertices (k,V,(Complex_of {A1})) -defined Vertices (k,V,(Complex_of {C})) -defined C -valued Function-like Element of bool [:(Vertices (k,V,(Complex_of {C}))),C:]
dom (F | (Vertices (k,V,(Complex_of {A1})))) is Element of bool (Vertices (k,V,(Complex_of {C})))
FA is Element of Vertices (k,V,(Complex_of {A1}))
(F | (Vertices (k,V,(Complex_of {A1})))) . FA is set
XXa is Element of bool the carrier of V
conv XXa is convex Element of bool the carrier of V
F . FA is set
rng (F | (Vertices (k,V,(Complex_of {A1})))) is finite Element of bool C
FA is set
XXa is set
(F | (Vertices (k,V,(Complex_of {A1})))) . XXa is set
n is Element of the carrier of (k,V,(Complex_of {A1}))
cnc is Element of bool the carrier of (k,V,(Complex_of {A1}))
(V,(k,V,(Complex_of {A1})),cnc) is Element of bool the carrier of V
conv (V,(k,V,(Complex_of {A1})),cnc) is convex Element of bool the carrier of V
[:(Vertices (k,V,(Complex_of {A1}))),A1:] is Relation-like set
bool [:(Vertices (k,V,(Complex_of {A1}))),A1:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
FA is Relation-like Vertices (k,V,(Complex_of {A1})) -defined A1 -valued Function-like quasi_total Element of bool [:(Vertices (k,V,(Complex_of {A1}))),A1:]
{ b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {A1})) : FA .: b₁ = A1 } is set
card { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {A1})) : FA .: b₁ = A1 } is epsilon-transitive epsilon-connected ordinal cardinal set
n is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
2 * n is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
(2 * n) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
x - {} is V21() V29() ext-real non negative set
- {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
K87({}) is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() set
x + (- {}) is V21() V29() ext-real non negative set
K85(x,(- {})) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
K89(x,{}) is V21() V29() V30() ext-real non negative finite set
x + (- 1) is V21() V29() V30() ext-real finite Element of REAL
cnc is set
Im (R,cnc) is set
card (Im (R,cnc)) is epsilon-transitive epsilon-connected ordinal cardinal set
c₃₉ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: c₃₉ is finite Element of bool C
(V,(k,V,(Complex_of {C})),c₃₉) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {C})),c₃₉) is convex Element of bool the carrier of V
{ b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : c₃₉ c= b₁ } is set
{c₃₉} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of (k,V,(Complex_of {C})))
R .: {c₃₉} is set
w is set
W is set
[W,w] is V15() set
{W,w} is non empty finite set
{W} is non empty trivial finite 1 -element set
{{W,w},{W}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
W is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
w is set
W is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
[c₃₉,W] is V15() Element of [:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):]
[:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):] is Relation-like non empty set
{c₃₉,W} is non empty finite finite-membered set
{c₃₉} is non empty trivial finite finite-membered 1 -element set
{{c₃₉,W},{c₃₉}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
card { b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : c₃₉ c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
degree (k,V,(Complex_of {C})) is V21() V29() V30() ext-real finite set
M is ext-real set
M + 1 is ext-real set
cnc is set
Im (R1,cnc) is set
card (Im (R1,cnc)) is epsilon-transitive epsilon-connected ordinal cardinal set
c₃₉ is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: c₃₉ is finite Element of bool C
F | c₃₉ is Relation-like Vertices (k,V,(Complex_of {C})) -defined c₃₉ -defined Vertices (k,V,(Complex_of {C})) -defined C -valued Function-like finite Element of bool [:(Vertices (k,V,(Complex_of {C}))),C:]
rng (F | c₃₉) is finite Element of bool C
{cnc} is non empty trivial finite 1 -element set
dom (F | c₃₉) is finite Element of bool c₃₉
bool c₃₉ is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card c₃₉ is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
[:c₃₉,A1:] is Relation-like finite set
bool [:c₃₉,A1:] is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
w is Relation-like c₃₉ -defined A1 -valued Function-like quasi_total finite Element of bool [:c₃₉,A1:]
W is set
