:: CHAIN_1 semantic presentation

REAL is non empty non trivial non finite V153() V154() V155() V159() set
NAT is non empty non trivial ordinal limit_ordinal non finite cardinal limit_cardinal V153() V154() V155() V156() V157() V158() V159() Element of bool REAL
bool REAL is non empty non trivial non finite set
omega is non empty non trivial ordinal limit_ordinal non finite cardinal limit_cardinal V153() V154() V155() V156() V157() V158() V159() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
[:1,1:] is non empty Relation-like RAT -valued INT -valued finite V143() V144() V145() V146() set
RAT is non empty non trivial non finite V153() V154() V155() V156() V159() set
INT is non empty non trivial non finite V153() V154() V155() V156() V157() V159() set
bool [:1,1:] is non empty finite V28() set
[:[:1,1:],1:] is non empty Relation-like RAT -valued INT -valued finite V143() V144() V145() V146() set
bool [:[:1,1:],1:] is non empty finite V28() set
[:[:1,1:],REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:[:1,1:],REAL:] is non empty non trivial non finite set
[:REAL,REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
[:[:REAL,REAL:],REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
2 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
[:2,2:] is non empty Relation-like RAT -valued INT -valued finite V143() V144() V145() V146() set
[:[:2,2:],REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:[:2,2:],REAL:] is non empty non trivial non finite set
COMPLEX is non empty non trivial non finite V153() V159() set
bool [:REAL,REAL:] is non empty non trivial non finite set
K388(2) is non empty V81() right_complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V177() L15()
the carrier of K388(2) is non empty set
{} is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element FinSequence-like FinSequence-membered ext-real non positive non negative V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() set
3 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
card {} is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element FinSequence-like FinSequence-membered ext-real non positive non negative V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() set
0 is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of NAT
G is complex real ext-real set
d is complex real ext-real set
k is complex real ext-real set
f is complex real ext-real Element of REAL
d is complex real ext-real set
G is complex real ext-real set
k is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
max (k,f) is complex real ext-real set
(max (k,f)) + 1 is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
k + 0 is complex real ext-real Element of REAL
f + 0 is complex real ext-real Element of REAL
F₁() is non empty set
F₂() is non empty set
{ F₃(b₁,b₂) where b₁, b₂ is Element of F₂() : P₁[b₁,b₂] } is set
d is set
G is Element of F₂()
k is Element of F₂()
F₃(G,k) is Element of F₁()
d is set
bool d is non empty set
G is Element of bool d
bool G is non empty set
bool (bool d) is non empty set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
1 + 0 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is set
G is set
{d,G} is non empty finite set
{d,d} is non empty finite set
{d} is non empty trivial finite 1 -element set
{0,1} is non empty finite V28() V153() V154() V155() V156() V157() V158() Element of bool NAT
d is non empty non trivial set
G is set
d \/ G is non empty set
k is set
f is set
G \/ d is non empty non trivial set
d is non empty non trivial set
bool d is non empty set
G is set
k is set
{G,k} is non empty finite set
d is set
G is set
{G} is non empty trivial finite 1 -element set
d \/ {G} is non empty set
k is set
{k} is non empty trivial finite 1 -element set
F₁() is non empty set
bool F₁() is non empty set
F₂() is non empty finite Element of bool F₁()
d is set
G is set
{d} is non empty trivial finite 1 -element set
G \/ {d} is non empty set
k is Element of bool F₁()
k is Element of bool F₁()
k is Element of bool F₁()
k is Element of bool F₁()
F₁() is non empty non trivial set
bool F₁() is non empty set
F₂() is non empty non trivial finite Element of bool F₁()
d is set
G is set
{d} is non empty trivial finite 1 -element set
G \/ {d} is non empty set
k is Element of bool F₁()
k \/ {d} is non empty set
k is Element of bool F₁()
k is Element of bool F₁()
k is Element of bool F₁()
k \/ {d} is non empty set
f is set
{f} is non empty trivial finite 1 -element set
{d,f} is non empty finite set
k is Element of bool F₁()
k \/ {d} is non empty set
d is set
card d is ordinal cardinal set
G is finite set
k is set
f is set
{k,f} is non empty finite set
g is set
A is set
A9 is set
G is set
k is set
f is set
g is set
{G,k} is non empty finite set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is finite set
G is finite set
card d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d \/ G is finite set
card (d \/ G) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card d) + (card G) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is finite set
card d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
G is finite set
card G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d \+\ G is finite set
card (d \+\ G) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d \ G is finite Element of bool d
bool d is non empty finite V28() set
d /\ G is finite set
(d \ G) \/ (d /\ G) is finite set
G \ d is finite Element of bool G
bool G is non empty finite V28() set
(G \ d) \/ (d /\ G) is finite set
(d \ G) \/ (G \ d) is finite set
card (d \ G) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (d /\ G) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (G \ d) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
[:(Seg d),REAL:] is Relation-like V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty set
G is set
d -tuples_on REAL is functional FinSequence-membered FinSequenceSet of REAL
Funcs ((Seg d),REAL) is non empty FUNCTION_DOMAIN of Seg d, REAL
G is FinSequenceSet of REAL
k is set
f is set
k is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
[:(Seg d),(bool REAL):] is non empty non trivial Relation-like non finite set
bool [:(Seg d),(bool REAL):] is non empty non trivial non finite set
G is set
the non empty non trivial finite V153() V154() V155() Element of bool REAL is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is Relation-like Function-like set
dom G is set
k is set
G . k is set
k is Relation-like Seg d -defined bool REAL -valued Function-like finite quasi_total Element of bool [:(Seg d),(bool REAL):]
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . f is V153() V154() V155() Element of bool REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . g is V153() V154() V155() Element of bool REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite quasi_total (d)
k is set
G . k is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . k is finite set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
product k is finite set
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
dom G is finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
dom k is finite set
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . f is complex real ext-real Element of REAL
(d,k,f) is non empty non trivial finite V153() V154() V155() Element of bool REAL
f is set
G . f is complex real ext-real Element of REAL
k . f is finite set
d is non empty finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
k is complex real ext-real Element of REAL
G is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
k is non empty finite V153() V154() V155() Element of bool REAL
k \/ {G} is non empty finite V153() V154() V155() Element of bool REAL
f is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
d is non empty finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
k is complex real ext-real Element of REAL
G is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
k is non empty finite V153() V154() V155() Element of bool REAL
k \/ {G} is non empty finite V153() V154() V155() Element of bool REAL
f is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
{f,g} is non empty finite V153() V154() V155() Element of bool REAL
A is complex real ext-real Element of REAL
{G,k} is non empty finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
k is non empty non trivial finite V153() V154() V155() Element of bool REAL
k \/ {G} is non empty non trivial finite V153() V154() V155() Element of bool REAL
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
d is non empty finite V153() V154() V155() Element of bool REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
f is set
g is set
A is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
f is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
[G,k] is non empty non natural Element of [:REAL,REAL:]
{G,k} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,k},{G}} is non empty finite V28() set
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
[G,k] is non empty non natural Element of [:REAL,REAL:]
{G,k} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,k},{G}} is non empty finite V28() set
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
[f,g] is non empty non natural Element of [:REAL,REAL:]
{f,g} is non empty finite V153() V154() V155() set
{f} is non empty trivial finite 1 -element V153() V154() V155() set
{{f,g},{f}} is non empty finite V28() set
A is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
