:: SCMYCIEL semantic presentation

REAL is set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V268() Element of bool REAL
bool REAL is non empty V233() V267() subset-closed V329() V332() set
omega is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal V268() set
bool omega is non empty non trivial non finite V233() V267() V268() subset-closed V329() V332() set
bool NAT is non empty non trivial non finite V233() V267() V268() subset-closed V329() V332() set
K226() is set
{} is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
the functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
{{}} is non empty trivial finite finite-membered 1 -element V233() V267() subset-closed V329() V332() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
{{}} * is functional non empty FinSequence-membered FinSequenceSet of {{}}
[:({{}} *),{{}}:] is non empty set
bool [:({{}} *),{{}}:] is non empty V233() V267() subset-closed V329() V332() set
K36(({{}} *),{{}}) is functional non empty set
COMPLEX is set
RAT is set
INT is set
2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
{{},1} is non empty finite finite-membered set
K386(NAT) is non empty V235() V330() V331() set
[:NAT,REAL:] is set
bool [:NAT,REAL:] is non empty V233() V267() subset-closed V329() V332() set
1 -tuples_on NAT is FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 1 } is set
[:REAL,REAL:] is set
bool [:REAL,REAL:] is non empty V233() V267() subset-closed V329() V332() set
Z_2 is V48() V52() V96() V116() V121() V122() V123() V132() V134() V139() V140() L13()
the U1 of Z_2 is set
[:COMPLEX,COMPLEX:] is set
bool [:COMPLEX,COMPLEX:] is non empty V233() V267() subset-closed V329() V332() set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty V233() V267() subset-closed V329() V332() set
[:[:REAL,REAL:],REAL:] is set
bool [:[:REAL,REAL:],REAL:] is non empty V233() V267() subset-closed V329() V332() set
[:RAT,RAT:] is set
bool [:RAT,RAT:] is non empty V233() V267() subset-closed V329() V332() set
[:[:RAT,RAT:],RAT:] is set
bool [:[:RAT,RAT:],RAT:] is non empty V233() V267() subset-closed V329() V332() set
[:INT,INT:] is set
bool [:INT,INT:] is non empty V233() V267() subset-closed V329() V332() set
[:[:INT,INT:],INT:] is set
bool [:[:INT,INT:],INT:] is non empty V233() V267() subset-closed V329() V332() set
[:NAT,NAT:] is non empty non trivial non finite V268() set
[:[:NAT,NAT:],NAT:] is non empty non trivial non finite V268() set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite V233() V267() V268() subset-closed V329() V332() set
bool (bool REAL) is non empty V233() V267() subset-closed V329() V332() set
3 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
0 is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of NAT
len {} is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
K653(1,2) is V83() ext-real non negative V216() Element of RAT
n is set
G is set
[n,G] is non empty set
{n,G} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,G},{n}} is non empty finite finite-membered V267() V268() set
n is set
G is set
[n,G] is non empty set
{n,G} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,G},{n}} is non empty finite finite-membered V267() V268() set
n is set
G is set
[n,G] is non empty set
{n,G} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,G},{n}} is non empty finite finite-membered V267() V268() set
n is set
MG is set
S is set
G is set
[G,MG] is non empty set
{G,MG} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,MG},{G}} is non empty finite finite-membered V267() V268() set
{n,[G,MG]} is non empty finite set
c₄ is set
[c₄,MG] is non empty set
{c₄,MG} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,MG},{c₄}} is non empty finite finite-membered V267() V268() set
{S,[c₄,MG]} is non empty finite set
n is set
len n is epsilon-transitive epsilon-connected ordinal cardinal set
G is set
S is set
c₄ is set
MG is set
n is set
{n} is non empty trivial finite 1 -element set
singletons {n} is Element of bool the U1 of (bspace {n})
bspace {n} is V48() L15( Z_2 )
bool {n} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
bspace-sum {n} is Relation-like [:(bool {n}),(bool {n}):] -defined bool {n} -valued Function-like non empty V14([:(bool {n}),(bool {n}):]) quasi_total finite Element of bool [:[:(bool {n}),(bool {n}):],(bool {n}):]
[:(bool {n}),(bool {n}):] is non empty finite set
[:[:(bool {n}),(bool {n}):],(bool {n}):] is non empty finite set
bool [:[:(bool {n}),(bool {n}):],(bool {n}):] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{} {n} is functional empty trivial proper epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool {n}
bspace-scalar-mult {n} is Relation-like [: the U1 of Z_2,(bool {n}):] -defined bool {n} -valued Function-like V14([: the U1 of Z_2,(bool {n}):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool {n}):],(bool {n}):]
[: the U1 of Z_2,(bool {n}):] is set
[:[: the U1 of Z_2,(bool {n}):],(bool {n}):] is set
bool [:[: the U1 of Z_2,(bool {n}):],(bool {n}):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool {n}),(bspace-sum {n}),({} {n}),(bspace-scalar-mult {n})) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace {n}) is set
bool the U1 of (bspace {n}) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool {n} : b₁ is 1 -element } is set
{{n}} is non empty trivial finite finite-membered 1 -element V267() V268() set
G is set
S is trivial finite Element of bool {n}
G is set
<*{}*> is Relation-like NAT -defined Function-like non empty trivial finite 1 -element FinSequence-like FinSubsequence-like set
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
G is set
rng n is finite set
n is non empty finite set
card n is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
G is non empty finite finite-membered V267() V268() a_partition of n
card G is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
proj G is Relation-like n -defined G -valued Function-like non empty V14(n) quasi_total finite Element of bool [:n,G:]
[:n,G:] is non empty finite set
bool [:n,G:] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S is set
c₄ is set
(proj G) . S is set
(proj G) . c₄ is set
MG is set
union {{}} is finite set
n is set
{n} is non empty trivial finite 1 -element set
{{},{n}} is non empty finite finite-membered set
union {{},{n}} is finite set
bool {n} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
union (bool {n}) is finite set
n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is Element of bool n
S is Element of G
{S} is non empty trivial finite 1 -element set
n is set
{ {n,[b₁,n]} where b₁ is Element of n : b₁ in n } is set
len { {n,[b₁,n]} where b₁ is Element of n : b₁ in n } is epsilon-transitive epsilon-connected ordinal cardinal set
len n is epsilon-transitive epsilon-connected ordinal cardinal set
c₄ is Relation-like Function-like set
dom c₄ is set
MG is set
n is set
c₄ . MG is set
c₄ . n is set
[MG,n] is non empty set
{MG,n} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{{MG,n},{MG}} is non empty finite finite-membered V267() V268() set
{n,[MG,n]} is non empty finite set
[n,n] is non empty set
{n,n} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,n},{n}} is non empty finite finite-membered V267() V268() set
{n,[n,n]} is non empty finite set
rng c₄ is set
MG is set
n is set
c₄ . n is set
[n,n] is non empty set
{n,n} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,n},{n}} is non empty finite finite-membered V267() V268() set
{n,[n,n]} is non empty finite set
MG is set
n is Element of n
[n,n] is non empty set
{n,n} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,n},{n}} is non empty finite finite-membered V267() V268() set
{n,[n,n]} is non empty finite set
c₄ . n is set
n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is Element of bool n
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
G is Element of bool n
S is Element of bool n
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
n is set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
union n is set
G is set
len G is epsilon-transitive epsilon-connected ordinal cardinal set
S is set
c₄ is set
{S,c₄} is non empty finite set
G is set
S is set
{G,S} is non empty finite set
n is set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
card {G,S} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
G is set
S is set
{G,S} is non empty finite set
n is set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
union n is set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
n is set
G is set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
(G) is Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
n is finite set
union n is set
(n) is finite Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
len { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is epsilon-transitive epsilon-connected ordinal cardinal set
card (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 * (card (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
canFS (n) is Relation-like NAT -defined (n) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of (n)
the Relation-like NAT -defined (n) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like FinSequence of (n) is Relation-like NAT -defined (n) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like FinSequence of (n)
len (canFS (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
rng (canFS (n)) is finite set
H is set
E is set
S is set
C is set
EuG is set
{C,EuG} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{EuG,[C,(union n)]} is non empty finite set
{{C,[EuG,(union n)]},{EuG,[C,(union n)]}} is non empty finite finite-membered V267() V268() set
H is set
E is set
S is set
{E,S} is non empty finite set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{E,[S,(union n)]} is non empty finite set
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{S,[E,(union n)]} is non empty finite set
{{E,[S,(union n)]},{S,[E,(union n)]}} is non empty finite finite-membered V267() V268() set
C is non empty finite finite-membered V267() V268() set
EuG is set
EuG is set
{EuG,EuG} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{{EuG,[EuG,(union n)]},{EuG,[EuG,(union n)]}} is non empty finite finite-membered V267() V268() set
H is Relation-like Function-like set
dom H is set
E is set
(canFS (n)) * H is Relation-like finite set
rng ((canFS (n)) * H) is finite set
rng H is set
S is set
H . S is set
C is set
EuG is set
{C,EuG} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{EuG,[C,(union n)]} is non empty finite set
{{C,[EuG,(union n)]},{EuG,[C,(union n)]}} is non empty finite finite-membered V267() V268() set
E is Relation-like NAT -defined Function-like finite finite-yielding FinSequence-like FinSubsequence-like set
dom E is finite Element of bool NAT
dom (canFS (n)) is finite Element of bool NAT
len E is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
S is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V218() V219() V220() V221() FinSequence of NAT
len S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
dom S is finite Element of bool NAT
C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
E . C is finite set
S . C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (E . C) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
C is set
EuG is set
(canFS (n)) . C is set
(canFS (n)) . EuG is set
EuG is set
uG is set
{EuG,uG} is non empty finite set
se is set
sev is set
{se,sev} is non empty finite set
E . C is finite set
H . ((canFS (n)) . C) is set
E . EuG is finite set
H . ((canFS (n)) . EuG) is set
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{EuG,[uG,(union n)]} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{uG,[EuG,(union n)]} is non empty finite set
{{EuG,[uG,(union n)]},{uG,[EuG,(union n)]}} is non empty finite finite-membered V267() V268() set
[sev,(union n)] is non empty set
{sev,(union n)} is non empty finite set
{sev} is non empty trivial finite 1 -element set
{{sev,(union n)},{sev}} is non empty finite finite-membered V267() V268() set
{se,[sev,(union n)]} is non empty finite set
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{sev,[se,(union n)]} is non empty finite set
{{se,[sev,(union n)]},{sev,[se,(union n)]}} is non empty finite finite-membered V267() V268() set
csev is set
E . C is finite set
E . EuG is finite set
Union E is set
rng E is finite finite-membered set
union (rng E) is finite set
card (union (rng E)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
Sum S is V216() set
(len S) |-> 2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V218() V219() V220() V221() Element of (len S) -tuples_on NAT
(len S) -tuples_on NAT is FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len S } is set
Seg (len S) is finite len S -element Element of bool NAT
K197((Seg (len S)),2) is Relation-like Seg (len S) -defined {2} -valued Function-like V14( Seg (len S)) quasi_total finite V218() V219() V220() V221() Element of bool [:(Seg (len S)),{2}:]
{2} is non empty trivial finite finite-membered 1 -element V267() V268() set
[:(Seg (len S)),{2}:] is finite set
bool [:(Seg (len S)),{2}:] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
dom ((len S) |-> 2) is finite Element of bool NAT
C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
E . C is finite set
(canFS (n)) . C is set
H . ((canFS (n)) . C) is set
EuG is set
EuG is set
{EuG,EuG} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{{EuG,[EuG,(union n)]},{EuG,[EuG,(union n)]}} is non empty finite finite-membered V267() V268() set
S . C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (E . C) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
((len S) |-> 2) . C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
C is set
EuG is set
EuG is set
E . EuG is finite set
(canFS (n)) . EuG is set
H . ((canFS (n)) . EuG) is set
uG is set
se is set
{uG,se} is non empty finite set
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{uG,[se,(union n)]} is non empty finite set
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{se,[uG,(union n)]} is non empty finite set
{{uG,[se,(union n)]},{se,[uG,(union n)]}} is non empty finite finite-membered V267() V268() set
C is set
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
uG is set
(canFS (n)) . uG is set
H . ((canFS (n)) . uG) is set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{{EuG,[EuG,(union n)]},{EuG,[EuG,(union n)]}} is non empty finite finite-membered V267() V268() set
n is finite set
union n is set
(n) is finite Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
len { [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is epsilon-transitive epsilon-connected ordinal cardinal set
card (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 * (card (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
MG is set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
[n,H] is non empty set
{n} is non empty trivial finite 1 -element set
{{n,H},{n}} is non empty finite finite-membered V267() V268() set
MG is set
n is Element of union n
H is Element of union n
[n,H] is non empty set
{n,H} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,H},{n}} is non empty finite finite-membered V267() V268() set
E is set
S is Element of union n
C is Element of union n
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{S,[C,(union n)]} is non empty finite set
{S,C} is non empty finite set
[S,C] is non empty set
{S} is non empty trivial finite 1 -element set
{{S,C},{S}} is non empty finite finite-membered V267() V268() set
EuG is non empty set
EuG is Element of union n
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{EuG,[uG,(union n)]} is non empty finite set
[EuG,uG] is non empty set
{EuG,uG} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,uG},{EuG}} is non empty finite finite-membered V267() V268() set
[: { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } , { [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } :] is set
bool [: { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } , { [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } :] is non empty V233() V267() subset-closed V329() V332() set
E is Relation-like { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } -defined { [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } -valued Function-like quasi_total Element of bool [: { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } , { [b₁,b₂] where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } :]
dom E is set
S is set
C is set
E . S is set
E . C is set
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
uG is Element of union n
se is Element of union n
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{uG,[se,(union n)]} is non empty finite set
{uG,se} is non empty finite set
[EuG,EuG] is non empty set
{EuG} is non empty trivial finite 1 -element set
{{EuG,EuG},{EuG}} is non empty finite finite-membered V267() V268() set
[uG,se] is non empty set
{uG} is non empty trivial finite 1 -element set
{{uG,se},{uG}} is non empty finite finite-membered V267() V268() set
rng E is set
S is set
C is set
E . C is set
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
[EuG,EuG] is non empty set
{EuG} is non empty trivial finite 1 -element set
{{EuG,EuG},{EuG}} is non empty finite finite-membered V267() V268() set
S is set
C is Element of union n
EuG is Element of union n