{W} is non empty trivial finite 1 -element set
w " {W} is finite Element of bool c₃₉
card (w " {W}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
W is set
s is set
{W,s} is non empty finite set
{W} is non empty trivial finite 1 -element set
c₃₉ \ {W} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
{s} is non empty trivial finite 1 -element set
c₃₉ \ {s} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
FS is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
s is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
card s is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card FS is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
w . s is set
F . s is set
R1 .: {cnc} is set
S1 is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
S2 is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
{S1,S2} is non empty finite finite-membered Element of bool (bool the carrier of (k,V,(Complex_of {C})))
z is set
c is set
[c,z] is V15() set
{c,z} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,z},{c}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
W is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: W is finite Element of bool C
w .: W is finite Element of bool A1
dom w is finite Element of bool c₃₉
w is set
w . w is set
{w} is non empty trivial finite 1 -element set
c₃₉ \ {W,s} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
s is set
w . s is set
w is set
w . w is set
{(w . s)} is non empty trivial finite 1 -element set
w " {(w . s)} is finite Element of bool c₃₉
card (w " {(w . s)}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
z is set
{z} is non empty trivial finite 1 -element set
card W is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
c₃₉ /\ {W} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
c₃₉ /\ {s} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
(c₃₉ \ {W,s}) \/ {w} is non empty finite set
{W,s} \ {W} is finite Element of bool {W,s}
bool {W,s} is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
c₃₉ \ ({W,s} \ {W}) is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
(c₃₉ \ {W,s}) \/ {w} is non empty finite set
{W,s} \ {s} is finite Element of bool {W,s}
bool {W,s} is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
c₃₉ \ ({W,s} \ {s}) is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
w . W is set
F . W is set
F .: S1 is finite Element of bool C
z is set
c is set
F . c is set
F .: S2 is finite Element of bool C
z is set
c is set
F . c is set
[c₃₉,S1] is V15() Element of [:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):]
[:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):] is Relation-like non empty set
{c₃₉,S1} is non empty finite finite-membered set
{c₃₉} is non empty trivial finite finite-membered 1 -element set
{{c₃₉,S1},{c₃₉}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
[c₃₉,S2] is V15() Element of [:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):]
{c₃₉,S2} is non empty finite finite-membered set
{{c₃₉,S2},{c₃₉}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
card (R1 .: {cnc}) is epsilon-transitive epsilon-connected ordinal cardinal set
M - 1 is ext-real set
M + (- 1) is ext-real set
cnc is set
c₃₉ is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: c₃₉ is finite Element of bool C
F | c₃₉ is Relation-like Vertices (k,V,(Complex_of {C})) -defined c₃₉ -defined Vertices (k,V,(Complex_of {C})) -defined C -valued Function-like finite Element of bool [:(Vertices (k,V,(Complex_of {C}))),C:]
rng (F | c₃₉) is finite Element of bool C
card c₃₉ is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
dom (F | c₃₉) is finite Element of bool c₃₉
bool c₃₉ is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:c₃₉,C:] is Relation-like finite set
bool [:c₃₉,C:] is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
w is Relation-like c₃₉ -defined C -valued Function-like quasi_total finite Element of bool [:c₃₉,C:]
dom w is finite Element of bool c₃₉
W is set
w . W is set
{W} is non empty trivial finite 1 -element set
c₃₉ \ {W} is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
w .: (c₃₉ \ {W}) is finite Element of bool C
w .: c₃₉ is finite Element of bool C
w .: {W} is finite Element of bool C
(w .: c₃₉) \ (w .: {W}) is finite Element of bool C
C \ (w .: {W}) is finite Element of bool the carrier of V
Im (w,W) is set
C \ (Im (w,W)) is finite Element of bool the carrier of V