{G,k} is non empty finite V153() V154() V155() Element of bool REAL
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
[f,g] is non empty non natural Element of [:REAL,REAL:]
{f,g} is non empty finite V153() V154() V155() set
{f} is non empty trivial finite 1 -element V153() V154() V155() set
{{f,g},{f}} is non empty finite V28() set
A is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
{A9,A99} is non empty finite V153() V154() V155() Element of bool REAL
[A9,A99] is non empty non natural Element of [:REAL,REAL:]
{A9,A99} is non empty finite V153() V154() V155() set
{A9} is non empty trivial finite 1 -element V153() V154() V155() set
{{A9,A99},{A9}} is non empty finite V28() set
B9 is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
{ H₁(b₁) where b₁ is complex real ext-real Element of REAL : ( H₁(b₁) in d & S₁[b₁] ) } is set
f is finite V153() V154() V155() Element of bool REAL
g is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
[G,g] is non empty non natural Element of [:REAL,REAL:]
{G,g} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,g},{G}} is non empty finite V28() set
A is set
A9 is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
A is complex real ext-real Element of REAL
f is finite V153() V154() V155() Element of bool REAL
g is non empty finite V153() V154() V155() Element of bool REAL
A is complex real ext-real Element of REAL
[G,A] is non empty non natural Element of [:REAL,REAL:]
{G,A} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,A},{G}} is non empty finite V28() set
A9 is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
f is finite V153() V154() V155() Element of bool REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
{ H₁(b₁) where b₁ is complex real ext-real Element of REAL : ( H₁(b₁) in d & S₁[b₁] ) } is set
f is finite V153() V154() V155() Element of bool REAL
g is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
[g,G] is non empty non natural Element of [:REAL,REAL:]
{g,G} is non empty finite V153() V154() V155() set
{g} is non empty trivial finite 1 -element V153() V154() V155() set
{{g,G},{g}} is non empty finite V28() set
A is set
A9 is complex real ext-real Element of REAL
{G} is non empty trivial finite 1 -element V153() V154() V155() Element of bool REAL
A is complex real ext-real Element of REAL
f is finite V153() V154() V155() Element of bool REAL
g is non empty finite V153() V154() V155() Element of bool REAL
A is complex real ext-real Element of REAL
[A,G] is non empty non natural Element of [:REAL,REAL:]
{A,G} is non empty finite V153() V154() V155() set
{A} is non empty trivial finite 1 -element V153() V154() V155() set
{{A,G},{A}} is non empty finite V28() set
A9 is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
f is finite V153() V154() V155() Element of bool REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
[G,k] is non empty non natural Element of [:REAL,REAL:]
{G,k} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,k},{G}} is non empty finite V28() set
f is complex real ext-real Element of REAL
[G,f] is non empty non natural Element of [:REAL,REAL:]
{G,f} is non empty finite V153() V154() V155() set
{{G,f},{G}} is non empty finite V28() set
g is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
[G,k] is non empty non natural Element of [:REAL,REAL:]
{G,k} is non empty finite V153() V154() V155() set
{G} is non empty trivial finite 1 -element V153() V154() V155() set
{{G,k},{G}} is non empty finite V28() set
f is complex real ext-real Element of REAL
[f,k] is non empty non natural Element of [:REAL,REAL:]
{f,k} is non empty finite V153() V154() V155() set
{f} is non empty trivial finite 1 -element V153() V154() V155() set
{{f,k},{f}} is non empty finite V28() set
g is complex real ext-real Element of REAL
A is complex real ext-real Element of REAL
d is non empty non trivial finite V153() V154() V155() Element of bool REAL
G is complex real ext-real Element of REAL
k is complex real ext-real Element of REAL
[k,G] is non empty non natural Element of [:REAL,REAL:]
{k,G} is non empty finite V153() V154() V155() set
{k} is non empty trivial finite 1 -element V153() V154() V155() set
{{k,G},{k}} is non empty finite V28() set
f is complex real ext-real Element of REAL
g is complex real ext-real Element of REAL
[g,f] is non empty non natural Element of [:REAL,REAL:]
{g,f} is non empty finite V153() V154() V155() set
{g} is non empty trivial finite 1 -element V153() V154() V155() set
{{g,f},{g}} is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : S₁[b₁] } is set
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
k . g is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,k,f) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : S₁[b₁] } is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
f . g is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
f . g is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,k,G) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= G . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not k . b₂ <= G . b₂ & ( b₁ . b₂ <= G . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . g is complex real ext-real Element of REAL
G . g is complex real ext-real Element of REAL
f . g is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A is complex real ext-real Element of REAL
G . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . f is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
k . g is complex real ext-real Element of REAL
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . f is complex real ext-real Element of REAL
k . f is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . g is complex real ext-real Element of REAL
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . f is complex real ext-real Element of REAL
k . f is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . g is complex real ext-real Element of REAL
k . g is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,G) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= G . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= G . b₂ & ( b₁ . b₂ <= G . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
{G} is non empty trivial functional finite V28() 1 -element FinSequence-membered Element of bool (REAL d)
k is set
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
f is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
g is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
g . A is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A99 . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
A is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 . A99 is complex real ext-real Element of REAL
g . A99 is complex real ext-real Element of REAL
k . A99 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,k,G) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= G . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not k . b₂ <= G . b₂ & ( b₁ . b₂ <= G . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
k . A is complex real ext-real Element of REAL
G . A is complex real ext-real Element of REAL
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A99 . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
G . A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A is set
the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
g . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
f . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
k . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . B9 is complex real ext-real Element of REAL
G . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,f) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
A9 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
f . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
G . A is complex real ext-real Element of REAL
A9 is set
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 . A is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
k . A99 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . A9 is complex real ext-real Element of REAL
A99 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 . A9 is complex real ext-real Element of REAL
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A9 . A9 is complex real ext-real Element of REAL
B9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B9 . A9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A99 is complex real ext-real Element of REAL
G . A99 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . A9 is complex real ext-real Element of REAL
A99 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 . A9 is complex real ext-real Element of REAL
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A9 . A9 is complex real ext-real Element of REAL
B9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B9 . A9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (Seg d) is non empty finite V28() set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
{ (d,b₁,b₂) where b₁, b₂ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( ex b₃ being finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d) st
( card b₃ = k & ( for b₄ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( ( b₄ in b₃ & not b₂ . b₄ <= b₁ . b₄ & [(b₁ . b₄),(b₂ . b₄)] is ((d,G,b₄)) ) or ( not b₄ in b₃ & b₁ . b₄ = b₂ . b₄ & b₁ . b₄ in (d,G,b₄) ) ) ) ) or ( k = d & ( for b₃ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( not b₁ . b₃ <= b₂ . b₃ & [(b₁ . b₃),(b₂ . b₃)] is ((d,G,b₃)) ) ) ) ) } is set
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