[C,EuG] is non empty set
{C,EuG} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,EuG},{C}} is non empty finite finite-membered V267() V268() set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
E . {C,[EuG,(union n)]} is set
len { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is epsilon-transitive epsilon-connected ordinal cardinal set
n is finite set
(n) is finite Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n is set
n is finite () set
union n is set
n is set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
len G is epsilon-transitive epsilon-connected ordinal cardinal set
n is set
union n is set
G is set
S is set
S is set
n is set
union n is set
len (union n) is epsilon-transitive epsilon-connected ordinal cardinal set
(n) is Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
S is set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
n is set
{n} is non empty trivial finite 1 -element set
n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
union { b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
S is set
c₄ is set
MG is finite Element of bool n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is set
{S} is non empty trivial finite 1 -element set
card {S} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
n is finite-membered set
G is set
n /\ G is set
S is set
n \ G is finite-membered Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is set
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
n is non empty V233() V267() subset-closed set
G is set
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is finite-membered (n) set
S is finite Element of G
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is finite-membered (n) set
S is finite-membered (n) set
G \/ S is finite-membered set
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is finite-membered (n) set
S is set
G /\ S is finite-membered set
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
G \ S is finite-membered Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is finite-membered (n) set
bool G is non empty V233() V267() subset-closed V329() V332() set
S is finite-membered Element of bool G
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is set
n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
{G} is non empty trivial finite 1 -element set
S is set
n is set
{n} is non empty trivial finite 1 -element set
{{},{n}} is non empty finite finite-membered set
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
S is set
c₄ is set
n is finite finite-membered set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is finite set
(n) is finite Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite finite-membered V233() V267() subset-closed (1) () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card n is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total finite Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty finite set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty finite set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
len (singletons (union n)) is epsilon-transitive epsilon-connected ordinal cardinal set
S is set
c₄ is finite Element of bool (union n)
MG is set
{MG} is non empty trivial finite 1 -element set
n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
G is set
S is finite set
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
c₄ is set
{c₄} is non empty trivial finite 1 -element set
MG is Element of bool (union n)
G is set
S is set
S is Element of bool (union n)
c₄ is set
{c₄} is non empty trivial finite 1 -element set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
S is Element of bool (union n)
c₄ is set
{c₄} is non empty trivial finite 1 -element set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
{G} is non empty trivial finite 1 -element set
{{},{G}} is non empty finite finite-membered set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
S is set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{G}} is non empty trivial finite finite-membered 1 -element V267() V268() set
{{}} \/ (singletons (union n)) is non empty set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
n is set
singletons n is Element of bool the U1 of (bspace n)
bspace n is V48() L15( Z_2 )
bool n is non empty V233() V267() subset-closed V329() V332() set
bspace-sum n is Relation-like [:(bool n),(bool n):] -defined bool n -valued Function-like non empty V14([:(bool n),(bool n):]) quasi_total Element of bool [:[:(bool n),(bool n):],(bool n):]
[:(bool n),(bool n):] is non empty set
[:[:(bool n),(bool n):],(bool n):] is non empty set
bool [:[:(bool n),(bool n):],(bool n):] is non empty V233() V267() subset-closed V329() V332() set
{} n is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool n
bspace-scalar-mult n is Relation-like [: the U1 of Z_2,(bool n):] -defined bool n -valued Function-like V14([: the U1 of Z_2,(bool n):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool n):],(bool n):]
[: the U1 of Z_2,(bool n):] is set
[:[: the U1 of Z_2,(bool n):],(bool n):] is set
bool [:[: the U1 of Z_2,(bool n):],(bool n):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool n),(bspace-sum n),({} n),(bspace-scalar-mult n)) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace n) is set
bool the U1 of (bspace n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool n : b₁ is 1 -element } is set
{{}} \/ (singletons n) is non empty set
S is set
c₄ is set
MG is Element of bool n
n is set
{n} is non empty trivial finite 1 -element set
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
c₄ is Element of bool n
MG is set
{MG} is non empty trivial finite 1 -element set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
S is non empty finite-membered V233() V267() subset-closed (1) () set
union S is set
(S) is finite-membered (1) Element of bool S
bool S is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
len c₄ is epsilon-transitive epsilon-connected ordinal cardinal set
MG is Element of bool n
n is set
{n} is non empty trivial finite 1 -element set
c₄ is set
MG is set
n is Element of bool n
c₄ is set
{c₄} is non empty trivial finite 1 -element set
MG is Element of bool n
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is Element of union n
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of union n : {G,b₁} in (n) } is set
c₄ is set
MG is Element of union n
{G,MG} is non empty finite set
c₄ is Element of bool (union n)
MG is Element of union n
{G,MG} is non empty finite set
n is Element of union n
{G,n} is non empty finite set
S is Element of bool (union n)
c₄ is Element of bool (union n)
MG is set
{G,MG} is non empty finite set
MG is set
{G,MG} is non empty finite set
n is set
G is non empty finite-membered V233() V267() subset-closed (1) () set
union G is set
n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
S is set
c₄ is set
MG is finite Element of bool n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is finite Element of bool n
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is set
len S is epsilon-transitive epsilon-connected ordinal cardinal set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
c₄ is finite Element of bool n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
card {} is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of omega
S is non empty finite-membered V233() V267() subset-closed (1) () set
union S is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
S is set
c₄ is finite Element of bool (union n)
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is set
{MG} is non empty trivial finite 1 -element set
{MG,MG} is non empty finite set
MG is set
n is set
{MG,n} is non empty finite set
S is set
c₄ is finite set
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is finite set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
S is set
c₄ is finite Element of bool n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
G is set
{G} is non empty trivial finite 1 -element set
card {G} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
G is set
S is set
{G,S} is non empty finite set
card {G,S} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
({}) is non empty finite finite-membered V233() V267() subset-closed (1) () ( {} )
bool {} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {} : card b₁ <= 2 } is set
n is set
G is set
{n,G} is non empty finite set
n is set
{n} is non empty trivial finite 1 -element set
({n}) is non empty finite finite-membered V233() V267() subset-closed (1) () ({n})
bool {n} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {n} : card b₁ <= 2 } is set
{{},{n}} is non empty finite finite-membered set
G is set
S is trivial finite Element of bool {n}
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union ({n}) is finite set
G is set
n is set
G is set
{n,G} is non empty finite set
({n,G}) is non empty finite finite-membered V233() V267() subset-closed (1) () ({n,G})
bool {n,G} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {n,G} : card b₁ <= 2 } is set
{n} is non empty trivial finite 1 -element set
{G} is non empty trivial finite 1 -element set
{{},{n},{G},{n,G}} is non empty finite set
S is set
c₄ is finite Element of bool {n,G}
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union ({n,G}) is finite set
card {n,G} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
S is set
n is set
G is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
(G) is non empty finite-membered V233() V267() subset-closed (1) () (G)
bool G is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool G : card b₁ <= 2 } is set
S is set
c₄ is finite Element of bool n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
{G} is non empty trivial finite 1 -element set
({G}) is non empty finite finite-membered V233() V267() subset-closed (1) () ({G})
bool {G} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {G} : card b₁ <= 2 } is set
{{},{G}} is non empty finite finite-membered set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is finite-membered (1) Element of bool n
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
c₄ is non empty set
MG is set
n is set
len n is epsilon-transitive epsilon-connected ordinal cardinal set
H is finite set
card H is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (((union n)) \ (n)) is set
S is set
{S} is non empty trivial finite 1 -element set
{S,S} is non empty finite set
union ((union n)) is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((((union n)) \ (n))) is finite-membered (1) Element of bool (((union n)) \ (n))
bool (((union n)) \ (n)) is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
G is non empty finite-membered V233() V267() subset-closed (1) () set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union G is set
((union G)) is non empty finite-membered V233() V267() subset-closed (1) () ( union G)
bool (union G) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union G) : card b₁ <= 2 } is set
(G) is finite-membered (1) Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
((union G)) \ (G) is finite-membered (1) Element of bool ((union G))
bool ((union G)) is non empty V233() V267() subset-closed V329() V332() set
union (((union G)) \ (G)) is set
singletons (union (((union G)) \ (G))) is Element of bool the U1 of (bspace (union (((union G)) \ (G))))
bspace (union (((union G)) \ (G))) is V48() L15( Z_2 )
bool (union (((union G)) \ (G))) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union (((union G)) \ (G))) is Relation-like [:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):] -defined bool (union (((union G)) \ (G))) -valued Function-like non empty V14([:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):]) quasi_total Element of bool [:[:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):]
[:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):] is non empty set
[:[:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):] is non empty set
bool [:[:(bool (union (((union G)) \ (G)))),(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):] is non empty V233() V267() subset-closed V329() V332() set
{} (union (((union G)) \ (G))) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union (((union G)) \ (G)))
bspace-scalar-mult (union (((union G)) \ (G))) is Relation-like [: the U1 of Z_2,(bool (union (((union G)) \ (G)))):] -defined bool (union (((union G)) \ (G))) -valued Function-like V14([: the U1 of Z_2,(bool (union (((union G)) \ (G)))):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):]
[: the U1 of Z_2,(bool (union (((union G)) \ (G)))):] is set
[:[: the U1 of Z_2,(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):] is set
bool [:[: the U1 of Z_2,(bool (union (((union G)) \ (G)))):],(bool (union (((union G)) \ (G)))):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union (((union G)) \ (G)))),(bspace-sum (union (((union G)) \ (G)))),({} (union (((union G)) \ (G)))),(bspace-scalar-mult (union (((union G)) \ (G))))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union (((union G)) \ (G)))) is set
bool the U1 of (bspace (union (((union G)) \ (G)))) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union (((union G)) \ (G))) : b₁ is 1 -element } is set
{{}} \/ (singletons (union (((union G)) \ (G)))) is non empty set
((((union G)) \ (G))) is finite-membered (1) Element of bool (((union G)) \ (G))
bool (((union G)) \ (G)) is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ (singletons (union (((union G)) \ (G))))) \/ ((((union G)) \ (G))) is non empty set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
c₄ is set
MG is set
n is set
{MG,n} is non empty finite set
MG is set
n is set
{MG,n} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () set
S is non empty finite-membered V233() V267() subset-closed (1) () set
union S is set
((union S)) is non empty finite-membered V233() V267() subset-closed (1) () ( union S)
bool (union S) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union S) : card b₁ <= 2 } is set
(S) is finite-membered (1) Element of bool S
bool S is non empty V233() V267() subset-closed V329() V332() set
((union S)) \ (S) is finite-membered (1) Element of bool ((union S))
bool ((union S)) is non empty V233() V267() subset-closed V329() V332() set
union G is set
((union G)) is non empty finite-membered V233() V267() subset-closed (1) () ( union G)
bool (union G) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union G) : card b₁ <= 2 } is set
(G) is finite-membered (1) Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
((union G)) \ (G) is finite-membered (1) Element of bool ((union G))
bool ((union G)) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
G is set
S is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{G,S} is non empty finite set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
bool n is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is set
(n,G) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
c₄ is non empty set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n,(union n)) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool (union n)) is finite-membered V233() subset-closed (1) set
G is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
G /\ (union n) is set
(n,(G /\ (union n))) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool (G /\ (union n)) is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool (G /\ (union n))) is finite-membered V233() subset-closed (1) set
S is set
S is set
n is non empty finite finite-membered V233() V267() subset-closed (1) () set
G is set
(n,G) is non empty finite finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is finite set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(G,(union n)) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G /\ (bool (union n)) is finite-membered V233() subset-closed (1) set
c₄ is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
union (n,G) is set
c₄ is set
MG is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
union (n,G) is set
S is set
{S} is non empty trivial finite 1 -element set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
union (n,G) is set
(union n) /\ G is set
union (n /\ (bool G)) is set
union (bool G) is set
(union n) /\ (union (bool G)) is set
n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
union (n,G) is set
(union n) /\ G is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
S is set
G is set
c₄ is set
{S,c₄} is non empty finite set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
G is set
{G} is non empty trivial finite 1 -element set
(n,{G}) is non empty finite finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool {G} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool {G}) is finite finite-membered V233() subset-closed (1) set
{{},{G}} is non empty finite finite-membered set
c₄ is set
c₄ is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
{G,S} is non empty finite set
{G} is non empty trivial finite 1 -element set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
((union {{}})) is non empty finite finite-membered V233() V267() subset-closed (1) () ( union {{}})
bool (union {{}}) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union {{}}) : card b₁ <= 2 } is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
union (n) is set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () (n)