card (c₃₉ \ {W}) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
s is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
s is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
w .: s is finite Element of bool C
F .: s is finite Element of bool C
[c₃₉,s] is V15() Element of [:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):]
[:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):] is Relation-like non empty set
{c₃₉,s} is non empty finite finite-membered set
{c₃₉} is non empty trivial finite finite-membered 1 -element set
{{c₃₉,s},{c₃₉}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
FS is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: FS is finite Element of bool C
cnc is set
Im ((R1 | ((dom R1) \ YA)),cnc) is set
card (Im ((R1 | ((dom R1) \ YA)),cnc)) is epsilon-transitive epsilon-connected ordinal cardinal set
c₃₉ is set
[cnc,c₃₉] is V15() set
{cnc,c₃₉} is non empty finite set
{cnc} is non empty trivial finite 1 -element set
{{cnc,c₃₉},{cnc}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
XX is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
w is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: w is finite Element of bool C
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(R1 | ((dom R1) \ YA)) .: {cnc} is set
{c₃₉} is non empty trivial finite 1 -element set
W is set
F | XX is Relation-like Vertices (k,V,(Complex_of {C})) -defined XX -defined Vertices (k,V,(Complex_of {C})) -defined C -valued Function-like finite Element of bool [:(Vertices (k,V,(Complex_of {C}))),C:]
s is set
[s,W] is V15() set
{s,W} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,W},{s}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
s is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: s is finite Element of bool C
dom (F | XX) is finite Element of bool XX
bool XX is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
rng (F | XX) is finite Element of bool C
FS is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
F .: FS is finite Element of bool C
[:XX,C:] is Relation-like finite set
bool [:XX,C:] is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
FS is Relation-like XX -defined C -valued Function-like quasi_total finite Element of bool [:XX,C:]
FS .: w is finite Element of bool C
FS .: s is finite Element of bool C
card (R1 | ((dom R1) \ YA)) is epsilon-transitive epsilon-connected ordinal cardinal set
card {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of omega
card ((dom R1) \ YA) is epsilon-transitive epsilon-connected ordinal cardinal set
1 *` (card ((dom R1) \ YA)) is epsilon-transitive epsilon-connected ordinal cardinal set
(card {}) +` (1 *` (card ((dom R1) \ YA))) is epsilon-transitive epsilon-connected ordinal cardinal set
card XX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
card (card XX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card (card XX)) +` (card (2 * (card YA))) is epsilon-transitive epsilon-connected ordinal cardinal set
(card XX) + (2 * (card YA)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
card ((card XX) + (2 * (card YA))) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(V,(k,V,(Complex_of {C}))) is Element of bool the carrier of V
conv C is convex Element of bool the carrier of V
cnc is set
c₃₉ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: c₃₉ is finite Element of bool C
(V,(k,V,(Complex_of {C})),c₃₉) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {C})),c₃₉) is convex Element of bool the carrier of V
XX is Element of bool the carrier of V
conv XX is convex Element of bool the carrier of V
w is finite Element of bool the carrier of V
card w is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
W is set
W is set
F . W is set
card A1 is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
the topology of (k,V,(Complex_of {A1})) is Element of bool (bool the carrier of (k,V,(Complex_of {A1})))
bool (bool the carrier of (k,V,(Complex_of {A1}))) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card c₃₉ is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
center_of_mass V is Relation-like BOOL the carrier of V -defined the carrier of V -valued Function-like quasi_total Element of bool [:(BOOL the carrier of V), the carrier of V:]
BOOL the carrier of V is set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool the carrier of V)