{ H₂(b₁,b₂) where b₁, b₂ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : S₁[b₁,b₂] } is set
Seg k is finite k -element V153() V154() V155() V156() V157() V158() Element of bool NAT
g is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
(d,G,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
[A9,A99] is non empty non natural Element of [:REAL,REAL:]
{A9,A99} is non empty finite V153() V154() V155() set
{A9} is non empty trivial finite 1 -element V153() V154() V155() set
{{A9,A99},{A9}} is non empty finite V28() set
the complex real ext-real Element of (d,G,A) is complex real ext-real Element of (d,G,A)
A99 is complex real ext-real Element of REAL
[A99,A99] is non empty non natural Element of [:REAL,REAL:]
{A99,A99} is non empty finite V153() V154() V155() set
{A99} is non empty trivial finite 1 -element V153() V154() V155() set
{{A99,A99},{A99}} is non empty finite V28() set
B9 is Element of [:REAL,REAL:]
[:(Seg d),[:REAL,REAL:]:] is non empty non trivial Relation-like non finite set
bool [:(Seg d),[:REAL,REAL:]:] is non empty non trivial non finite set
A is Relation-like Seg d -defined [:REAL,REAL:] -valued Function-like finite quasi_total Element of bool [:(Seg d),[:REAL,REAL:]:]
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
B9 . B9 is complex real ext-real Element of REAL
A . B9 is Element of [:REAL,REAL:]
(A . B9) `1 is complex real ext-real Element of REAL
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 . B9 is complex real ext-real Element of REAL
(A . B9) `2 is complex real ext-real Element of REAL
[(B9 . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(B9 . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B9),(A9 . B9)},{(B9 . B9)}} is non empty finite V28() set
(d,B9,A9) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( B9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not B9 . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or B9 . b₂ <= b₁ . b₂ ) ) ) } is set
B9 is non empty functional FinSequence-membered Element of bool (REAL d)
A1 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A2 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,B1,A2) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( B1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A2 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not B1 . b₂ <= A2 . b₂ & ( b₁ . b₂ <= A2 . b₂ or B1 . b₂ <= b₁ . b₂ ) ) ) } is set
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . B2 is Element of [:REAL,REAL:]
(d,G,B2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
r9 is complex real ext-real Element of REAL
X is complex real ext-real Element of REAL
[r9,X] is non empty non natural Element of [:REAL,REAL:]
{r9,X} is non empty finite V153() V154() V155() set
{r9} is non empty trivial finite 1 -element V153() V154() V155() set
{{r9,X},{r9}} is non empty finite V28() set
B1 . B2 is complex real ext-real Element of REAL
A2 . B2 is complex real ext-real Element of REAL
[(B1 . B2),(A2 . B2)] is non empty non natural Element of [:REAL,REAL:]
{(B1 . B2),(A2 . B2)} is non empty finite V153() V154() V155() set
{(B1 . B2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B1 . B2),(A2 . B2)},{(B1 . B2)}} is non empty finite V28() set
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . B2 is Element of [:REAL,REAL:]
(d,G,B2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
r9 is complex real ext-real Element of REAL
[r9,r9] is non empty non natural Element of [:REAL,REAL:]
{r9,r9} is non empty finite V153() V154() V155() set
{r9} is non empty trivial finite 1 -element V153() V154() V155() set
{{r9,r9},{r9}} is non empty finite V28() set
B1 . B2 is complex real ext-real Element of REAL
A2 . B2 is complex real ext-real Element of REAL
[(B1 . B2),(A2 . B2)] is non empty non natural Element of [:REAL,REAL:]
{(B1 . B2),(A2 . B2)} is non empty finite V153() V154() V155() set
{(B1 . B2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B1 . B2),(A2 . B2)},{(B1 . B2)}} is non empty finite V28() set
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A2 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,B1,A2) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( B1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A2 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not B1 . b₂ <= A2 . b₂ & ( b₁ . b₂ <= A2 . b₂ or B1 . b₂ <= b₁ . b₂ ) ) ) } is set
B9 is non empty functional FinSequence-membered Element of bool (REAL d)
A1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
B1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,A1,B1) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B1 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A1 . b₂ <= B1 . b₂ & ( b₁ . b₂ <= B1 . b₂ or A1 . b₂ <= b₁ . b₂ ) ) ) } is set
A2 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
product G is finite set
[:(REAL d),(REAL d):] is non empty Relation-like set
B9 is set
A1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
B1 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,A1,B1) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B1 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A1 . b₂ <= B1 . b₂ & ( b₁ . b₂ <= B1 . b₂ or A1 . b₂ <= b₁ . b₂ ) ) ) } is set
[A1,B1] is non empty non natural Element of [:(REAL d),(REAL d):]
{A1,B1} is non empty finite V28() set
{A1} is non empty trivial finite V28() 1 -element set
{{A1,B1},{A1}} is non empty finite V28() set
[0,[A1,B1]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
[:NAT,[:(REAL d),(REAL d):]:] is non empty non trivial Relation-like non finite set
{0,[A1,B1]} is non empty finite set
{0} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{0,[A1,B1]},{0}} is non empty finite V28() set
[1,[A1,B1]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{1,[A1,B1]} is non empty finite set
{1} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{1,[A1,B1]},{1}} is non empty finite V28() set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
B1 . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
[(A1 . A2),(B1 . A2)] is non empty non natural Element of [:REAL,REAL:]
{(A1 . A2),(B1 . A2)} is non empty finite V153() V154() V155() set
{(A1 . A2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A1 . A2),(B1 . A2)},{(A1 . A2)}} is non empty finite V28() set
(d,G,A2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B2 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
[A1,B1] is non empty non natural Element of [:(REAL d),(REAL d):]
{A1,B1} is non empty finite V28() set
{A1} is non empty trivial finite V28() 1 -element set
{{A1,B1},{A1}} is non empty finite V28() set
[1,[A1,B1]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
[:NAT,[:(REAL d),(REAL d):]:] is non empty non trivial Relation-like non finite set
{1,[A1,B1]} is non empty finite set
{1} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{1,[A1,B1]},{1}} is non empty finite V28() set
[0,[A1,B1]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{0,[A1,B1]} is non empty finite set
{0} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{0,[A1,B1]},{0}} is non empty finite V28() set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 . A2 is complex real ext-real Element of REAL
B1 . A2 is complex real ext-real Element of REAL
[(A1 . A2),(B1 . A2)] is non empty non natural Element of [:REAL,REAL:]
{(A1 . A2),(B1 . A2)} is non empty finite V153() V154() V155() set
{(A1 . A2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A1 . A2),(B1 . A2)},{(A1 . A2)}} is non empty finite V28() set
(d,G,A2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 . A2 is complex real ext-real Element of REAL
B1 . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 . B2 is complex real ext-real Element of REAL
B1 . B2 is complex real ext-real Element of REAL
[(A1 . B2),(B1 . B2)] is non empty non natural Element of [:REAL,REAL:]
{(A1 . B2),(B1 . B2)} is non empty finite V153() V154() V155() set
{(A1 . B2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A1 . B2),(B1 . B2)},{(A1 . B2)}} is non empty finite V28() set
(d,G,B2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
r9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card r9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B9 is Relation-like Function-like set
dom B9 is set
A1 is set
B1 is set
B9 . A1 is set
B9 . B1 is set
A2 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
B2 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
[A2,B2] is non empty non natural Element of [:(REAL d),(REAL d):]
{A2,B2} is non empty finite V28() set
{A2} is non empty trivial finite V28() 1 -element set
{{A2,B2},{A2}} is non empty finite V28() set
[0,[A2,B2]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
[:NAT,[:(REAL d),(REAL d):]:] is non empty non trivial Relation-like non finite set
{0,[A2,B2]} is non empty finite set
{0} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{0,[A2,B2]},{0}} is non empty finite V28() set
(d,A2,B2) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A2 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B2 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A2 . b₂ <= B2 . b₂ & ( b₁ . b₂ <= B2 . b₂ or A2 . b₂ <= b₁ . b₂ ) ) ) } is set
[1,[A2,B2]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{1,[A2,B2]} is non empty finite set
{1} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{1,[A2,B2]},{1}} is non empty finite V28() set
r9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
X is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
[r9,X] is non empty non natural Element of [:(REAL d),(REAL d):]