bool n is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
{G} is non empty trivial finite 1 -element set
{{},{G}} is non empty finite finite-membered set
({G}) is non empty finite finite-membered V233() V267() subset-closed (1) () () ({G})
bool {G} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {G} : card b₁ <= 2 } is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
{G} is non empty trivial finite 1 -element set
{S} is non empty trivial finite 1 -element set
{{},{G},{S},{G,S}} is non empty finite set
({G,S}) is non empty finite finite-membered V233() V267() subset-closed (1) () () ({G,S})
bool {G,S} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {G,S} : card b₁ <= 2 } is set
MG is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
{G} is non empty trivial finite 1 -element set
({G}) is non empty finite finite-membered V233() V267() subset-closed (1) () () ({G})
bool {G} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {G} : card b₁ <= 2 } is set
{{},{G}} is non empty finite finite-membered set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
union c₄ is finite set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
G is set
S is set
{G,S} is non empty finite set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
the non empty finite finite-membered V233() V267() subset-closed () (1) () set is non empty finite finite-membered V233() V267() subset-closed () (1) () set
n is non empty finite finite-membered V233() V267() subset-closed () (1) () set
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is finite set
card (union G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union n is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () set
card G is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card c₄ is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
S is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is finite set
card (union G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is set
G is set
bool n is non empty V233() V267() subset-closed V329() V332() set
{G} is non empty trivial finite 1 -element set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
union S is finite set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite finite-membered V233() V267() subset-closed () (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is finite set
card (union G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union n is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
(n,{G,S}) is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
bool {G,S} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool {G,S}) is finite finite-membered V233() subset-closed (1) set
MG is set
union (n,{G,S}) is finite set
n is set
{MG,n} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{MG,n} is non empty finite set
((union (n,{G,S}))) is non empty finite finite-membered V233() V267() subset-closed (1) () () () ( union (n,{G,S}))
bool (union (n,{G,S})) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n,{G,S})) : card b₁ <= 2 } is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is set
union n is set
S is set
c₄ is set
{S,c₄} is non empty finite set
(n,{S,c₄}) is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
bool {S,c₄} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool {S,c₄}) is finite finite-membered V233() subset-closed (1) set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
union MG is finite set
(union n) /\ {S,c₄} is finite set
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
G is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
bool n is non empty V233() V267() subset-closed V329() V332() set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
bool G is non empty V233() V267() subset-closed V329() V332() set
n is finite set
(n) is non empty finite finite-membered V233() V267() subset-closed (1) () () () (n)
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n) is finite set
card (union (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
bool (n) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool (n)
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
S is V267() a_partition of union n
bool n is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
(n,c₄) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool c₄ is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool c₄) is finite-membered V233() subset-closed (1) set
SmallestPartition (union n) is V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() Element of bool [:(union n),(union n):]
[:(union n),(union n):] is set
bool [:(union n),(union n):] is non empty V233() V267() subset-closed V329() V332() set
Class K38((union n)) is V267() a_partition of union n
S is non empty set
{ {b₁} where b₁ is Element of S : verum } is set
MG is set
n is Element of S
{n} is non empty trivial finite 1 -element set
(n,MG) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool MG is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool MG) is finite-membered V233() subset-closed (1) set
{{},{n}} is non empty finite finite-membered set
n is non empty finite finite-membered V233() V267() subset-closed () (1) () () set
union n is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
G is finite finite-membered V267() a_partition of union n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S is set
(n,S) is non empty finite finite-membered V233() V267() subset-closed (1) () () Element of bool n
bool S is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool S) is finite finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
S is V267() a_partition of union n
bool n is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
(n,c₄) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool c₄ is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool c₄) is finite-membered V233() subset-closed (1) set
c₄ is V267() (n) a_partition of union n
SmallestPartition (union n) is V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() Element of bool [:(union n),(union n):]
[:(union n),(union n):] is set
bool [:(union n),(union n):] is non empty V233() V267() subset-closed V329() V332() set
Class K38((union n)) is V267() a_partition of union n
S is non empty finite set
{ H₁(b₁) where b₁ is Element of S : S₁[b₁] } is set
MG is set
n is Element of union n
{n} is non empty trivial finite 1 -element set
(n,MG) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool MG is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool MG) is finite-membered V233() subset-closed (1) set
{{},{n}} is non empty finite finite-membered set
MG is V267() (n) a_partition of union n
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is Element of bool (union n)
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
c₄ is V267() (n) a_partition of union n
union (n,G) is set
MG is Element of bool (union n)
c₄ | MG is V267() a_partition of MG
{ (b₁ /\ MG) where b₁ is Element of c₄ : not b₁ misses MG } is set
H is set
n is V267() a_partition of union (n,G)
((n,G),H) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool (n,G)
bool (n,G) is non empty V233() V267() subset-closed V329() V332() set
bool H is non empty V233() V267() subset-closed V329() V332() set
(n,G) /\ (bool H) is finite-membered V233() subset-closed (1) set
S is set
union ((n,G),H) is set
C is set
EuG is Element of c₄
EuG /\ MG is Element of bool (union n)
(n,EuG) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool EuG is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool EuG) is finite-membered V233() subset-closed (1) set
union (n,EuG) is set
((union (n,EuG))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n,EuG))
bool (union (n,EuG)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n,EuG)) : card b₁ <= 2 } is set
{S,C} is non empty finite set
((union ((n,G),H))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union ((n,G),H))
bool (union ((n,G),H)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union ((n,G),H)) : card b₁ <= 2 } is set
H is V267() ((n,G)) a_partition of union (n,G)
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
G is V267() (n) a_partition of union n
len G is epsilon-transitive epsilon-connected ordinal cardinal set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite V267() (n) a_partition of union n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite V267() (n) a_partition of union n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is Element of bool (union n)
S is set
{S} is non empty trivial finite 1 -element set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is Element of bool (union n)
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool (union n)
S is set
{S} is non empty trivial finite 1 -element set
S is set
{S} is non empty trivial finite 1 -element set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed (n) Element of bool (union n)
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
c₄ is Element of bool (union n)
MG is set
n is set
{MG,n} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
S is set
c₄ is set
{S,c₄} is non empty finite set
bool n is non empty V233() V267() subset-closed V329() V332() set
(n,{S,c₄}) is non empty finite finite-membered V233() V267() subset-closed (1) () () () Element of bool n
bool {S,c₄} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool {S,c₄}) is finite finite-membered V233() subset-closed (1) set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () Element of bool n
union MG is finite set
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed (n) Element of bool (union n)
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is (n) Element of bool (union n)
bool G is non empty V233() V267() subset-closed V329() V332() set
S is Element of bool G
MG is Element of bool (union n)
n is set
H is set
{n,H} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
SmallestPartition (union n) is V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() Element of bool [:(union n),(union n):]
[:(union n),(union n):] is set
bool [:(union n),(union n):] is non empty V233() V267() subset-closed V329() V332() set
Class K38((union n)) is V267() a_partition of union n
S is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
id (union n) is Relation-like V21() V23() V24() V28() set
c₄ is set
Im ((id (union n)),c₄) is set
MG is Element of bool (union n)
n is set
H is set
{n,H} is non empty finite set
[c₄,n] is non empty set
{c₄,n} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,n},{c₄}} is non empty finite finite-membered V267() V268() set
[c₄,H] is non empty set
{c₄,H} is non empty finite set
{{c₄,H},{c₄}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
SmallestPartition (union n) is V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() Element of bool [:(union n),(union n):]
[:(union n),(union n):] is set
bool [:(union n),(union n):] is non empty V233() V267() subset-closed V329() V332() set
Class K38((union n)) is V267() a_partition of union n
the non empty finite finite-membered V233() V267() subset-closed (1) () () () set is non empty finite finite-membered V233() V267() subset-closed (1) () () () set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () set
union n is finite set
SmallestPartition (union n) is finite finite-membered V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() finite Element of bool [:(union n),(union n):]
[:(union n),(union n):] is finite set
bool [:(union n),(union n):] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
Class K38((union n)) is finite finite-membered V267() a_partition of union n
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
SmallestPartition (union n) is V267() a_partition of union n
K38((union n)) is Relation-like union n -defined union n -valued V14( union n) quasi_total V21() V23() V24() V28() Element of bool [:(union n),(union n):]
[:(union n),(union n):] is set
bool [:(union n),(union n):] is non empty V233() V267() subset-closed V329() V332() set
Class K38((union n)) is V267() a_partition of union n
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () () Element of bool n
union G is set
S is set
(n,S) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool S is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool S) is finite-membered V233() subset-closed (1) set
MG is set
union (n,S) is set
n is set
{MG,n} is non empty finite set
((union (n,S))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n,S))
bool (union (n,S)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n,S)) : card b₁ <= 2 } is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is V267() (n) a_partition of union n
S is Element of bool (union n)
G | S is V267() a_partition of S
{ (b₁ /\ S) where b₁ is Element of G : not b₁ misses S } is set
(n,S) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool S is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool S) is finite-membered V233() subset-closed (1) set
union (n,S) is set
S /\ (union n) is Element of bool (union n)
MG is V267() a_partition of union (n,S)
n is set
bool (union (n,S)) is non empty V233() V267() subset-closed V329() V332() set
H is Element of bool (union (n,S))
E is Element of G
E /\ S is Element of bool (union n)
S is set
C is set
{S,C} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
G is set
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
union n is set
S is V267() (n) a_partition of union n
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G /\ (union n) is set
MG is Element of bool (union n)
(n,MG) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool MG is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool MG) is finite-membered V233() subset-closed (1) set
union (n,G) is set
c₄ is finite V267() (n) a_partition of union n
c₄ | MG is finite V267() a_partition of MG
{ (b₁ /\ MG) where b₁ is Element of c₄ : not b₁ misses MG } is set
n is V267() ((n,G)) a_partition of union (n,G)
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
G is V267() (n) a_partition of union n
len G is epsilon-transitive epsilon-connected ordinal cardinal set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite V267() (n) a_partition of union n
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite V267() (n) a_partition of union n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite V267() (n) a_partition of union n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
G is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is set
bool (union G) is non empty V233() V267() subset-closed V329() V332() set
union n is set
S is Element of bool (union G)
(G,S) is non empty finite-membered V233() V267() subset-closed (1) () () Element of bool G
bool G is non empty V233() V267() subset-closed V329() V332() set
bool S is non empty V233() V267() subset-closed V329() V332() set
G /\ (bool S) is finite-membered V233() subset-closed (1) set
MG is finite V267() (G) a_partition of union G
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
MG | S is finite V267() a_partition of S
{ (b₁ /\ S) where b₁ is Element of MG : not b₁ misses S } is set
E is set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed (n) Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed (n) Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
H is finite V267() (n) a_partition of union n
E is V267() a_partition of union n
S is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
C is Element of bool (union n)
EuG is set
uG is set
{EuG,uG} is non empty finite set
EuG is Element of bool (union n)
card H is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card (MG | S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is finite set
(n) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () (n)
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool n : card b₁ <= 2 } is set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
SmallestPartition n is finite finite-membered V267() a_partition of n
K38(n) is Relation-like n -defined n -valued V14(n) quasi_total V21() V23() V24() V28() finite Element of bool [:n,n:]
[:n,n:] is finite set
bool [:n,n:] is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
Class K38(n) is finite finite-membered V267() a_partition of n
card (SmallestPartition n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union (n) is finite set
MG is finite finite-membered V267() a_partition of union (n)
n is set
bool (union (n)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
H is finite Element of bool (union (n))
E is set
S is set
{E,S} is non empty finite set
{ {b₁} where b₁ is Element of n : verum } is set
C is Element of n
{C} is non empty trivial finite 1 -element set
n is finite finite-membered V267() ((n)) a_partition of union (n)
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
H is set
E is set
S is set
bool (union (n)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{E,S} is non empty finite set
C is finite Element of bool (union (n))
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G is finite V267() (n) a_partition of union n
card G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is set
union G is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
MG is Element of bool (union n)
{MG} is non empty trivial finite 1 -element set
G \ {MG} is finite Element of bool (bool (union n))
bool (bool (union n)) is non empty V233() V267() subset-closed V329() V332() set