(bool the carrier of V) \ {{}} is Element of bool (bool the carrier of V)
[:(BOOL the carrier of V), the carrier of V:] is Relation-like set
bool [:(BOOL the carrier of V), the carrier of V:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(center_of_mass V) . c₃₉ is set
Int (V,(k,V,(Complex_of {C})),c₃₉) is Element of bool the carrier of V
W is Element of bool the carrier of (k,V,(Complex_of {A1}))
(V,(k,V,(Complex_of {A1})),W) is Element of bool the carrier of V
conv (V,(k,V,(Complex_of {A1})),W) is convex Element of bool the carrier of V
W is finite non dependent Element of bool the carrier of (k,V,(Complex_of {C}))
(V,(k,V,(Complex_of {C})),W) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {C})),W) is convex Element of bool the carrier of V
s is Element of bool the carrier of (k,V,(Complex_of {A1}))
s is finite non dependent Element of bool the carrier of (k,V,(Complex_of {A1}))
FA .: s is finite Element of bool A1
degree (Complex_of {A1}) is V21() V29() V30() ext-real finite set
cnc is set
c₃₉ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {A1}))
FA .: c₃₉ is finite Element of bool A1
degree (k,V,(Complex_of {A1})) is V21() V29() V30() ext-real finite set
F .: c₃₉ is finite Element of bool C
(V,(k,V,(Complex_of {A1})),c₃₉) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {A1})),c₃₉) is convex Element of bool the carrier of V
XX is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
(V,(k,V,(Complex_of {C})),XX) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {C})),XX) is convex Element of bool the carrier of V
card DX is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
2 * (card DX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
cnc is set
Im ((R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )),cnc) is set
card (Im ((R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )),cnc)) is epsilon-transitive epsilon-connected ordinal cardinal set
c₃₉ is set
[cnc,c₃₉] is V15() set
{cnc,c₃₉} is non empty finite set
{cnc} is non empty trivial finite 1 -element set
{{cnc,c₃₉},{cnc}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
c₃₉ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C}))
F .: c₃₉ is finite Element of bool C
{ b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : c₃₉ c= b₁ } is set
(V,(k,V,(Complex_of {C})),c₃₉) is affinely-independent Element of bool the carrier of V
conv (V,(k,V,(Complex_of {C})),c₃₉) is convex Element of bool the carrier of V
card { b₁ where b₁ is finite non dependent Simplex of x,(k,V,(Complex_of {C})) : c₃₉ c= b₁ } is epsilon-transitive epsilon-connected ordinal cardinal set
{c₃₉} is non empty trivial finite finite-membered 1 -element Element of bool (bool the carrier of (k,V,(Complex_of {C})))
(R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )) .: {c₃₉} is set
w is set
W is set
[W,w] is V15() set
{W,w} is non empty finite set
{W} is non empty trivial finite 1 -element set
{{W,w},{W}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
W is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
w is set
W is finite non dependent Simplex of x,(k,V,(Complex_of {C}))
[c₃₉,W] is V15() Element of [:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):]
[:(bool the carrier of (k,V,(Complex_of {C}))),(bool the carrier of (k,V,(Complex_of {C}))):] is Relation-like non empty set
{c₃₉,W} is non empty finite finite-membered set
{c₃₉} is non empty trivial finite finite-membered 1 -element set
{{c₃₉,W},{c₃₉}} is non empty finite finite-membered with_non-empty_elements non empty-membered set
card (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )) is epsilon-transitive epsilon-connected ordinal cardinal set
card ((R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } )) | ((dom (R | ((dom R) \ { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ))) \ DX)) is epsilon-transitive epsilon-connected ordinal cardinal set
2 *` (card DX) is epsilon-transitive epsilon-connected ordinal cardinal set
(card ((R | ((dom R) \ { b<sub>1</sub> where b<sub>1</sub> is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b<sub>1</sub> = A1 & conv (V,(k,V,(Complex_of {C})),b<sub>1</sub>) misses Int C ) } )) | ((dom (R | ((dom R) \ { b<sub>1</sub> where b<sub>1</sub> is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b<sub>1</sub> = A1 & conv (V,(k,V,(Complex_of {C})),b<sub>1</sub>) misses Int C ) } ))) \ DX))) +` (2 *` (card DX)) is epsilon-transitive epsilon-connected ordinal cardinal set