{r9,X} is non empty finite V28() set
{r9} is non empty trivial finite V28() 1 -element set
{{r9,X},{r9}} is non empty finite V28() set
[0,[r9,X]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{0,[r9,X]} is non empty finite set
{{0,[r9,X]},{0}} is non empty finite V28() set
(d,r9,X) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( r9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= X . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not r9 . b₂ <= X . b₂ & ( b₁ . b₂ <= X . b₂ or r9 . b₂ <= b₁ . b₂ ) ) ) } is set
[1,[r9,X]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{1,[r9,X]} is non empty finite set
{{1,[r9,X]},{1}} is non empty finite V28() set
rng B9 is set
{0,1} is non empty finite V28() V153() V154() V155() V156() V157() V158() Element of bool NAT
A1 is finite set
[:A1,A1:] is Relation-like finite set
[:{0,1},[:A1,A1:]:] is Relation-like finite set
B1 is set
A2 is set
B9 . A2 is set
B2 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
r9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
[B2,r9] is non empty non natural Element of [:(REAL d),(REAL d):]
{B2,r9} is non empty finite V28() set
{B2} is non empty trivial finite V28() 1 -element set
{{B2,r9},{B2}} is non empty finite V28() set
[0,[B2,r9]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
[:NAT,[:(REAL d),(REAL d):]:] is non empty non trivial Relation-like non finite set
{0,[B2,r9]} is non empty finite set
{0} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{0,[B2,r9]},{0}} is non empty finite V28() set
(d,B2,r9) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( B2 . b₂ <= b₁ . b₂ & b₁ . b₂ <= r9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not B2 . b₂ <= r9 . b₂ & ( b₁ . b₂ <= r9 . b₂ or B2 . b₂ <= b₁ . b₂ ) ) ) } is set
[1,[B2,r9]] is non empty non natural Element of [:NAT,[:(REAL d),(REAL d):]:]
{1,[B2,r9]} is non empty finite set
{1} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() set
{{1,[B2,r9]},{1}} is non empty finite V28() set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (Seg G) is non empty finite V28() set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
{ (G,b₁,b₂) where b₁, b₂ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( ex b₃ being finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G) st
( card b₃ = d & ( for b₄ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( ( b₄ in b₃ & not b₂ . b₄ <= b₁ . b₄ & [(b₁ . b₄),(b₂ . b₄)] is ((G,k,b₄)) ) or ( not b₄ in b₃ & b₁ . b₄ = b₂ . b₄ & b₁ . b₄ in (G,k,b₄) ) ) ) ) or ( d = G & ( for b₃ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( not b₁ . b₃ <= b₂ . b₃ & [(b₁ . b₃),(b₂ . b₃)] is ((G,k,b₃)) ) ) ) ) } is set
f is functional FinSequence-membered Element of bool (REAL G)
g is functional FinSequence-membered Element of bool (REAL G)
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
bool (Seg G) is non empty finite V28() set
k is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
f is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,k,f) is non empty functional FinSequence-membered Element of bool (REAL G)
bool (REAL G) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
g is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,g,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(G,g,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
[(k . B9),(f . B9)] is non empty non natural Element of [:REAL,REAL:]
{(k . B9),(f . B9)} is non empty finite V153() V154() V155() set
{(k . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . B9),(f . B9)},{(k . B9)}} is non empty finite V28() set
(G,g,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
f is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,k,f) is non empty functional FinSequence-membered Element of bool (REAL G)
bool (REAL G) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
g is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,g,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
bool (Seg G) is non empty finite V28() set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
f . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
[(k . A),(f . A)] is non empty non natural Element of [:REAL,REAL:]
{(k . A),(f . A)} is non empty finite V153() V154() V155() set
{(k . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A),(f . A)},{(k . A)}} is non empty finite V28() set
(G,g,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
[(k . A99),(f . A99)] is non empty non natural Element of [:REAL,REAL:]
{(k . A99),(f . A99)} is non empty finite V153() V154() V155() set
{(k . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A99),(f . A99)},{(k . A99)}} is non empty finite V28() set
(G,g,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
f . A is complex real ext-real Element of REAL
k . A is complex real ext-real Element of REAL
[(k . A),(f . A)] is non empty non natural Element of [:REAL,REAL:]
{(k . A),(f . A)} is non empty finite V153() V154() V155() set
{(k . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A),(f . A)},{(k . A)}} is non empty finite V28() set
(G,g,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
[(k . A99),(f . A99)] is non empty non natural Element of [:REAL,REAL:]
{(k . A99),(f . A99)} is non empty finite V153() V154() V155() set
{(k . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A99),(f . A99)},{(k . A99)}} is non empty finite V28() set
(G,g,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
bool (Seg G) is non empty finite V28() set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
[(k . A9),(f . A9)] is non empty non natural Element of [:REAL,REAL:]
{(k . A9),(f . A9)} is non empty finite V153() V154() V155() set
{(k . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A9),(f . A9)},{(k . A9)}} is non empty finite V28() set
(G,g,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
f is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,k,f) is non empty functional FinSequence-membered Element of bool (REAL G)
bool (REAL G) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
g is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,g,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A is complex real ext-real Element of REAL
(G,g,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
f . A is complex real ext-real Element of REAL
[(k . A),(f . A)] is non empty non natural Element of [:REAL,REAL:]
{(k . A),(f . A)} is non empty finite V153() V154() V155() set
{(k . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A),(f . A)},{(k . A)}} is non empty finite V28() set
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
f is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,k,f) is non empty functional FinSequence-membered Element of bool (REAL G)
bool (REAL G) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
g is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,g,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,0) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k is functional FinSequence-membered Element of bool (REAL d)
bool (Seg d) is non empty finite V28() set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A99 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A9 . B9 is complex real ext-real Element of REAL
A99 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,f) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
g is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
[(f . A),(f . A)] is non empty non natural Element of [:REAL,REAL:]
{(f . A),(f . A)} is non empty finite V153() V154() V155() set
{(f . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A),(f . A)},{(f . A)}} is non empty finite V28() set
(d,G,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
g is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,f,0) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A is complex real ext-real Element of REAL
(d,f,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k is functional FinSequence-membered Element of bool (REAL d)
bool (Seg d) is non empty finite V28() set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (Seg d) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
[(f . A9),(g . A9)] is non empty non natural Element of [:REAL,REAL:]
{(f . A9),(g . A9)} is non empty finite V153() V154() V155() set
{(f . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A9),(g . A9)},{(f . A9)}} is non empty finite V28() set
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
[(f . A),(g . A)] is non empty non natural Element of [:REAL,REAL:]
{(f . A),(g . A)} is non empty finite V153() V154() V155() set
{(f . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A),(g . A)},{(f . A)}} is non empty finite V28() set
(d,G,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A99 is complex real ext-real Element of REAL
g . A99 is complex real ext-real Element of REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A is complex real ext-real Element of REAL
g . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
[(f . A9),(g . A9)] is non empty non natural Element of [:REAL,REAL:]
{(f . A9),(g . A9)} is non empty finite V153() V154() V155() set
{(f . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A9),(g . A9)},{(f . A9)}} is non empty finite V28() set
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
[(f . A9),(g . A9)] is non empty non natural Element of [:REAL,REAL:]
{(f . A9),(g . A9)} is non empty finite V153() V154() V155() set