H is set
E is Element of union n
(n,E) is Element of bool (union n)
S is Element of G
card {MG} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
(card G) - (card {MG}) is V83() ext-real V216() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(card G) - 1 is V83() ext-real V216() set
((card G) - 1) + 1 is V83() ext-real V216() set
C is set
{C} is non empty trivial finite 1 -element set
E is set
S is Element of union n
(n,S) is Element of bool (union n)
C is Element of G
EuG is Element of union n
EuG is Element of union n
(n,EuG) is Element of bool (union n)
E is Relation-like Function-like set
dom E is set
H is non empty set
S is non empty finite set
{ H₁(b₁) where b₁ is Element of S : S₂[b₁] } is set
EuG is set
{EuG} is non empty trivial finite 1 -element set
E " {EuG} is set
EuG \/ (E " {EuG}) is set
EuG is set
uG is Element of S
{uG} is non empty trivial finite 1 -element set
E " {uG} is set
uG \/ (E " {uG}) is set
union { H₁(b₁) where b₁ is Element of S : S₂[b₁] } is set
EuG is set
uG is set
EuG is set
uG is set
se is Element of union n
E . se is set
{(E . se)} is non empty trivial finite 1 -element set
E " {(E . se)} is set
(E . se) \/ (E " {(E . se)}) is set
{uG} is non empty trivial finite 1 -element set
E " {uG} is set
uG \/ (E " {uG}) is set
EuG is Element of bool (union n)
uG is Element of S
{uG} is non empty trivial finite 1 -element set
E " {uG} is set
uG \/ (E " {uG}) is set
se is Element of bool (union n)
sev is Element of S
{sev} is non empty trivial finite 1 -element set
E " {sev} is set
sev \/ (E " {sev}) is set
csev is set
uG is set
EuG is V267() a_partition of union n
sev is Element of S
{sev} is non empty trivial finite 1 -element set
E " {sev} is set
sev \/ (E " {sev}) is set
se is Element of bool (union n)
csev is set
Ecse is set
{csev,Ecse} is non empty finite set
E . Ecse is set
sf is Element of union n
(n,sf) is Element of bool (union n)
Ecse is Element of union n
{Ecse,sf} is non empty finite set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
E . csev is set
Ecse is Element of union n
(n,Ecse) is Element of bool (union n)
sf is Element of union n
{sf,Ecse} is non empty finite set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
card S is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
card {MG} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
(card G) - (card {MG}) is V83() ext-real V216() set
(card S) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(card G) - 1 is V83() ext-real V216() set
((card G) - 1) + 1 is V83() ext-real V216() set
se is Relation-like Function-like set
dom se is set
rng se is set
uG is V267() (n) a_partition of union n
sev is set
csev is set
se . csev is set
{csev} is non empty trivial finite 1 -element set
E " {csev} is set
csev \/ (E " {csev}) is set
[:S,uG:] is set
bool [:S,uG:] is non empty V233() V267() subset-closed V329() V332() set
sev is Relation-like S -defined uG -valued Function-like quasi_total finite Element of bool [:S,uG:]
csev is set
dom sev is finite set
Ecse is set
sev . csev is set
sev . Ecse is set
{csev} is non empty trivial finite 1 -element set
E " {csev} is set
csev \/ (E " {csev}) is set
{Ecse} is non empty trivial finite 1 -element set
E " {Ecse} is set
Ecse \/ (E " {Ecse}) is set
Ecse is set
sf is Element of bool (union n)
Ecse is set
sf is Element of bool (union n)
rng sev is finite set
csev is set
Ecse is Element of S
{Ecse} is non empty trivial finite 1 -element set
E " {Ecse} is set
Ecse \/ (E " {Ecse}) is set
sev . Ecse is Element of uG
len uG is epsilon-transitive epsilon-connected ordinal cardinal set
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () set
union G is finite set
bool (union G) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S is finite set
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite (G) Element of bool (union G)
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
{} S is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of bool S
bool S is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
card {} is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of omega
c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
MG is finite (G) Element of bool (union G)
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite (n) Element of bool (union n)
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is finite (n) Element of bool (union n)
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is finite (n) Element of bool (union n)
card G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is (n) Element of bool (union n)
bool S is non empty V233() V267() subset-closed V329() V332() set
c₄ is finite Element of bool S
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
{{},{{}}} is non empty finite finite-membered set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G is finite (n) Element of bool (union n)
card G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite (n) Element of bool (union n)
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is finite (n) Element of bool (union n)
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is finite (n) Element of bool (union n)
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite (n) Element of bool (union n)
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite (n) Element of bool (union n)
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed () (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is set
G is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{G} is non empty trivial finite 1 -element set
S is finite Element of bool (union n)
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is set
S is set
c₄ is set
{S,c₄} is non empty finite set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
card {S,c₄} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
S is non empty finite-membered V233() V267() subset-closed (1) () () () set
bool S is non empty V233() V267() subset-closed V329() V332() set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool S
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
union S is set
bool (union S) is non empty V233() V267() subset-closed V329() V332() set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
n is finite (S) Element of bool (union S)
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
C is finite Element of bool (union n)
card C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
C is finite Element of bool (union n)
card C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(S,C) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool S
bool C is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S /\ (bool C) is finite finite-membered V233() subset-closed (1) set
((S,C)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (S,C) is finite set
card (union (S,C)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
max ((S),(S)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(max ((S),(S))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
EuG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
EuG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
uG is finite Element of bool (union n)
card uG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
H is finite Element of bool (union n)
uG \/ H is finite Element of bool (union n)
MG is finite Element of bool (union n)
(uG \/ H) \/ MG is finite Element of bool (union n)
uG \/ n is finite set
(uG \/ n) \/ MG is finite set
sev is finite Element of bool (union n)
card sev is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(S) is finite-membered (1) Element of bool S
{ {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & {b₁,b₂} in (S) ) } is set
{ {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & not {b₁,b₂} in (S) ) } is set
the_subsets_of_card (2,sev) is finite finite-membered Element of bool (bool sev)
bool sev is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
bool (bool sev) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool sev : len b₁ = 2 } is set
{ { {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & {b₁,b₂} in (S) ) } , { {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & not {b₁,b₂} in (S) ) } } is non empty finite set
w is set
S is Element of union n
S is Element of union n
{S,S} is non empty finite set
len w is epsilon-transitive epsilon-connected ordinal cardinal set
w is set
S is Element of union n
S is Element of union n
{S,S} is non empty finite set
len w is epsilon-transitive epsilon-connected ordinal cardinal set
w is set
S is set
S is Element of union n
p is Element of union n
{S,p} is non empty finite set
x is Element of union n
y is Element of union n
{x,y} is non empty finite set
bool (the_subsets_of_card (2,sev)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
w is set
union { { {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & {b₁,b₂} in (S) ) } , { {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & not {b₁,b₂} in (S) ) } } is set
w is set
S is set
w is set
S is finite Element of bool sev
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is set
p is set
{S,p} is non empty finite set
x is Element of union n
y is Element of union n
{x,y} is non empty finite set
w is finite finite-membered Element of bool (the_subsets_of_card (2,sev))
S is set
S is set
p is Element of union n
x is Element of union n
{p,x} is non empty finite set
S is set
S is set
p is Element of union n
x is Element of union n
{p,x} is non empty finite set
S is finite finite-membered Element of bool (the_subsets_of_card (2,sev))
{ {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & {b₁,b₂} in (S) ) } /\ { {b₁,b₂} where b₁, b₂ is Element of union n : ( not b₁ = b₂ & b₁ in sev & b₂ in sev & not {b₁,b₂} in (S) ) } is set
S is set
p is Element of union n
x is Element of union n
{p,x} is non empty finite set
y is Element of union n
xx is Element of union n
{y,xx} is non empty finite set
w is finite finite-membered V267() a_partition of the_subsets_of_card (2,sev)
card w is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is finite Element of bool sev
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is finite Element of bool (union n)
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
the_subsets_of_card (2,S) is finite finite-membered Element of bool (bool S)
bool S is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
bool (bool S) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool S : len b₁ = 2 } is set
p is finite Element of w
(S,S) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool S
S /\ (bool S) is finite finite-membered V233() subset-closed (1) set
union (S,S) is finite set
y is set
xx is set
{y,xx} is non empty finite set
card {y,xx} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
yy is Element of union n
yy is Element of union n
{yy,yy} is non empty finite set
((S,S)) is finite finite-membered (1) Element of bool (S,S)
bool (S,S) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
((S,S)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union (S,S)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
x is set
y is set
{x,y} is non empty finite set
card {x,y} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
xx is Element of union n
yy is Element of union n
{xx,yy} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
bool n is non empty V233() V267() subset-closed V329() V332() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () () Element of bool n
union G is set
MG is set
n is set
c₄ is Element of bool (union (n))
{MG,n} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite-membered V233() V267() subset-closed (1) () () Element of bool (n)
union G is set
((n)) is non empty finite-membered V233() V267() subset-closed (1) () set
union (n) is set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
union ((n)) is set
bool (union ((n))) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is (n) Element of bool (union n)
((n),G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool (n)
bool G is non empty V233() V267() subset-closed V329() V332() set
(n) /\ (bool G) is finite-membered V233() subset-closed (1) set
union ((n),G) is set
n is set
H is set
{n,H} is non empty finite set
(((n),G)) is finite-membered (1) Element of bool ((n),G)
bool ((n),G) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
G is ((n)) Element of bool (union (n))
(n,G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
bool G is non empty V233() V267() subset-closed V329() V332() set
n /\ (bool G) is finite-membered V233() subset-closed (1) set
((n)) is non empty finite-membered V233() V267() subset-closed (1) () set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
(((n)),G) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool ((n))
bool ((n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) /\ (bool G) is finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool n
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union (n) is set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
MG is finite ((n)) Element of bool (union (n))
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is finite ((n)) Element of bool (union (n))
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n,n) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool n) is finite finite-membered V233() subset-closed (1) set
((n,n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n,n) is finite set
card (union (n,n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
S is finite (n) Element of bool (union n)
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union (n) is set
((n),S) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
bool S is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
(n) /\ (bool S) is finite finite-membered V233() subset-closed (1) set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool n
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union (n) is set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
n is finite ((n)) Element of bool (union (n))
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
H is finite ((n)) Element of bool (union (n))
(n,H) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool n
bool H is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool H) is finite finite-membered V233() subset-closed (1) set
((n,H)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n,H) is finite set
card (union (n,H)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card H is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((n)) is non empty finite-membered V233() V267() subset-closed (1) () () set
union (n) is set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
G is V267() ((n)) a_partition of union (n)
c₄ is set
((n),c₄) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
bool c₄ is non empty V233() V267() subset-closed V329() V332() set
(n) /\ (bool c₄) is finite-membered V233() subset-closed (1) set
union ((n),c₄) is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
G is V267() (n) a_partition of union n
((n)) is non empty finite-membered V233() V267() subset-closed (1) () set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
G is V267() (n) a_partition of union n
c₄ is set
((n),c₄) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
bool c₄ is non empty V233() V267() subset-closed V329() V332() set
(n) /\ (bool c₄) is finite-membered V233() subset-closed (1) set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
G is V267() ((n)) a_partition of union (n)
((n)) is non empty finite-membered V233() V267() subset-closed (1) () set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
G is V267() (n) a_partition of union n
union (n) is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
G is V267() (n) a_partition of union n
union (n) is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n) is set
c₄ is finite V267() ((n)) a_partition of union (n)
card c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is finite V267() (n) a_partition of union n
card MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
((union n)) is non empty finite-membered V233() V267() subset-closed (1) () () ( union n)
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union n) : card b₁ <= 2 } is set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
((union n)) \ (n) is finite-membered (1) Element of bool ((union n))
bool ((union n)) is non empty V233() V267() subset-closed V329() V332() set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((n)) is non empty finite-membered V233() V267() subset-closed (1) () () set
union (n) is set
((union (n))) is non empty finite-membered V233() V267() subset-closed (1) () () ( union (n))
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (n)) : card b₁ <= 2 } is set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
((union (n))) \ ((n)) is finite-membered (1) Element of bool ((union (n)))
bool ((union (n))) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
E is set
S is set
C is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{C} is non empty trivial finite 1 -element set
C is set
EuG is set
{C,EuG} is non empty finite set
{C} is non empty trivial finite 1 -element set
{EuG} is non empty trivial finite 1 -element set
C is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
{C,EuG} is non empty finite set
{C} is non empty trivial finite 1 -element set