{} +` (2 *` (card DX)) is epsilon-transitive epsilon-connected ordinal cardinal set
card { b<sub>1</sub> where b<sub>1</sub> is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b<sub>1</sub> = A1 & conv (V,(k,V,(Complex_of {C})),b<sub>1</sub>) misses Int C ) } is epsilon-transitive epsilon-connected ordinal cardinal set
1 *` (card { b₁ where b₁ is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b₁ = A1 & conv (V,(k,V,(Complex_of {C})),b₁) misses Int C ) } ) is epsilon-transitive epsilon-connected ordinal cardinal set
(2 *` (card DX)) +` (1 *` (card { b<sub>1</sub> where b<sub>1</sub> is finite non dependent Simplex of x - 1,(k,V,(Complex_of {C})) : ( F .: b<sub>1</sub> = A1 & conv (V,(k,V,(Complex_of {C})),b<sub>1</sub>) misses Int C ) } )) is epsilon-transitive epsilon-connected ordinal cardinal set
(2 *` (card DX)) +` ((2 * n) + 1) is epsilon-transitive epsilon-connected ordinal cardinal set
card (card DX) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card 2) *` (card (card DX)) is epsilon-transitive epsilon-connected ordinal cardinal set
((card 2) *` (card (card DX))) +` ((2 * n) + 1) is epsilon-transitive epsilon-connected ordinal cardinal set
card (2 * (card DX)) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card (2 * (card DX))) +` ((2 * n) + 1) is epsilon-transitive epsilon-connected ordinal cardinal set
card ((2 * n) + 1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
(card (2 * (card DX))) +` (card ((2 * n) + 1)) is epsilon-transitive epsilon-connected ordinal cardinal set
(2 * (card DX)) + ((2 * n) + 1) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
card ((2 * (card DX)) + ((2 * n) + 1)) is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of omega
(card DX) + n is V21() V29() ext-real non negative set
K85((card DX),n) is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
((card DX) + n) - (card YA) is V21() V29() ext-real set
- (card YA) is V21() V29() ext-real non positive set
K87((card YA)) is V21() V29() V30() ext-real non positive finite set
((card DX) + n) + (- (card YA)) is V21() V29() ext-real set
K85(((card DX) + n),(- (card YA))) is V21() V29() ext-real set
K89(((card DX) + n),(card YA)) is V21() V29() ext-real set
2 * (((card DX) + n) - (card YA)) is V21() V29() ext-real Element of REAL
(2 * (((card DX) + n) - (card YA))) + 1 is V21() V29() ext-real Element of REAL
(- 1) / 2 is non empty V21() V29() ext-real non positive negative Element of REAL
cnc is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of NAT
2 * cnc is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
(2 * cnc) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
x is finite affinely-independent Element of bool the carrier of V
card x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
{x} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
Complex_of {x} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {x} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {x}) #) is strict TopStruct
(k,V,(Complex_of {x})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (V) (V) (V, Complex_of {x})
Vertices (k,V,(Complex_of {x})) is Element of bool the carrier of (k,V,(Complex_of {x}))
the carrier of (k,V,(Complex_of {x})) is set
bool the carrier of (k,V,(Complex_of {x})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:(Vertices (k,V,(Complex_of {x}))),x:] is Relation-like set
bool [:(Vertices (k,V,(Complex_of {x}))),x:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
(card x) - 1 is V21() V29() V30() ext-real finite Element of REAL
(V,(Complex_of {x})) is Element of bool the carrier of V
[#] V is non empty non proper Element of bool the carrier of V
[#] (Complex_of {x}) is non proper Element of bool the carrier of (Complex_of {x})
the carrier of (Complex_of {x}) is set
bool the carrier of (Complex_of {x}) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
F is Relation-like Vertices (k,V,(Complex_of {x})) -defined x -valued Function-like quasi_total Element of bool [:(Vertices (k,V,(Complex_of {x}))),x:]
{ b₁ where b₁ is finite non dependent Simplex of (card x) - 1,(k,V,(Complex_of {x})) : F .: b₁ = x } is set
card { b₁ where b₁ is finite non dependent Simplex of (card x) - 1,(k,V,(Complex_of {x})) : F .: b₁ = x } is epsilon-transitive epsilon-connected ordinal cardinal set