{(f . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A9),(g . A9)},{(f . A9)}} is non empty finite V28() set
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,f,d) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,A) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
[(G . A9),(k . A9)] is non empty non natural Element of [:REAL,REAL:]
{(G . A9),(k . A9)} is non empty finite V153() V154() V155() set
{(G . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(G . A9),(k . A9)},{(G . A9)}} is non empty finite V28() set
(d,f,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A99 is complex real ext-real Element of REAL
G . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . B9 is complex real ext-real Element of REAL
k . B9 is complex real ext-real Element of REAL
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
f is functional FinSequence-membered Element of bool (REAL G)
bool (Seg G) is non empty finite V28() set
g is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,g,A) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(Seg G) \ B9 is finite V153() V154() V155() V156() V157() V158() Element of bool NAT
card ((Seg G) \ B9) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (Seg G) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card (Seg G)) - (card B9) is complex real ext-real set
G - d is complex real ext-real Element of REAL
A9 is set
{A9} is non empty trivial finite 1 -element set
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
(G,A9,A99) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A99 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A9 . b₂ <= A99 . b₂ & ( b₁ . b₂ <= A99 . b₂ or A9 . b₂ <= b₁ . b₂ ) ) ) } is set
A9 . A1 is complex real ext-real Element of REAL
A99 . A1 is complex real ext-real Element of REAL
(G,k,A1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B1 is complex real ext-real Element of REAL
A9 . B1 is complex real ext-real Element of REAL
[(A9 . B1),(A99 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(A9 . B1),(A99 . B1)} is non empty finite V153() V154() V155() set
{(A9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A9 . B1),(A99 . B1)},{(A9 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
g is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,g,A) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
(G,k,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
{A9} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
(Seg G) \ {A9} is finite V153() V154() V155() V156() V157() V158() Element of bool NAT
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card {A9} is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card (Seg G)) - (card {A9}) is complex real ext-real set
G - (card {A9}) is complex real ext-real Element of REAL
G - 1 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
[(g . B9),(A . B9)] is non empty non natural Element of [:REAL,REAL:]
{(g . B9),(A . B9)} is non empty finite V153() V154() V155() set
{(g . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(g . B9),(A . B9)},{(g . B9)}} is non empty finite V28() set
(G,k,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
f is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,k,f) is non empty functional FinSequence-membered Element of bool (REAL G)
bool (REAL G) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( k . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not k . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or k . b₂ <= b₁ . b₂ ) ) ) } is set
g is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,g,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
(G,g,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A9 is complex real ext-real Element of REAL
A9 . A9 is complex real ext-real Element of REAL
k . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
(G,g,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
f . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
[(k . A9),(f . A9)] is non empty non natural Element of [:REAL,REAL:]
{(k . A9),(f . A9)} is non empty finite V153() V154() V155() set
{(k . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(k . A9),(f . A9)},{(k . A9)}} is non empty finite V28() set
(G,g,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
k . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
(G,g,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,1) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k is functional FinSequence-membered Element of bool (REAL d)
bool (Seg d) is non empty finite V28() set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,A,A9) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B9 is set
{B9} is non empty trivial finite 1 -element set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A9 . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
[(A . A9),(A9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A . A9),(A9 . A9)} is non empty finite V153() V154() V155() set
{(A . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . A9),(A9 . A9)},{(A . A9)}} is non empty finite V28() set
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
(d,G,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A9 . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
[(A . A99),(A9 . A99)] is non empty non natural Element of [:REAL,REAL:]
{(A . A99),(A9 . A99)} is non empty finite V153() V154() V155() set
{(A . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . A99),(A9 . A99)},{(A . A99)}} is non empty finite V28() set
(d,G,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
(d,G,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(d,G,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
[(f . A),(g . A)] is non empty non natural Element of [:REAL,REAL:]
{(f . A),(g . A)} is non empty finite V153() V154() V155() set
{(f . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A),(g . A)},{(f . A)}} is non empty finite V28() set
(d,G,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
{A} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
card {A} is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
f . A99 is complex real ext-real Element of REAL
[(f . A99),(g . A99)] is non empty non natural Element of [:REAL,REAL:]
{(f . A99),(g . A99)} is non empty finite V153() V154() V155() set
{(f . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A99),(g . A99)},{(f . A99)}} is non empty finite V28() set
(d,G,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A99 is complex real ext-real Element of REAL
g . A99 is complex real ext-real Element of REAL
[(f . A99),(g . A99)] is non empty non natural Element of [:REAL,REAL:]
{(f . A99),(g . A99)} is non empty finite V153() V154() V155() set
{(f . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A99),(g . A99)},{(f . A99)}} is non empty finite V28() set
(d,G,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,f,1) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,A) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
[(g . A9),(A . A9)] is non empty non natural Element of [:REAL,REAL:]
{(g . A9),(A . A9)} is non empty finite V153() V154() V155() set
{(g . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(g . A9),(A . A9)},{(g . A9)}} is non empty finite V28() set
(d,f,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . g is complex real ext-real Element of REAL
G . g is complex real ext-real Element of REAL
[(G . g),(k . g)] is non empty non natural Element of [:REAL,REAL:]
{(G . g),(k . g)} is non empty finite V153() V154() V155() set
{(G . g)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(G . g),(k . g)},{(G . g)}} is non empty finite V28() set
(d,f,g) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL k is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg k is non empty finite k -element V153() V154() V155() V156() V157() V158() Element of bool NAT
f is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
g is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
(k,f,g) is non empty functional FinSequence-membered Element of bool (REAL k)
bool (REAL k) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
A9 is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
(k,A,A9) is non empty functional FinSequence-membered Element of bool (REAL k)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is Relation-like Seg k -defined bool REAL -valued Function-like finite finite-yielding quasi_total (k)
(k,A99,d) is non empty finite Element of bool (bool (REAL k))
bool (bool (REAL k)) is non empty set
(k,A99,G) is non empty finite Element of bool (bool (REAL k))
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
f . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
A . A9 is complex real ext-real Element of REAL
A9 . A9 is complex real ext-real Element of REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(k,A99,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(k,A99,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(k,A99,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
[(f . B9),(g . B9)] is non empty non natural Element of [:REAL,REAL:]
{(f . B9),(g . B9)} is non empty finite V153() V154() V155() set
{(f . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . B9),(g . B9)},{(f . B9)}} is non empty finite V28() set
(k,A99,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
g . A1 is complex real ext-real Element of REAL
f . A1 is complex real ext-real Element of REAL
[(f . A1),(g . A1)] is non empty non natural Element of [:REAL,REAL:]
{(f . A1),(g . A1)} is non empty finite V153() V154() V155() set