{[EuG,(union n)]} is non empty trivial finite 1 -element V267() V268() set
C is Element of union n
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{(union n),[C,(union n)]} is non empty finite set
{[C,(union n)]} is non empty trivial finite 1 -element V267() V268() set
H is non empty set
E is set
len E is epsilon-transitive epsilon-connected ordinal cardinal set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
S is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{S} is non empty trivial finite 1 -element set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
S is Element of union n
C is Element of union n
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{S,[C,(union n)]} is non empty finite set
{S,C} is non empty finite set
card {S,[C,(union n)]} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
S is Element of union n
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{(union n),[S,(union n)]} is non empty finite set
card {(union n),[S,(union n)]} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
c₄ is set
singletons (union n) is Element of bool the U1 of (bspace (union n))
bspace (union n) is V48() L15( Z_2 )
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union n) is Relation-like [:(bool (union n)),(bool (union n)):] -defined bool (union n) -valued Function-like non empty V14([:(bool (union n)),(bool (union n)):]) quasi_total Element of bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):]
[:(bool (union n)),(bool (union n)):] is non empty set
[:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty set
bool [:[:(bool (union n)),(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
{} (union n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed (n) Element of bool (union n)
bspace-scalar-mult (union n) is Relation-like [: the U1 of Z_2,(bool (union n)):] -defined bool (union n) -valued Function-like V14([: the U1 of Z_2,(bool (union n)):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):]
[: the U1 of Z_2,(bool (union n)):] is set
[:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is set
bool [:[: the U1 of Z_2,(bool (union n)):],(bool (union n)):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union n)),(bspace-sum (union n)),({} (union n)),(bspace-scalar-mult (union n))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union n)) is set
bool the U1 of (bspace (union n)) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union n) : b₁ is 1 -element } is set
{{}} \/ (singletons (union n)) is non empty set
({{}} \/ (singletons (union n))) \/ (n) is non empty set
MG is Element of bool (union n)
n is set
{n} is non empty trivial finite 1 -element set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
union (n) is set
G is set
n is set
H is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{H} is non empty trivial finite 1 -element set
E is set
S is set
[E,S] is non empty set
{E,S} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,S},{E}} is non empty finite finite-membered V267() V268() set
H is set
E is set
{H,E} is non empty finite set
H is Element of union n
E is Element of union n
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{H,[E,(union n)]} is non empty finite set
{H,E} is non empty finite set
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{(union n),[H,(union n)]} is non empty finite set
n is set
{n} is non empty trivial finite 1 -element set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
union (n) is set
MG is set
[:(union n),{(union n)}:] \/ {(union n)} is non empty set
(union n) \/ ([:(union n),{(union n)}:] \/ {(union n)}) is non empty set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
MG is set
n is set
H is set
[n,H] is non empty set
{n,H} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,H},{n}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
union (n) is set
n is non empty finite finite-membered V233() V267() subset-closed () (1) () () () () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
{(union n)} is non empty trivial finite finite-membered 1 -element set
[:(union n),{(union n)}:] is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed set
(union n) \/ [:(union n),{(union n)}:] is finite finite-membered V233() subset-closed set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty finite finite-membered set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite finite-membered (1) Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
{{},{(union n)}} is non empty finite finite-membered set
S is set
c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal FinSequence-like V83() ext-real non negative V216() Element of union n
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite finite-membered set
{c₄} is non empty trivial finite finite-membered 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
{(union n),[c₄,(union n)]} is non empty finite set
S is set
c₄ is epsilon-transitive epsilon-connected ordinal natural finite cardinal FinSequence-like V83() ext-real non negative V216() Element of union n
MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal FinSequence-like V83() ext-real non negative V216() Element of union n
[MG,(union n)] is non empty set
{MG,(union n)} is non empty finite finite-membered set
{MG} is non empty trivial finite finite-membered 1 -element set
{{MG,(union n)},{MG}} is non empty finite finite-membered V267() V268() set
{c₄,[MG,(union n)]} is non empty finite set
{c₄,MG} is non empty finite finite-membered set
S is set
c₄ is set
MG is set
{c₄,MG} is non empty finite set
{{(union n)}} is non empty trivial finite finite-membered 1 -element V267() V268() set
S is set
c₄ is finite Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{c₄} is non empty trivial finite finite-membered 1 -element set
S is set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is finite set
{(union n)} is non empty trivial finite finite-membered 1 -element set
[:(union n),{(union n)}:] is finite set
(union n) \/ [:(union n),{(union n)}:] is finite set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty finite set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite finite-membered (1) Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
{{},{(union n)}} is non empty finite finite-membered set
H is non empty set
{ H₁(b₁) where b₁ is Element of H : b₁ in H } is set
{ H₂(b₁,b₂) where b₁, b₂ is Element of H : ( b₁ in union n & b₂ in union n ) } is set
S is set
C is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
{C,EuG} is non empty finite set
{ H₃(b₁) where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : S₁[b₁] } is set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union n is finite set
{(union n)} is non empty trivial finite finite-membered 1 -element set
[:(union n),{(union n)}:] is finite set
(union n) \/ [:(union n),{(union n)}:] is finite set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty finite set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite finite-membered (1) Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n) is finite set
card (union (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 * (n)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
card [:(union n),{(union n)}:] is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is set
MG is set
n is set
[MG,n] is non empty set
{MG,n} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{{MG,n},{MG}} is non empty finite finite-membered V267() V268() set
c₄ is set
MG is set
[c₄,MG] is non empty set
{c₄,MG} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,MG},{c₄}} is non empty finite finite-membered V267() V268() set
card (((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
card ((union n) \/ [:(union n),{(union n)}:]) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(card ((union n) \/ [:(union n),{(union n)}:])) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(card (union n)) + (card [:(union n),{(union n)}:]) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((card (union n)) + (card [:(union n),{(union n)}:])) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is set
union (n) is set
E is set
S is set
{E,S} is non empty finite set
C is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{C} is non empty trivial finite 1 -element set
len G is epsilon-transitive epsilon-connected ordinal cardinal set
E is Element of union n
S is Element of union n
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{E,[S,(union n)]} is non empty finite set
{E,S} is non empty finite set
E is Element of union n
S is Element of union n
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{E,[S,(union n)]} is non empty finite set
{E,S} is non empty finite set
len G is epsilon-transitive epsilon-connected ordinal cardinal set
E is Element of union n
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{(union n),[E,(union n)]} is non empty finite set
E is Element of union n
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{(union n),[E,(union n)]} is non empty finite set
len G is epsilon-transitive epsilon-connected ordinal cardinal set
E is Element of union n
S is Element of union n
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{E,[S,(union n)]} is non empty finite set
{E,S} is non empty finite set
C is Element of union n
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{(union n),[C,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
(n) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
((n) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
MG is set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{(union n),[n,(union n)]} is non empty finite set
MG is set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{(union n),[n,(union n)]} is non empty finite set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union n is finite set
{(union n)} is non empty trivial finite finite-membered 1 -element set
[:(union n),{(union n)}:] is finite set
(union n) \/ [:(union n),{(union n)}:] is finite set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty finite set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite finite-membered (1) Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((n)) is finite finite-membered (1) Element of bool (n)
bool (n) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card ((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(3 * (n)) + (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
{{},{(union n)}} is non empty finite finite-membered set
n is set
union (n) is finite set
H is set
E is set
{H,E} is non empty finite set
n is non empty set
{ H₁(b₁) where b₁ is Element of n : b₁ in n } is set
{ H₂(b₁,b₂) where b₁, b₂ is Element of n : ( b₁ in union n & b₂ in union n ) } is set
S is set
C is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
{C,EuG} is non empty finite set
H is finite set
card H is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
S is finite set
card S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(n) \/ S is finite set
C is set
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{(union n),[EuG,(union n)]} is non empty finite set
EuG is set
uG is set
{EuG,uG} is non empty finite set
EuG is Element of union n
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{EuG,[uG,(union n)]} is non empty finite set
{EuG,uG} is non empty finite set
C is set
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
uG is set
se is set
{uG,se} is non empty finite set
((n) \/ S) \/ H is finite set
card (((n) \/ S) \/ H) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card ((n) \/ S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(card ((n) \/ S)) + (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(card (n)) + (2 * (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((card (n)) + (2 * (n))) + (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is set
S is set
{G,S} is non empty finite set
MG is Element of union n
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{MG,[n,(union n)]} is non empty finite set
{MG,n} is non empty finite set
MG is Element of union n
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{MG,[n,(union n)]} is non empty finite set
{MG,n} is non empty finite set
MG is Element of union n
[MG,(union n)] is non empty set
{MG,(union n)} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{{MG,(union n)},{MG}} is non empty finite finite-membered V267() V268() set
{(union n),[MG,(union n)]} is non empty finite set
MG is Element of union n
[MG,(union n)] is non empty set
{MG,(union n)} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{{MG,(union n)},{MG}} is non empty finite finite-membered V267() V268() set
{(union n),[MG,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is set
{(union n),G} is non empty finite set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
{[G,(union n)]} is non empty trivial finite 1 -element V267() V268() set
{[G,(union n)],(union n)} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
{[G,(union n)],(union n)} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
S is set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{[G,(union n)],[S,(union n)]} is non empty finite V267() V268() set
{{G},{G,(union n)}} is non empty finite finite-membered V267() V268() set
{{S},{S,(union n)}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
S is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{[G,(union n)],[S,(union n)]} is non empty finite V267() V268() set
card {[G,(union n)],[S,(union n)]} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
S is set
{[G,(union n)],S} is non empty finite set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
E is set
S is set
{E,S} is non empty finite set
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{(union n),[n,(union n)]} is non empty finite set
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{(union n),[n,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{G} is non empty trivial finite 1 -element set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
S is set
{[G,(union n)],S} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
H is set
n is set
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{[n,(union n)],H} is non empty finite set
{n,H} is non empty finite set
union (n) is set
singletons (union (n)) is Element of bool the U1 of (bspace (union (n)))
bspace (union (n)) is V48() L15( Z_2 )
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
bspace-sum (union (n)) is Relation-like [:(bool (union (n))),(bool (union (n))):] -defined bool (union (n)) -valued Function-like non empty V14([:(bool (union (n))),(bool (union (n))):]) quasi_total Element of bool [:[:(bool (union (n))),(bool (union (n))):],(bool (union (n))):]
[:(bool (union (n))),(bool (union (n))):] is non empty set
[:[:(bool (union (n))),(bool (union (n))):],(bool (union (n))):] is non empty set
bool [:[:(bool (union (n))),(bool (union (n))):],(bool (union (n))):] is non empty V233() V267() subset-closed V329() V332() set
{} (union (n)) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed ((n)) Element of bool (union (n))
bspace-scalar-mult (union (n)) is Relation-like [: the U1 of Z_2,(bool (union (n))):] -defined bool (union (n)) -valued Function-like V14([: the U1 of Z_2,(bool (union (n))):]) quasi_total Element of bool [:[: the U1 of Z_2,(bool (union (n))):],(bool (union (n))):]
[: the U1 of Z_2,(bool (union (n))):] is set
[:[: the U1 of Z_2,(bool (union (n))):],(bool (union (n))):] is set
bool [:[: the U1 of Z_2,(bool (union (n))):],(bool (union (n))):] is non empty V233() V267() subset-closed V329() V332() set
G15(Z_2,(bool (union (n))),(bspace-sum (union (n))),({} (union (n))),(bspace-scalar-mult (union (n)))) is V142( Z_2 ) L15( Z_2 )
the U1 of (bspace (union (n))) is set
bool the U1 of (bspace (union (n))) is non empty V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is Element of bool (union (n)) : b₁ is 1 -element } is set
{{}} \/ (singletons (union (n))) is non empty set
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ (singletons (union (n)))) \/ ((n)) is non empty set
E is Element of bool (union (n))
S is set
{S} is non empty trivial finite 1 -element set
len E is epsilon-transitive epsilon-connected ordinal cardinal set
(n) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
((n) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
E is Element of union n
S is Element of union n
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{E,[S,(union n)]} is non empty finite set
{E,S} is non empty finite set
E is Element of union n
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{(union n),[E,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
S is set
c₄ is set
{S,c₄} is non empty finite set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
{[S,(union n)],c₄} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
S is set
{G,S} is non empty finite set
{G} is non empty trivial finite 1 -element set
card {G,S} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
((n)) is finite-membered (1) Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
c₄ is set
[c₄,(union n)] is non empty set
{c₄,(union n)} is non empty finite set
{c₄} is non empty trivial finite 1 -element set
{{c₄,(union n)},{c₄}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n),(union n)) is non empty finite-membered V233() V267() subset-closed (1) () Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