2 * {} is Relation-like non-empty empty-yielding RAT -valued Function-like one-to-one constant functional empty V21() epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V29() V30() ext-real non positive non negative finite finite-yielding finite-membered cardinal {} -element V47() V50() V51() V52() V53() subset-closed Function-yielding V197() Element of REAL
(2 * {}) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
XX is ext-real set
XX - 1 is ext-real set
XX + (- 1) is ext-real set
degree (Complex_of {x}) is V21() V29() V30() ext-real finite set
the topology of (Complex_of {x}) is Element of bool (bool the carrier of (Complex_of {x}))
bool (bool the carrier of (Complex_of {x})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool x is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed Element of bool (bool x)
bool x is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
bool (bool x) is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set
A1 is set
S is finite non dependent Simplex of (card x) - 1,(k,V,(Complex_of {x}))
F .: S is finite Element of bool x
A1 is finite non dependent Element of bool the carrier of (Complex_of {x})
F .: A1 is finite Element of bool x
x is Relation-like Vertices (k,V,(Complex_of {Aff})) -defined Aff -valued Function-like quasi_total Element of bool [:(Vertices (k,V,(Complex_of {Aff}))),Aff:]
{ b₁ where b₁ is finite non dependent Simplex of (card Aff) - 1,(k,V,(Complex_of {Aff})) : x .: b₁ = Aff } is set
card { b₁ where b₁ is finite non dependent Simplex of (card Aff) - 1,(k,V,(Complex_of {Aff})) : x .: b₁ = Aff } is epsilon-transitive epsilon-connected ordinal cardinal set
k is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
V is non empty V97() V98() Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V130() RLSStruct
the carrier of V is non empty set
bool the carrier of V is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Aff is finite affinely-independent Element of bool the carrier of V
{Aff} is non empty trivial finite finite-membered 1 -element affinely-independent Element of bool (bool the carrier of V)
bool (bool the carrier of V) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
Complex_of {Aff} is strict non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total total (V) (V) SimplicialComplexStr of the carrier of V
subset-closed_closure_of {Aff} is non empty finite-membered V47() subset-closed non with_non-empty_elements Element of bool (bool the carrier of V)
TopStruct(# the carrier of V,(subset-closed_closure_of {Aff}) #) is strict TopStruct
(k,V,(Complex_of {Aff})) is non void subset-closed finite-membered finite-degree finite-vertices locally-finite non with_non-empty_elements total (V) (V) (V, Complex_of {Aff})
Vertices (k,V,(Complex_of {Aff})) is Element of bool the carrier of (k,V,(Complex_of {Aff}))
the carrier of (k,V,(Complex_of {Aff})) is set
bool the carrier of (k,V,(Complex_of {Aff})) is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
[:(Vertices (k,V,(Complex_of {Aff}))),Aff:] is Relation-like set
bool [:(Vertices (k,V,(Complex_of {Aff}))),Aff:] is non empty V47() subset-closed non with_non-empty_elements V290() d.union-closed set
card Aff is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of omega
(card Aff) - 1 is V21() V29() V30() ext-real finite Element of REAL
F is Relation-like Vertices (k,V,(Complex_of {Aff})) -defined Aff -valued Function-like quasi_total Element of bool [:(Vertices (k,V,(Complex_of {Aff}))),Aff:]
{ b₁ where b₁ is finite non dependent Simplex of (card Aff) - 1,(k,V,(Complex_of {Aff})) : F .: b₁ = Aff } is set
card { b₁ where b₁ is finite non dependent Simplex of (card Aff) - 1,(k,V,(Complex_of {Aff})) : F .: b₁ = Aff } is epsilon-transitive epsilon-connected ordinal cardinal set
x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal set
2 * x is V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real non negative finite cardinal Element of REAL
(2 * x) + 1 is non empty V21() epsilon-transitive epsilon-connected ordinal natural V29() V30() ext-real positive non negative finite cardinal Element of REAL
x is set
C is finite non dependent Simplex of (card Aff) - 1,(k,V,(Complex_of {Aff}))
F .: C is finite Element of bool Aff
bool Aff is non empty finite finite-membered V47() subset-closed non with_non-empty_elements V290() d.union-closed set