{(f . A1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A1),(g . A1)},{(f . A1)}} is non empty finite V28() set
(k,A99,A1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
A9 . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
[(A . A9),(A9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A . A9),(A9 . A9)} is non empty finite V153() V154() V155() set
{(A . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . A9),(A9 . A9)},{(A . A9)}} is non empty finite V28() set
(k,A99,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
A . A1 is complex real ext-real Element of REAL
A9 . A1 is complex real ext-real Element of REAL
[(A . A1),(A9 . A1)] is non empty non natural Element of [:REAL,REAL:]
{(A . A1),(A9 . A1)} is non empty finite V153() V154() V155() set
{(A . A1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . A1),(A9 . A1)},{(A . A1)}} is non empty finite V28() set
(k,A99,A1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL k is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg k is non empty finite k -element V153() V154() V155() V156() V157() V158() Element of bool NAT
f is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
g is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
(k,f,g) is non empty functional FinSequence-membered Element of bool (REAL k)
bool (REAL k) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
A9 is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
(k,A,A9) is non empty functional FinSequence-membered Element of bool (REAL k)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is Relation-like Seg k -defined bool REAL -valued Function-like finite finite-yielding quasi_total (k)
(k,A99,d) is non empty finite Element of bool (bool (REAL k))
bool (bool (REAL k)) is non empty set
(k,A99,G) is non empty finite Element of bool (bool (REAL k))
d + 0 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
bool (Seg k) is non empty finite V28() set
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg k)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
k - d is complex real ext-real Element of REAL
(Seg k) \ B9 is finite V153() V154() V155() V156() V157() V158() Element of bool NAT
card ((Seg k) \ B9) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (Seg k) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card (Seg k)) - (card B9) is complex real ext-real set
A9 is set
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
f . A1 is complex real ext-real Element of REAL
g . A1 is complex real ext-real Element of REAL
A9 . A1 is complex real ext-real Element of REAL
A . A1 is complex real ext-real Element of REAL
(k,A99,A1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
[(A . A1),(A9 . A1)] is non empty non natural Element of [:REAL,REAL:]
{(A . A1),(A9 . A1)} is non empty finite V153() V154() V155() set
{(A . A1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . A1),(A9 . A1)},{(A . A1)}} is non empty finite V28() set
B1 is complex real ext-real Element of REAL
[:(Seg k),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg k),REAL:] is non empty non trivial non finite set
A1 is Relation-like Seg k -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg k),REAL:]
B1 is Relation-like NAT -defined REAL -valued Function-like finite k -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL k
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg k
A . A2 is complex real ext-real Element of REAL
A9 . A2 is complex real ext-real Element of REAL
B1 . A2 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
bool (Seg d) is non empty finite V28() set
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
A9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
A99 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg d)
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
B9 is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
k . A9 is complex real ext-real Element of REAL
G . A9 is complex real ext-real Element of REAL
f . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
card A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card (Seg d) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(d,A,(card A9)) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
(d,A,(card A99)) is non empty finite Element of bool (bool (REAL d))
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
k . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . B9 is complex real ext-real Element of REAL
f . B9 is complex real ext-real Element of REAL
k . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,d) is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
f is finite Element of bool (d,G,k)
g is finite Element of bool (d,G,k)
f \+\ g is finite set
f \+\ g is finite Element of bool (d,G,k)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
(d,G,k) is non empty non trivial finite V153() V154() V155() Element of bool REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
k is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
(d,G,g) is non empty non trivial finite V153() V154() V155() Element of bool REAL
g is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A . A9 is complex real ext-real Element of REAL
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
f . A9 is complex real ext-real Element of REAL
A99 is set
B9 is set
A9 is complex real ext-real Element of REAL
B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A . A9 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
[(f . A9),(A . A9)] is non empty non natural Element of [:REAL,REAL:]
{(f . A9),(A . A9)} is non empty finite V153() V154() V155() set
{(f . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A9),(A . A9)},{(f . A9)}} is non empty finite V28() set
(d,f,A) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is Element of (d,G,d)
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
[(f . B9),(A . B9)] is non empty non natural Element of [:REAL,REAL:]
{(f . B9),(A . B9)} is non empty finite V153() V154() V155() set
{(f . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . B9),(A . B9)},{(f . B9)}} is non empty finite V28() set
(d,G,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
k is Element of (d,G,d)
f is Element of (d,G,d)
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,A) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A99 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,A9,A99) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A99 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A9 . b₂ <= A99 . b₂ & ( b₁ . b₂ <= A99 . b₂ or A9 . b₂ <= b₁ . b₂ ) ) ) } is set
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
B9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
B9 . B1 is complex real ext-real Element of REAL
A9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A9 . B1 is complex real ext-real Element of REAL
[(B9 . B1),(A9 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B1),(A9 . B1)} is non empty finite V153() V154() V155() set
{(B9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B1),(A9 . B1)},{(B9 . B1)}} is non empty finite V28() set
(d,G,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B9 . B1 is complex real ext-real Element of REAL
A1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A1 . B1 is complex real ext-real Element of REAL
[(B9 . B1),(A1 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B1),(A1 . B1)} is non empty finite V153() V154() V155() set
{(B9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B1),(A1 . B1)},{(B9 . B1)}} is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
bool (REAL d) is non empty set
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,f,d) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
(d,f) is Element of (d,f,d)
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,A,A9) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A99 is complex real ext-real Element of REAL
k . A99 is complex real ext-real Element of REAL
the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
k . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d is complex real ext-real Element of REAL
g is Element of (d,f,d)
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A99 is complex real ext-real Element of REAL
k . A99 is complex real ext-real Element of REAL
[(G . A99),(k . A99)] is non empty non natural Element of [:REAL,REAL:]
{(G . A99),(k . A99)} is non empty finite V153() V154() V155() set
{(G . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(G . A99),(k . A99)},{(G . A99)}} is non empty finite V28() set
(d,f,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
k is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,G,k) is non empty functional FinSequence-membered Element of bool (REAL d)
bool (REAL d) is non empty set
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( G . b₂ <= b₁ . b₂ & b₁ . b₂ <= k . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not G . b₂ <= k . b₂ & ( b₁ . b₂ <= k . b₂ or G . b₂ <= b₁ . b₂ ) ) ) } is set
f is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,f) is Element of (d,f,d)
(d,f,d) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
A is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,g,A) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A9 is complex real ext-real Element of REAL
k . A9 is complex real ext-real Element of REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
G . A99 is complex real ext-real Element of REAL
k . A99 is complex real ext-real Element of REAL