(n) /\ (bool (union n)) is finite-membered V233() subset-closed (1) set
c₄ is set
n is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{n} is non empty trivial finite 1 -element set
n is Element of union n
H is Element of union n
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
{n,[H,(union n)]} is non empty finite set
{n,H} is non empty finite set
n is Element of union n
[n,(union n)] is non empty set
{n,(union n)} is non empty finite set
{n} is non empty trivial finite 1 -element set
{{n,(union n)},{n}} is non empty finite finite-membered V267() V268() set
{(union n),[n,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is finite set
card (union G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
c₄ is Element of union G
MG is set
n is set
{c₄,MG} is non empty finite set
{c₄,n} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
H is set
[H,(union n)] is non empty set
{H,(union n)} is non empty finite set
{H} is non empty trivial finite 1 -element set
{{H,(union n)},{H}} is non empty finite finite-membered V267() V268() set
E is set
[E,(union n)] is non empty set
{E,(union n)} is non empty finite set
{E} is non empty trivial finite 1 -element set
{{E,(union n)},{E}} is non empty finite finite-membered V267() V268() set
{MG,n} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
{{},{(union n)}} is non empty finite finite-membered set
union {{},{(union n)}} is finite set
{} \/ {(union n)} is non empty finite set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card {(union n)} is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
G is set
{G} is non empty trivial finite 1 -element set
[G,(union n)] is non empty set
{G,(union n)} is non empty finite set
{{G,(union n)},{G}} is non empty finite finite-membered V267() V268() set
{[G,(union n)]} is non empty trivial finite 1 -element V267() V268() set
{[G,(union n)],(union n)} is non empty finite set
{{},{G},{[G,(union n)]},{(union n)},{[G,(union n)],(union n)}} is non empty finite set
len (union n) is epsilon-transitive epsilon-connected ordinal cardinal set
E is set
{E} is non empty trivial finite 1 -element set
S is set
C is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{C} is non empty trivial finite 1 -element set
C is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{C,[EuG,(union n)]} is non empty finite set
{C,EuG} is non empty finite set
C is Element of union n
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
{(union n),[C,(union n)]} is non empty finite set
S is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
union (n) is set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
H is set
E is set
S is set
{H,E} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
EuG is Element of union n
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{EuG,[EuG,(union n)]} is non empty finite set
{EuG,EuG} is non empty finite set
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{(union n),[EuG,(union n)]} is non empty finite set
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{(union n),[EuG,(union n)]} is non empty finite set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union G is finite set
card (union G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union (n) is set
(G) is finite finite-membered (1) Element of bool G
bool G is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
((n)) is finite-membered (1) Element of bool (n)
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
MG is set
n is set
{n} is non empty trivial finite 1 -element set
n is set
H is set
{n,H} is non empty finite set
n is set
H is set
E is set
{n,H} is non empty finite set
{n,E} is non empty finite set
{H,E} is non empty finite set
S is set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
C is set
[C,(union n)] is non empty set
{C,(union n)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(union n)},{C}} is non empty finite finite-membered V267() V268() set
S is set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
S is set
[S,(union n)] is non empty set
{S,(union n)} is non empty finite set
{S} is non empty trivial finite 1 -element set
{{S,(union n)},{S}} is non empty finite finite-membered V267() V268() set
(G,(union n)) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool G
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
G /\ (bool (union n)) is finite finite-membered V233() subset-closed (1) set
union (G,(union n)) is finite set
{S} \/ (union (G,(union n))) is non empty finite set
(n,({S} \/ (union (G,(union n))))) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () Element of bool n
bool ({S} \/ (union (G,(union n)))) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
n /\ (bool ({S} \/ (union (G,(union n))))) is finite finite-membered V233() subset-closed (1) set
EuG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union EuG is finite set
EuG is set
(union n) /\ ({S} \/ (union (G,(union n)))) is finite set
EuG is set
(union n) /\ ({S} \/ (union (G,(union n)))) is finite set
(union G) /\ (union n) is finite set
(union n) /\ ({S} \/ (union (G,(union n)))) is finite set
{[S,(union n)],S} is non empty finite set
uG is set
se is set
EuG is non empty finite-membered V233() V267() subset-closed (1) () Element of bool n
union EuG is set
(union n) /\ ({S} \/ (union (G,(union n)))) is finite set
{uG,se} is non empty finite set
(union G) /\ (union n) is finite set
{[S,(union n)],se} is non empty finite set
{S,se} is non empty finite set
(union G) /\ (union n) is finite set
{[S,(union n)],uG} is non empty finite set
{S,uG} is non empty finite set
(union G) /\ (union n) is finite set
(EuG) is finite-membered (1) Element of bool EuG
bool EuG is non empty V233() V267() subset-closed V329() V332() set
{n} is non empty trivial finite 1 -element set
{n} \/ (union (G,(union n))) is non empty finite set
uG is set
{uG,[S,(union n)]} is non empty finite set
uG is set
(EuG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union EuG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
card (union (G,(union n))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
1 + (card (union (G,(union n)))) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
card ({n} \/ (union (G,(union n)))) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (n) is set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
union c₄ is finite set
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
MG is set
[MG,(union n)] is non empty set
{MG,(union n)} is non empty finite set
{MG} is non empty trivial finite 1 -element set
{{MG,(union n)},{MG}} is non empty finite finite-membered V267() V268() set
union (n) is set
{[MG,(union n)],(union n)} is non empty finite set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
{[MG,(union n)]} is non empty trivial finite 1 -element V267() V268() set
{{},{[MG,(union n)]},{(union n)},{[MG,(union n)],(union n)}} is non empty finite set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
({[MG,(union n)],(union n)}) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () ({[MG,(union n)],(union n)})
bool {[MG,(union n)],(union n)} is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool {[MG,(union n)],(union n)} : card b₁ <= 2 } is set
union n is finite set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
H is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(H) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union H is finite set
card (union H) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
c₄ is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool n
(c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union c₄ is finite set
card (union c₄) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
bool (n) is non empty V233() V267() subset-closed V329() V332() set
MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union MG is finite set
card (union MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
bool (n) is non empty V233() V267() subset-closed V329() V332() set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () Element of bool (n)
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union S is finite set
card (union S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
union (n) is set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
1 + (n) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is finite V267() (n) a_partition of union n
card n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in a₁ } is set
H is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in H } is set
H is Relation-like Function-like set
dom H is set
{ (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } is set
len { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } is epsilon-transitive epsilon-connected ordinal cardinal set
S is set
H . S is set
S \/ (H . S) is set
S is set
H . S is set
S \/ (H . S) is set
S is Relation-like Function-like set
dom S is set
rng S is set
C is set
EuG is set
S . EuG is set
H . EuG is set
EuG \/ (H . EuG) is set
[:n, { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } :] is set
bool [:n, { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } :] is non empty V233() V267() subset-closed V329() V332() set
C is Relation-like n -defined { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } -valued Function-like quasi_total finite Element of bool [:n, { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } :]
rng C is finite set
EuG is set
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
C . EuG is set
EuG is set
dom C is finite set
EuG is set
C . EuG is set
C . EuG is set
H . EuG is set
EuG \/ (H . EuG) is set
H . EuG is set
EuG \/ (H . EuG) is set
uG is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
se is Element of union n
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
uG is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
se is Element of union n
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{{(union n)}} is non empty trivial finite finite-membered 1 -element V267() V268() set
{ (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } \/ {{(union n)}} is non empty set
union ( { (b₁ \/ (H . b₁)) where b₁ is Element of n : b₁ in n } \/ {{(union n)}}) is set
C is set
EuG is set
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{C} is non empty trivial finite 1 -element set
C is set
EuG is set
union n is set
EuG is set
H . EuG is set
EuG \/ (H . EuG) is set
EuG is set
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
union n is set
uG is set
H . uG is set
uG \/ (H . uG) is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in uG } is set
EuG is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)}
{EuG} is non empty trivial finite 1 -element set
uG is set
se is set
[uG,se] is non empty set
{uG,se} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,se},{uG}} is non empty finite finite-membered V267() V268() set
EuG is set
uG is set
{EuG,uG} is non empty finite set
EuG is Element of union n
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{EuG,[uG,(union n)]} is non empty finite set
{EuG,uG} is non empty finite set
EuG is Element of union n
[EuG,(union n)] is non empty set
{EuG,(union n)} is non empty finite set
{EuG} is non empty trivial finite 1 -element set
{{EuG,(union n)},{EuG}} is non empty finite finite-membered V267() V268() set
{(union n),[EuG,(union n)]} is non empty finite set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
C is Element of bool (union (n))
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
EuG is Element of bool (union (n))
uG is Element of n
H . uG is set
uG \/ (H . uG) is set
se is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in uG } is set
sev is Element of union n
[sev,(union n)] is non empty set
{sev,(union n)} is non empty finite set
{sev} is non empty trivial finite 1 -element set
{{sev,(union n)},{sev}} is non empty finite finite-membered V267() V268() set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
sev is Element of union n
[sev,(union n)] is non empty set
{sev,(union n)} is non empty finite set
{sev} is non empty trivial finite 1 -element set
{{sev,(union n)},{sev}} is non empty finite finite-membered V267() V268() set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
sev is Element of union n
[sev,(union n)] is non empty set
{sev,(union n)} is non empty finite set
{sev} is non empty trivial finite 1 -element set
{{sev,(union n)},{sev}} is non empty finite finite-membered V267() V268() set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in uG } is set
csev is Element of union n
[csev,(union n)] is non empty set
{csev,(union n)} is non empty finite set
{csev} is non empty trivial finite 1 -element set
{{csev,(union n)},{csev}} is non empty finite finite-membered V267() V268() set
uG is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
se is Element of union n
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } is set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } \/ {{(union n)}} is non empty set
EuG is Element of bool (union (n))
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
uG is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
se is Element of union n
[se,(union n)] is non empty set
{se,(union n)} is non empty finite set
{se} is non empty trivial finite 1 -element set
{{se,(union n)},{se}} is non empty finite finite-membered V267() V268() set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } is set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } \/ {{(union n)}} is non empty set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } is set
{ (b₂ \/ (H . b₂)) where b₁ is Element of n : b₂ in n } \/ {{(union n)}} is non empty set
C is set
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
EuG is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
{EuG} is non empty trivial finite 1 -element set
C is V267() a_partition of union (n)
EuG is set
EuG is Element of bool (union (n))
uG is set
se is set
{uG,se} is non empty finite set
sev is Element of n
H . sev is set
sev \/ (H . sev) is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in sev } is set
csev is Element of union n
[csev,(union n)] is non empty set
{csev,(union n)} is non empty finite set
{csev} is non empty trivial finite 1 -element set
{{csev,(union n)},{csev}} is non empty finite finite-membered V267() V268() set
{csev,uG} is non empty finite set
csev is Element of union n
[csev,(union n)] is non empty set
{csev,(union n)} is non empty finite set
{csev} is non empty trivial finite 1 -element set
{{csev,(union n)},{csev}} is non empty finite finite-membered V267() V268() set
{csev,se} is non empty finite set
csev is Element of union n
[csev,(union n)] is non empty set
{csev,(union n)} is non empty finite set
{csev} is non empty trivial finite 1 -element set
{{csev,(union n)},{csev}} is non empty finite finite-membered V267() V268() set
Ecse is Element of union n
[Ecse,(union n)] is non empty set
{Ecse,(union n)} is non empty finite set
{Ecse} is non empty trivial finite 1 -element set
{{Ecse,(union n)},{Ecse}} is non empty finite finite-membered V267() V268() set
EuG is V267() ((n)) a_partition of union (n)
len EuG is epsilon-transitive epsilon-connected ordinal cardinal set
EuG is Element of n
H . EuG is set
EuG \/ (H . EuG) is set
{ [b₁,(union n)] where b₁ is Element of union n : b₁ in EuG } is set
uG is Element of union n
[uG,(union n)] is non empty set
{uG,(union n)} is non empty finite set
{uG} is non empty trivial finite 1 -element set
{{uG,(union n)},{uG}} is non empty finite finite-membered V267() V268() set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
union (n) is set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
1 + (n) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
G is V267() ((n)) a_partition of union (n)
len G is epsilon-transitive epsilon-connected ordinal cardinal set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
((n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
1 + (n) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
union (n) is set
H is V267() ((n)) a_partition of union (n)
len H is epsilon-transitive epsilon-connected ordinal cardinal set
E is finite V267() ((n)) a_partition of union (n)
card E is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
union E is set
((n),(union n)) is non empty finite-membered V233() V267() subset-closed (1) () () Element of bool (n)
bool (n) is non empty V233() V267() subset-closed V329() V332() set
bool (union n) is non empty V233() V267() subset-closed V329() V332() set
(n) /\ (bool (union n)) is finite-membered V233() subset-closed (1) set
bool (union (n)) is non empty V233() V267() subset-closed V329() V332() set
S is Element of bool (union (n))
E | S is finite V267() a_partition of S
{ (b₁ /\ S) where b₁ is Element of E : not b₁ misses S } is set
C is finite V267() (n) a_partition of union n
card C is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
EuG is set
EuG is Element of bool (union (n))
EuG /\ S is Element of bool (union (n))
sev is Element of union n
(n,sev) is Element of bool (union n)
uG is Element of union (n)
[sev,uG] is non empty set
{sev,uG} is non empty finite set
{sev} is non empty trivial finite 1 -element set
{{sev,uG},{sev}} is non empty finite finite-membered V267() V268() set
{[sev,uG]} is non empty trivial finite 1 -element V267() V268() set
csev is Element of union (n)
Ecse is set
Ecse is Element of bool (union (n))
Ecse /\ S is Element of bool (union (n))
w is Element of union n
{sev,w} is non empty finite set
{csev,w} is non empty finite set
{csev,uG} is non empty finite set
((n)) is finite-membered (1) Element of bool (n)
n is non empty finite-membered V233() V267() subset-closed (1) () set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
S is set
c₄ is non empty finite-membered V233() V267() subset-closed (1) () set
(c₄) is non empty finite-membered V233() V267() subset-closed (1) () set
union c₄ is set
{(union c₄)} is non empty trivial finite 1 -element set
[:(union c₄),{(union c₄)}:] is set
(union c₄) \/ [:(union c₄),{(union c₄)}:] is set
((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} is non empty set
{ {b₁} where b₁ is Element of ((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} : verum } is non empty set
(c₄) is finite-membered (1) Element of bool c₄
bool c₄ is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} : verum } ) \/ (c₄) is non empty set
{ {b₁,[b₂,(union c₄)]} where b₁, b₂ is Element of union c₄ : {b₁,b₂} in (c₄) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} : verum } ) \/ (c₄)) \/ { {b₁,[b₂,(union c₄)]} where b₁, b₂ is Element of union c₄ : {b₁,b₂} in (c₄) } is non empty set
{ {(union c₄),[b₁,(union c₄)]} where b₁ is Element of union c₄ : b₁ in union c₄ } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union c₄) \/ [:(union c₄),{(union c₄)}:]) \/ {(union c₄)} : verum } ) \/ (c₄)) \/ { {b₁,[b₂,(union c₄)]} where b₁, b₂ is Element of union c₄ : {b₁,b₂} in (c₄) } ) \/ { {(union c₄),[b₁,(union c₄)]} where b₁ is Element of union c₄ : b₁ in union c₄ } is non empty set
G is Relation-like Function-like set
dom G is set
G . 0 is set
S is Relation-like NAT -defined Function-like set
c₄ is Relation-like NAT -defined Function-like V14( NAT ) set
MG is Relation-like NAT -defined Function-like V14( NAT ) set
MG . 0 is set
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
MG . n is set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
MG . (n + 1) is set
H is non empty finite-membered V233() V267() subset-closed (1) () set
(H) is non empty finite-membered V233() V267() subset-closed (1) () set
union H is set
{(union H)} is non empty trivial finite 1 -element set
[:(union H),{(union H)}:] is set
(union H) \/ [:(union H),{(union H)}:] is set
((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} is non empty set
{ {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } is non empty set
(H) is finite-membered (1) Element of bool H
bool H is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H) is non empty set
{ {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H)) \/ { {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } is non empty set
{ {(union H),[b₁,(union H)]} where b₁ is Element of union H : b₁ in union H } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H)) \/ { {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } ) \/ { {(union H),[b₁,(union H)]} where b₁ is Element of union H : b₁ in union H } is non empty set
c₄ . n is set
c₄ . (n + 1) is set
E is non empty finite-membered V233() V267() subset-closed (1) () set
(E) is non empty finite-membered V233() V267() subset-closed (1) () set
union E is set
{(union E)} is non empty trivial finite 1 -element set
[:(union E),{(union E)}:] is set
(union E) \/ [:(union E),{(union E)}:] is set
((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} is non empty set
{ {b₁} where b₁ is Element of ((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} : verum } is non empty set
(E) is finite-membered (1) Element of bool E
bool E is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} : verum } ) \/ (E) is non empty set
{ {b₁,[b₂,(union E)]} where b₁, b₂ is Element of union E : {b₁,b₂} in (E) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} : verum } ) \/ (E)) \/ { {b₁,[b₂,(union E)]} where b₁, b₂ is Element of union E : {b₁,b₂} in (E) } is non empty set
{ {(union E),[b₁,(union E)]} where b₁ is Element of union E : b₁ in union E } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union E) \/ [:(union E),{(union E)}:]) \/ {(union E)} : verum } ) \/ (E)) \/ { {b₁,[b₂,(union E)]} where b₁, b₂ is Element of union E : {b₁,b₂} in (E) } ) \/ { {(union E),[b₁,(union E)]} where b₁ is Element of union E : b₁ in union E } is non empty set
G is Relation-like NAT -defined Function-like V14( NAT ) set
S is Relation-like NAT -defined Function-like V14( NAT ) set
c₄ is Relation-like Function-like set
c₄ . 0 is set
MG is Relation-like Function-like set
MG . 0 is set
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
c₄ . n is set
MG . n is set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
c₄ . (n + 1) is set
MG . (n + 1) is set
H is non empty finite-membered V233() V267() subset-closed (1) () set
(H) is non empty finite-membered V233() V267() subset-closed (1) () set
union H is set
{(union H)} is non empty trivial finite 1 -element set
[:(union H),{(union H)}:] is set
(union H) \/ [:(union H),{(union H)}:] is set
((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} is non empty set
{ {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } is non empty set
(H) is finite-membered (1) Element of bool H
bool H is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H) is non empty set
{ {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H)) \/ { {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } is non empty set
{ {(union H),[b₁,(union H)]} where b₁ is Element of union H : b₁ in union H } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union H) \/ [:(union H),{(union H)}:]) \/ {(union H)} : verum } ) \/ (H)) \/ { {b₁,[b₂,(union H)]} where b₁, b₂ is Element of union H : {b₁,b₂} in (H) } ) \/ { {(union H),[b₁,(union H)]} where b₁ is Element of union H : b₁ in union H } is non empty set
n is set
c₄ . n is set
MG . n is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
(n) . 0 is set
G is Relation-like Function-like set
G . 0 is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is set
c₄ is Relation-like Function-like set
c₄ . 0 is set
(n) . 0 is set
MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . MG is set
MG + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (MG + 1) is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is set
n is non empty finite-membered V233() V267() subset-closed (1) () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (S + 1) is non empty finite-membered V233() V267() subset-closed (1) () set
(n) . S is non empty finite-membered V233() V267() subset-closed (1) () set
(((n) . S)) is non empty finite-membered V233() V267() subset-closed (1) () set
union ((n) . S) is set
{(union ((n) . S))} is non empty trivial finite 1 -element set
[:(union ((n) . S)),{(union ((n) . S))}:] is set
(union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:] is set
((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} is non empty set
{ {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } is non empty set
(((n) . S)) is finite-membered (1) Element of bool ((n) . S)
bool ((n) . S) is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S)) is non empty set
{ {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S))) \/ { {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } is non empty set
{ {(union ((n) . S)),[b₁,(union ((n) . S))]} where b₁ is Element of union ((n) . S) : b₁ in union ((n) . S) } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S))) \/ { {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } ) \/ { {(union ((n) . S)),[b₁,(union ((n) . S))]} where b₁ is Element of union ((n) . S) : b₁ in union ((n) . S) } is non empty set
MG is Relation-like Function-like set
MG . 0 is set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is non empty finite-membered V233() V267() subset-closed (1) () set
c₄ is Relation-like Function-like set
c₄ . 0 is set
(n) . 0 is non empty finite-membered V233() V267() subset-closed (1) () set
MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . MG is non empty finite-membered V233() V267() subset-closed (1) () set
MG + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (MG + 1) is non empty finite-membered V233() V267() subset-closed (1) () set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is non empty finite-membered V233() V267() subset-closed (1) () set
c₄ is Relation-like Function-like set
c₄ . 0 is set
(n) . 0 is non empty finite-membered V233() V267() subset-closed (1) () set
MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . MG is non empty finite-membered V233() V267() subset-closed (1) () set
MG + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (MG + 1) is non empty finite-membered V233() V267() subset-closed (1) () set
n is non empty finite-membered V233() V267() subset-closed (1) () () set
(n) is non empty finite-membered V233() V267() subset-closed (1) () () set
union n is set
{(union n)} is non empty trivial finite 1 -element set
[:(union n),{(union n)}:] is set
(union n) \/ [:(union n),{(union n)}:] is set
((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} is non empty set
{ {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } is non empty set
(n) is finite-membered (1) Element of bool n
bool n is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n) is non empty set
{ {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } is non empty set
{ {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union n) \/ [:(union n),{(union n)}:]) \/ {(union n)} : verum } ) \/ (n)) \/ { {b₁,[b₂,(union n)]} where b₁, b₂ is Element of union n : {b₁,b₂} in (n) } ) \/ { {(union n),[b₁,(union n)]} where b₁ is Element of union n : b₁ in union n } is non empty set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is non empty finite-membered V233() V267() subset-closed (1) () () () set
(n) . 0 is non empty finite-membered V233() V267() subset-closed (1) () () () set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . S is non empty finite-membered V233() V267() subset-closed (1) () () () set
S + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (S + 1) is non empty finite-membered V233() V267() subset-closed (1) () () () set
(((n) . S)) is non empty finite-membered V233() V267() subset-closed (1) () () () set
union ((n) . S) is set
{(union ((n) . S))} is non empty trivial finite 1 -element set
[:(union ((n) . S)),{(union ((n) . S))}:] is set
(union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:] is set
((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} is non empty set
{ {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } is non empty set
(((n) . S)) is finite-membered (1) Element of bool ((n) . S)
bool ((n) . S) is non empty V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S)) is non empty set
{ {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S))) \/ { {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } is non empty set
{ {(union ((n) . S)),[b₁,(union ((n) . S))]} where b₁ is Element of union ((n) . S) : b₁ in union ((n) . S) } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . S)) \/ [:(union ((n) . S)),{(union ((n) . S))}:]) \/ {(union ((n) . S))} : verum } ) \/ (((n) . S))) \/ { {b₁,[b₂,(union ((n) . S))]} where b₁, b₂ is Element of union ((n) . S) : {b₁,b₂} in (((n) . S)) } ) \/ { {(union ((n) . S)),[b₁,(union ((n) . S))]} where b₁ is Element of union ((n) . S) : b₁ in union ((n) . S) } is non empty set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union ((n) . G) is finite set
card (union ((n) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 |^ G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(2 |^ G) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2 |^ G) * (n)) + (2 |^ G) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((2 |^ G) * (n)) + (2 |^ G)) - 1 is V83() ext-real V216() set
(n) . 0 is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union ((n) . 0) is finite set
card (union ((n) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(2 |^ 0) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2 |^ 0) * (n)) + (2 |^ 0) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((2 |^ 0) * (n)) + (2 |^ 0)) - 1 is V83() ext-real V216() set
(n) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((n) + 1) - 1 is V83() ext-real V216() set
1 * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
2 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(1 * (n)) + (2 |^ 0) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((1 * (n)) + (2 |^ 0)) - 1 is V83() ext-real V216() set
(2 |^ 0) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 |^ 0) * (n)) + (2 |^ 0) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((2 |^ 0) * (n)) + (2 |^ 0)) - 1 is V83() ext-real V216() set
MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . MG is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . MG)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union ((n) . MG) is finite set
card (union ((n) . MG)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 |^ MG is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(2 |^ MG) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2 |^ MG) * (n)) + (2 |^ MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((2 |^ MG) * (n)) + (2 |^ MG)) - 1 is V83() ext-real V216() set
MG + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n) . (MG + 1) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . (MG + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union ((n) . (MG + 1)) is finite set
card (union ((n) . (MG + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 |^ (MG + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(2 |^ (MG + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1))) - 1 is V83() ext-real V216() set
(((n) . MG)) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
{(union ((n) . MG))} is non empty trivial finite finite-membered 1 -element set
[:(union ((n) . MG)),{(union ((n) . MG))}:] is finite set
(union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:] is finite set
((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} is non empty finite set
{ {b₁} where b₁ is Element of ((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} : verum } is non empty set
(((n) . MG)) is finite finite-membered (1) Element of bool ((n) . MG)
bool ((n) . MG) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} : verum } ) \/ (((n) . MG)) is non empty set
{ {b₁,[b₂,(union ((n) . MG))]} where b₁, b₂ is Element of union ((n) . MG) : {b₁,b₂} in (((n) . MG)) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} : verum } ) \/ (((n) . MG))) \/ { {b₁,[b₂,(union ((n) . MG))]} where b₁, b₂ is Element of union ((n) . MG) : {b₁,b₂} in (((n) . MG)) } is non empty set
{ {(union ((n) . MG)),[b₁,(union ((n) . MG))]} where b₁ is Element of union ((n) . MG) : b₁ in union ((n) . MG) } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . MG)) \/ [:(union ((n) . MG)),{(union ((n) . MG))}:]) \/ {(union ((n) . MG))} : verum } ) \/ (((n) . MG))) \/ { {b₁,[b₂,(union ((n) . MG))]} where b₁, b₂ is Element of union ((n) . MG) : {b₁,b₂} in (((n) . MG)) } ) \/ { {(union ((n) . MG)),[b₁,(union ((n) . MG))]} where b₁ is Element of union ((n) . MG) : b₁ in union ((n) . MG) } is non empty set
((((n) . MG))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union (((n) . MG)) is finite set
card (union (((n) . MG))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
2 * ((((2 |^ MG) * (n)) + (2 |^ MG)) - 1) is V83() ext-real V216() set
(2 * ((((2 |^ MG) * (n)) + (2 |^ MG)) - 1)) + 1 is V83() ext-real V216() set
2 * (2 |^ MG) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 * (2 |^ MG)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 * (2 |^ MG)) * (n)) + (2 * (2 |^ MG)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((2 * (2 |^ MG)) * (n)) + (2 * (2 |^ MG))) - (2 * 1) is V83() ext-real V216() set
((((2 * (2 |^ MG)) * (n)) + (2 * (2 |^ MG))) - (2 * 1)) + 1 is V83() ext-real V216() set
2 |^ (MG + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 |^ (MG + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 |^ (MG + 1)) * (n)) + (2 * (2 |^ MG)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((2 |^ (MG + 1)) * (n)) + (2 * (2 |^ MG))) - (2 * 1) is V83() ext-real V216() set
((((2 |^ (MG + 1)) * (n)) + (2 * (2 |^ MG))) - (2 * 1)) + 1 is V83() ext-real V216() set
((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1))) - 2 is V83() ext-real V216() set
((((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1))) - 2) + 1 is V83() ext-real V216() set
(((2 |^ (MG + 1)) * (n)) + (2 |^ (MG + 1))) - 1 is V83() ext-real V216() set
n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(n) is Relation-like NAT -defined Function-like V14( NAT ) set
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) is finite finite-membered (1) Element of bool n
bool n is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
union n is finite set
card (union n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((n) . G)) is finite finite-membered (1) Element of bool ((n) . G)
bool ((n) . G) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (((n) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 |^ G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ G) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