[(G . A99),(k . A99)] is non empty non natural Element of [:REAL,REAL:]
{(G . A99),(k . A99)} is non empty finite V153() V154() V155() set
{(G . A99)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(G . A99),(k . A99)},{(G . A99)}} is non empty finite V28() set
(d,f,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
F₁() is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg F₁() is non empty finite F₁() -element V153() V154() V155() V156() V157() V158() Element of bool NAT
F₂() is Relation-like Seg F₁() -defined bool REAL -valued Function-like finite finite-yielding quasi_total (F₁())
F₃() is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(F₁(),F₂(),F₃()) is non empty finite Element of bool (bool (REAL F₁()))
REAL F₁() is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL F₁()) is non empty set
bool (bool (REAL F₁())) is non empty set
bool (F₁(),F₂(),F₃()) is non empty finite V28() set
(F₁(),F₂(),F₃()) is finite Element of bool (F₁(),F₂(),F₃())
F₄() is finite Element of bool (F₁(),F₂(),F₃())
d is set
G is set
{d} is non empty trivial finite 1 -element set
G \/ {d} is non empty set
g is set
G /\ {d} is finite set
f is finite Element of bool (F₁(),F₂(),F₃())
k is Element of (F₁(),F₂(),F₃())
{k} is non empty trivial finite 1 -element Element of bool (F₁(),F₂(),F₃())
(F₁(),F₂(),F₃(),f,{k}) is finite Element of bool (F₁(),F₂(),F₃())
(G \/ {d}) \ {} is Element of bool (G \/ {d})
bool (G \/ {d}) is non empty set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
f is Element of (d,G,k)
{ b₁ where b₁ is Element of (d,G,(k + 1)) : f c= b₁ } is set
bool (d,G,(k + 1)) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,(k + 1)) : S₁[b₁] } is set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
f is Element of (G,k,d)
(G,k,d,f) is finite Element of bool (G,k,(d + 1))
bool (G,k,(d + 1)) is non empty finite V28() set
{ b₁ where b₁ is Element of (G,k,(d + 1)) : f c= b₁ } is set
g is Element of (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : S₁[b₁] } is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,(k + 1)) is non empty finite V28() set
(d,G,k) is non empty finite Element of bool (bool (REAL d))
f is finite Element of bool (d,G,(k + 1))
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ f) is even ) } is set
bool (d,G,k) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,k) : S₁[b₁] } is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,(k + 1)) is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,(k + 1)) is non empty finite V28() set
(d,G,k) is non empty finite Element of bool (bool (REAL d))
bool (d,G,k) is non empty finite V28() set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
bool (G,k,(d + 1)) is non empty finite V28() set
f is Element of (G,k,d)
(G,k,d,f) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : f c= b₁ } is set
g is finite Element of bool (G,k,(d + 1))
(G,k,d,g) is finite Element of bool (G,k,d)
bool (G,k,d) is non empty finite V28() set
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ g) is even ) } is set
(G,k,d,f) /\ g is finite Element of bool (G,k,(d + 1))
card ((G,k,d,f) /\ g) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
{ b₁ where b₁ is Element of (G,k,d) : S₁[b₁] } is set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,(d + 1)) is non empty finite V28() set
(G,k,d) is finite Element of bool (G,k,d)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (G,k,d) is non empty finite V28() set
f is finite Element of bool (G,k,(d + 1))
(G,k,d,f) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ f) is even ) } is set
g is set
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
f is Element of (G,k,d)
g is Element of (G,k,(d + 1))
{g} is non empty trivial finite 1 -element Element of bool (G,k,(d + 1))
bool (G,k,(d + 1)) is non empty finite V28() set
(G,k,d,{g}) is finite Element of bool (G,k,d)
bool (G,k,d) is non empty finite V28() set
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ {g}) is even ) } is set
(G,k,d,f) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : f c= b₁ } is set
(G,k,d,f) /\ {g} is finite Element of bool (G,k,(d + 1))
card ((G,k,d,f) /\ {g}) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A9 is set
2 * 0 is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered even ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of NAT
(2 * 0) + 1 is non empty ordinal natural complex real finite cardinal V43() non even ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 is set
2 * 0 is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered even ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of NAT
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (bool (REAL G)) is non empty set
f is Element of (G,k,d)
(G,k,d,f) is finite Element of bool (G,k,(d + 1))
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
bool (G,k,(d + 1)) is non empty finite V28() set
{ b₁ where b₁ is Element of (G,k,(d + 1)) : f c= b₁ } is set
card (G,k,d,f) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
g is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,g,A) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= A . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not g . b₂ <= A . b₂ & ( b₁ . b₂ <= A . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
(G,k,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . A99 is complex real ext-real Element of REAL
A . A99 is complex real ext-real Element of REAL
A99 is complex real ext-real Element of REAL
[A99,(g . A9)] is non empty non natural Element of [:REAL,REAL:]
{A99,(g . A9)} is non empty finite V153() V154() V155() set
{A99} is non empty trivial finite 1 -element V153() V154() V155() set
{{A99,(g . A9)},{A99}} is non empty finite V28() set
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . B9 is complex real ext-real Element of REAL
[:(Seg G),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg G),REAL:] is non empty non trivial non finite set
B9 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 . A9 is complex real ext-real Element of REAL
[(A9 . A9),(A . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A9 . A9),(A . A9)} is non empty finite V153() V154() V155() set
{(A9 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A9 . A9),(A . A9)},{(A9 . A9)}} is non empty finite V28() set
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . A1 is complex real ext-real Element of REAL
A . A1 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . B1 is complex real ext-real Element of REAL
A . B1 is complex real ext-real Element of REAL
[(A9 . B1),(A . B1)] is non empty non natural Element of [:REAL,REAL:]
{(A9 . B1),(A . B1)} is non empty finite V153() V154() V155() set
{(A9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A9 . B1),(A . B1)},{(A9 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
(G,k) is Element of (G,k,G)
(G,k,G) is non empty finite Element of bool (bool (REAL G))
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,B9,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( B9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not B9 . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or B9 . b₂ <= b₁ . b₂ ) ) ) } is set
B9 . A9 is complex real ext-real Element of REAL
A9 . A9 is complex real ext-real Element of REAL
[(B9 . A9),(A9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . A9),(A9 . A9)} is non empty finite V153() V154() V155() set
{(B9 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . A9),(A9 . A9)},{(B9 . A9)}} is non empty finite V28() set
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B9 is complex real ext-real Element of REAL
g . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
A . B9 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A1 is complex real ext-real Element of REAL
A9 . A1 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B1 is complex real ext-real Element of REAL
A9 . B1 is complex real ext-real Element of REAL
[(B9 . B1),(A9 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B1),(A9 . B1)} is non empty finite V153() V154() V155() set
{(B9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B1),(A9 . B1)},{(B9 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A99 . A9 is complex real ext-real Element of REAL
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 . A9 is complex real ext-real Element of REAL
[(A99 . A9),(B9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A99 . A9),(B9 . A9)} is non empty finite V153() V154() V155() set
{(A99 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A99 . A9),(B9 . A9)},{(A99 . A9)}} is non empty finite V28() set
A99 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A99 . A9 is complex real ext-real Element of REAL
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 . A9 is complex real ext-real Element of REAL
[(A99 . A9),(B9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A99 . A9),(B9 . A9)} is non empty finite V153() V154() V155() set
{(A99 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A99 . A9),(B9 . A9)},{(A99 . A9)}} is non empty finite V28() set
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
B9 . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
[(A99 . A9),(B9 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A99 . A9),(B9 . A9)} is non empty finite V153() V154() V155() set
{(A99 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A99 . A9),(B9 . A9)},{(A99 . A9)}} is non empty finite V28() set
(G,k,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