2 |^ G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ G) - (2 |^ G) is V83() ext-real V216() set
((3 |^ G) - (2 |^ G)) * (n) is V83() ext-real V216() set
((3 |^ G) * (n)) + (((3 |^ G) - (2 |^ G)) * (n)) is V83() ext-real V216() set
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(G + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ G) * (n)) + (((3 |^ G) - (2 |^ G)) * (n))) + ((G + 1) block 3) is V83() ext-real V216() set
(n) . 0 is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((n) . 0)) is finite finite-membered (1) Element of bool ((n) . 0)
bool ((n) . 0) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (((n) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ 0) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
2 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ 0) - (2 |^ 0) is V83() ext-real V216() set
((3 |^ 0) - (2 |^ 0)) * (n) is V83() ext-real V216() set
((3 |^ 0) * (n)) + (((3 |^ 0) - (2 |^ 0)) * (n)) is V83() ext-real V216() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(0 + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ 0) * (n)) + (((3 |^ 0) - (2 |^ 0)) * (n))) + ((0 + 1) block 3) is V83() ext-real V216() set
1 * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
0 * (n) is functional empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural finite finite-membered cardinal {} -element FinSequence-membered V83() ext-real non positive non negative V216() V233() subset-closed Element of NAT
(1 * (n)) + (0 * (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((1 * (n)) + (0 * (n))) + 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
3 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(3 |^ 0) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
1 - 1 is V83() ext-real V216() set
(1 - 1) * (n) is V83() ext-real V216() set
((3 |^ 0) * (n)) + ((1 - 1) * (n)) is V83() ext-real V216() set
(((3 |^ 0) * (n)) + ((1 - 1) * (n))) + 0 is V83() ext-real V216() set
(3 |^ 0) - 1 is V83() ext-real V216() set
((3 |^ 0) - 1) * (n) is V83() ext-real V216() set
((3 |^ 0) * (n)) + (((3 |^ 0) - 1) * (n)) is V83() ext-real V216() set
(((3 |^ 0) * (n)) + (((3 |^ 0) - 1) * (n))) + 0 is V83() ext-real V216() set
2 |^ 0 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(3 |^ 0) - (2 |^ 0) is V83() ext-real V216() set
((3 |^ 0) - (2 |^ 0)) * (n) is V83() ext-real V216() set
((3 |^ 0) * (n)) + (((3 |^ 0) - (2 |^ 0)) * (n)) is V83() ext-real V216() set
(((3 |^ 0) * (n)) + (((3 |^ 0) - (2 |^ 0)) * (n))) + 0 is V83() ext-real V216() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(0 + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ 0) * (n)) + (((3 |^ 0) - (2 |^ 0)) * (n))) + ((0 + 1) block 3) is V83() ext-real V216() set
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(n) . n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((n) . n)) is finite finite-membered (1) Element of bool ((n) . n)
bool ((n) . n) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (((n) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 |^ n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ n) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
2 |^ n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ n) - (2 |^ n) is V83() ext-real V216() set
((3 |^ n) - (2 |^ n)) * (n) is V83() ext-real V216() set
((3 |^ n) * (n)) + (((3 |^ n) - (2 |^ n)) * (n)) is V83() ext-real V216() set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ n) * (n)) + (((3 |^ n) - (2 |^ n)) * (n))) + ((n + 1) block 3) is V83() ext-real V216() set
(n) . (n + 1) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(((n) . (n + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((n) . (n + 1))) is finite finite-membered (1) Element of bool ((n) . (n + 1))
bool ((n) . (n + 1)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card (((n) . (n + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 |^ (n + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ (n + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
2 |^ (n + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of REAL
(3 |^ (n + 1)) - (2 |^ (n + 1)) is V83() ext-real V216() set
((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + (((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n)) is V83() ext-real V216() set
(n + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((n + 1) + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ (n + 1)) * (n)) + (((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
1 / 2 is V83() ext-real non negative V216() set
2 |^ (n + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 |^ (n + 1)) - 2 is V83() ext-real V216() set
(1 / 2) * ((2 |^ (n + 1)) - 2) is V83() ext-real V216() set
2 * (2 |^ n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
2 * 1 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 * (2 |^ n)) - (2 * 1) is V83() ext-real V216() set
(1 / 2) * ((2 * (2 |^ n)) - (2 * 1)) is V83() ext-real V216() set
(2 |^ n) - 1 is V83() ext-real V216() set
(((n) . n)) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union ((n) . n) is finite set
{(union ((n) . n))} is non empty trivial finite finite-membered 1 -element set
[:(union ((n) . n)),{(union ((n) . n))}:] is finite set
(union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:] is finite set
((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} is non empty finite set
{ {b₁} where b₁ is Element of ((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} : verum } is non empty set
({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} : verum } ) \/ (((n) . n)) is non empty set
{ {b₁,[b₂,(union ((n) . n))]} where b₁, b₂ is Element of union ((n) . n) : {b₁,b₂} in (((n) . n)) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} : verum } ) \/ (((n) . n))) \/ { {b₁,[b₂,(union ((n) . n))]} where b₁, b₂ is Element of union ((n) . n) : {b₁,b₂} in (((n) . n)) } is non empty set
{ {(union ((n) . n)),[b₁,(union ((n) . n))]} where b₁ is Element of union ((n) . n) : b₁ in union ((n) . n) } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union ((n) . n)) \/ [:(union ((n) . n)),{(union ((n) . n))}:]) \/ {(union ((n) . n))} : verum } ) \/ (((n) . n))) \/ { {b₁,[b₂,(union ((n) . n))]} where b₁, b₂ is Element of union ((n) . n) : {b₁,b₂} in (((n) . n)) } ) \/ { {(union ((n) . n)),[b₁,(union ((n) . n))]} where b₁ is Element of union ((n) . n) : b₁ in union ((n) . n) } is non empty set
((((n) . n))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((n) . n))) is finite finite-membered (1) Element of bool (((n) . n))
bool (((n) . n)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
card ((((n) . n))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
3 * ((((3 |^ n) * (n)) + (((3 |^ n) - (2 |^ n)) * (n))) + ((n + 1) block 3)) is V83() ext-real V216() set
(((n) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
card (union ((n) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of omega
(3 * ((((3 |^ n) * (n)) + (((3 |^ n) - (2 |^ n)) * (n))) + ((n + 1) block 3))) + (((n) . n)) is V83() ext-real V216() set
3 * (3 |^ n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(3 * (3 |^ n)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
3 * ((3 |^ n) - (2 |^ n)) is V83() ext-real V216() set
(3 * ((3 |^ n) - (2 |^ n))) * (n) is V83() ext-real V216() set
((3 * (3 |^ n)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n)) is V83() ext-real V216() set
3 * ((n + 1) block 3) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 * (3 |^ n)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + (3 * ((n + 1) block 3)) is V83() ext-real V216() set
((((3 * (3 |^ n)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + (3 * ((n + 1) block 3))) + (((n) . n)) is V83() ext-real V216() set
3 |^ (n + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(3 |^ (n + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n)) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + (3 * ((n + 1) block 3)) is V83() ext-real V216() set
((((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + (3 * ((n + 1) block 3))) + (((n) . n)) is V83() ext-real V216() set
(2 |^ n) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2 |^ n) * (n)) + (2 |^ n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((2 |^ n) * (n)) + (2 |^ n)) - 1 is V83() ext-real V216() set
((((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + (3 * ((n + 1) block 3))) + ((((2 |^ n) * (n)) + (2 |^ n)) - 1) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + ((2 |^ n) * (n)) is V83() ext-real V216() set
(3 * ((n + 1) block 3)) + ((2 |^ n) - 1) is V83() ext-real V216() set
((((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + ((2 |^ n) * (n))) + ((3 * ((n + 1) block 3)) + ((2 |^ n) - 1)) is V83() ext-real V216() set
2 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n + 1) block (2 + 1) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 + 1) * ((n + 1) block (2 + 1)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(n + 1) block 2 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 + 1) * ((n + 1) block (2 + 1))) + ((n + 1) block 2) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((((3 |^ (n + 1)) * (n)) + ((3 * ((3 |^ n) - (2 |^ n))) * (n))) + ((2 |^ n) * (n))) + (((2 + 1) * ((n + 1) block (2 + 1))) + ((n + 1) block 2)) is V83() ext-real V216() set
(3 * (3 |^ n)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(2 * (2 |^ n)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 * (2 |^ n)) * (n)) + ((2 |^ n) * (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((3 * (3 |^ n)) * (n)) - (((2 * (2 |^ n)) * (n)) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(((3 * (3 |^ n)) * (n)) - (((2 * (2 |^ n)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n)) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + ((((3 * (3 |^ n)) * (n)) - (((2 * (2 |^ n)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(n + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((n + 1) + 1) block 3 is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
(((3 |^ (n + 1)) * (n)) + ((((3 * (3 |^ n)) * (n)) - (((2 * (2 |^ n)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n)))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
(2 |^ (n + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((2 |^ (n + 1)) * (n)) + ((2 |^ n) * (n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((3 * (3 |^ n)) * (n)) - (((2 |^ (n + 1)) * (n)) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(((3 * (3 |^ n)) * (n)) - (((2 |^ (n + 1)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n)) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + ((((3 * (3 |^ n)) * (n)) - (((2 |^ (n + 1)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + ((((3 * (3 |^ n)) * (n)) - (((2 |^ (n + 1)) * (n)) + ((2 |^ n) * (n)))) + ((2 |^ n) * (n)))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
((3 * (3 |^ n)) * (n)) - ((2 |^ (n + 1)) * (n)) is V83() ext-real V216() set
(((3 * (3 |^ n)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n)) is V83() ext-real V216() set
((((3 * (3 |^ n)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n)) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + (((((3 * (3 |^ n)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + (((((3 * (3 |^ n)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n)))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
(3 |^ (n + 1)) * (n) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() Element of NAT
((3 |^ (n + 1)) * (n)) - ((2 |^ (n + 1)) * (n)) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n)) is V83() ext-real V216() set
((((3 |^ (n + 1)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n)) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + (((((3 |^ (n + 1)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n))) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + (((((3 |^ (n + 1)) * (n)) - ((2 |^ (n + 1)) * (n))) - ((2 |^ n) * (n))) + ((2 |^ n) * (n)))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
(3 |^ (n + 1)) - (2 |^ (n + 1)) is V83() ext-real V216() set
((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n) is V83() ext-real V216() set
((3 |^ (n + 1)) * (n)) + (((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n)) is V83() ext-real V216() set
(((3 |^ (n + 1)) * (n)) + (((3 |^ (n + 1)) - (2 |^ (n + 1))) * (n))) + (((n + 1) + 1) block 3) is V83() ext-real V216() set
(2) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () (2)
bool 2 is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool 2 : card b₁ <= 2 } is set
((2)) is Relation-like NAT -defined Function-like V14( NAT ) set
card 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of omega
((2)) . 0 is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
((((2)) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((2)) . 0)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) . G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
((((2)) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((2)) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
G + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
G + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((2)) . (G + 1) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
((((2)) . (G + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((2)) . (G + 1))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(G + 1) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((((2)) . G)) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union (((2)) . G) is finite set
{(union (((2)) . G))} is non empty trivial finite finite-membered 1 -element set
[:(union (((2)) . G)),{(union (((2)) . G))}:] is finite set
(union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:] is finite set
((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} is non empty finite set
{ {b₁} where b₁ is Element of ((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} : verum } is set
{{}} \/ { {b₁} where b₁ is Element of ((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} : verum } is non empty set
((((2)) . G)) is finite finite-membered (1) Element of bool (((2)) . G)
bool (((2)) . G) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
({{}} \/ { {b₁} where b₁ is Element of ((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} : verum } ) \/ ((((2)) . G)) is non empty set
{ {b₁,[b₂,(union (((2)) . G))]} where b₁, b₂ is Element of union (((2)) . G) : {b₁,b₂} in ((((2)) . G)) } is set
(({{}} \/ { {b₁} where b₁ is Element of ((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} : verum } ) \/ ((((2)) . G))) \/ { {b₁,[b₂,(union (((2)) . G))]} where b₁, b₂ is Element of union (((2)) . G) : {b₁,b₂} in ((((2)) . G)) } is non empty set
{ {(union (((2)) . G)),[b₁,(union (((2)) . G))]} where b₁ is Element of union (((2)) . G) : b₁ in union (((2)) . G) } is set
((({{}} \/ { {b₁} where b₁ is Element of ((union (((2)) . G)) \/ [:(union (((2)) . G)),{(union (((2)) . G))}:]) \/ {(union (((2)) . G))} : verum } ) \/ ((((2)) . G))) \/ { {b₁,[b₂,(union (((2)) . G))]} where b₁, b₂ is Element of union (((2)) . G) : {b₁,b₂} in ((((2)) . G)) } ) \/ { {(union (((2)) . G)),[b₁,(union (((2)) . G))]} where b₁ is Element of union (((2)) . G) : b₁ in union (((2)) . G) } is non empty set
(((((2)) . G))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(((((2)) . G))) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
1 + (G + 2) is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(G + 1) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
G is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) . G is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
((((2)) . G)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) . S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
((((2)) . S)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
S + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) . n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((2)) . n is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
n + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
(n + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
n + 2 is non empty epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real positive non negative V216() Element of NAT
((((2)) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((2)) . n)) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
((((2)) . n)) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
union (((2)) . n) is finite set
((union (((2)) . n))) is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () () ( union (((2)) . n))
bool (union (((2)) . n)) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
{ b₁ where b₁ is finite Element of bool (union (((2)) . n)) : card b₁ <= 2 } is set
((((2)) . n)) is finite finite-membered (1) Element of bool (((2)) . n)
bool (((2)) . n) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
((union (((2)) . n))) \ ((((2)) . n)) is finite finite-membered (1) Element of bool ((union (((2)) . n)))
bool ((union (((2)) . n))) is non empty finite finite-membered V233() V267() subset-closed V329() V332() set
S is non empty finite finite-membered V233() V267() subset-closed (1) () () () () () set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set
(S) is epsilon-transitive epsilon-connected ordinal natural finite cardinal V83() ext-real non negative V216() set