B9 . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A9 is complex real ext-real Element of REAL
A99 . A9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B9 is complex real ext-real Element of REAL
B9 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A9 is complex real ext-real Element of REAL
A99 . A9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B9 is complex real ext-real Element of REAL
B9 . B9 is complex real ext-real Element of REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A9 is complex real ext-real Element of REAL
g . A9 is complex real ext-real Element of REAL
B9 . A9 is complex real ext-real Element of REAL
A . A9 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B9 is complex real ext-real Element of REAL
B9 . B9 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A1 is complex real ext-real Element of REAL
B9 . A1 is complex real ext-real Element of REAL
[(A99 . A1),(B9 . A1)] is non empty non natural Element of [:REAL,REAL:]
{(A99 . A1),(B9 . A1)} is non empty finite V153() V154() V155() set
{(A99 . A1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A99 . A1),(B9 . A1)},{(A99 . A1)}} is non empty finite V28() set
(G,k,A1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
(G,k,G) is non empty finite Element of bool (bool (REAL G))
(G,A99,B9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A99 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A99 . b₂ <= B9 . b₂ & ( b₁ . b₂ <= B9 . b₂ or A99 . b₂ <= b₁ . b₂ ) ) ) } is set
B9 is complex real ext-real Element of REAL
[(g . A9),B9] is non empty non natural Element of [:REAL,REAL:]
{(g . A9),B9} is non empty finite V153() V154() V155() set
{(g . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(g . A9),B9},{(g . A9)}} is non empty finite V28() set
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A1 is complex real ext-real Element of REAL
[:(Seg G),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg G),REAL:] is non empty non trivial non finite set
A1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
B1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B1 . A9 is complex real ext-real Element of REAL
[(g . A9),(B1 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(g . A9),(B1 . A9)} is non empty finite V153() V154() V155() set
{{(g . A9),(B1 . A9)},{(g . A9)}} is non empty finite V28() set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . A2 is complex real ext-real Element of REAL
B1 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . B2 is complex real ext-real Element of REAL
B1 . B2 is complex real ext-real Element of REAL
r9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . r9 is complex real ext-real Element of REAL
B1 . r9 is complex real ext-real Element of REAL
[(g . r9),(B1 . r9)] is non empty non natural Element of [:REAL,REAL:]
{(g . r9),(B1 . r9)} is non empty finite V153() V154() V155() set
{(g . r9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(g . r9),(B1 . r9)},{(g . r9)}} is non empty finite V28() set
(G,k,r9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
(G,k) is Element of (G,k,G)
A1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A1,B1) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B1 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A1 . b₂ <= B1 . b₂ & ( b₁ . b₂ <= B1 . b₂ or A1 . b₂ <= b₁ . b₂ ) ) ) } is set
A1 . A9 is complex real ext-real Element of REAL
B1 . A9 is complex real ext-real Element of REAL
[(A1 . A9),(B1 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(A1 . A9),(B1 . A9)} is non empty finite V153() V154() V155() set
{(A1 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A1 . A9),(B1 . A9)},{(A1 . A9)}} is non empty finite V28() set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
B1 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 . B2 is complex real ext-real Element of REAL
B1 . B2 is complex real ext-real Element of REAL
r9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 . r9 is complex real ext-real Element of REAL
B1 . r9 is complex real ext-real Element of REAL
[(A1 . r9),(B1 . r9)] is non empty non natural Element of [:REAL,REAL:]
{(A1 . r9),(B1 . r9)} is non empty finite V153() V154() V155() set
{(A1 . r9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A1 . r9),(B1 . r9)},{(A1 . r9)}} is non empty finite V28() set
(G,k,r9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 . A9 is complex real ext-real Element of REAL
A1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A1 . A9 is complex real ext-real Element of REAL
[(B9 . A9),(A1 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . A9),(A1 . A9)} is non empty finite V153() V154() V155() set
{(B9 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . A9),(A1 . A9)},{(B9 . A9)}} is non empty finite V28() set
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 . A9 is complex real ext-real Element of REAL
A1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A1 . A9 is complex real ext-real Element of REAL
[(B9 . A9),(A1 . A9)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . A9),(A1 . A9)} is non empty finite V153() V154() V155() set
{(B9 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . A9),(A1 . A9)},{(B9 . A9)}} is non empty finite V28() set
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B1 is complex real ext-real Element of REAL
g . B1 is complex real ext-real Element of REAL
A1 . B1 is complex real ext-real Element of REAL
A . B1 is complex real ext-real Element of REAL
[(B9 . B1),(A1 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B1),(A1 . B1)} is non empty finite V153() V154() V155() set
{(B9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B1),(A1 . B1)},{(B9 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B1 is complex real ext-real Element of REAL
g . B1 is complex real ext-real Element of REAL
A1 . B1 is complex real ext-real Element of REAL
A . B1 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 . B1 is complex real ext-real Element of REAL
B9 . B1 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A1 . B1 is complex real ext-real Element of REAL
B9 . B1 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B1 is complex real ext-real Element of REAL
g . B1 is complex real ext-real Element of REAL
A1 . B1 is complex real ext-real Element of REAL
A . B1 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B2 is complex real ext-real Element of REAL
A1 . B2 is complex real ext-real Element of REAL
[(B9 . B2),(A1 . B2)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . B2),(A1 . B2)} is non empty finite V153() V154() V155() set
{(B9 . B2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . B2),(A1 . B2)},{(B9 . B2)}} is non empty finite V28() set
(G,k,B2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
(G,B9,A1) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( B9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= A1 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not B9 . b₂ <= A1 . b₂ & ( b₁ . b₂ <= A1 . b₂ or B9 . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is Element of (G,k,G)
B1 is Element of (G,k,G)
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
B9 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
B9 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
B9 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B2 is complex real ext-real Element of REAL
B9 . B2 is complex real ext-real Element of REAL
r9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . r9 is complex real ext-real Element of REAL
B9 . r9 is complex real ext-real Element of REAL
[(A99 . r9),(B9 . r9)] is non empty non natural Element of [:REAL,REAL:]
{(A99 . r9),(B9 . r9)} is non empty finite V153() V154() V155() set
{(A99 . r9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A99 . r9),(B9 . r9)},{(A99 . r9)}} is non empty finite V28() set
(G,k,r9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . A2 is complex real ext-real Element of REAL
g . A2 is complex real ext-real Element of REAL
A1 . A2 is complex real ext-real Element of REAL
A . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . B2 is complex real ext-real Element of REAL
A1 . B2 is complex real ext-real Element of REAL
r9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
B9 . r9 is complex real ext-real Element of REAL
A1 . r9 is complex real ext-real Element of REAL
[(B9 . r9),(A1 . r9)] is non empty non natural Element of [:REAL,REAL:]
{(B9 . r9),(A1 . r9)} is non empty finite V153() V154() V155() set
{(B9 . r9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(B9 . r9),(A1 . r9)},{(B9 . r9)}} is non empty finite V28() set
(G,k,r9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . A2 is complex real ext-real Element of REAL
B9 . A2 is complex real ext-real Element of REAL
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A99 . B2 is complex real ext-real Element of REAL
B9 . B2 is complex real ext-real Element of REAL
A2 is set
B2 is Element of (G,k,G)
r9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
X is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,r9,X) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( r9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= X . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not r9 . b₂ <= X . b₂ & ( b₁ . b₂ <= X . b₂ or r9 . b₂ <= b₁ . b₂ ) ) ) } is set
r9 . A9 is complex real ext-real Element of REAL
X . A9 is complex real ext-real Element of REAL
[(r9 . A9),(X . A9)] is non empty non natural Element of [:REAL,REAL:]
{(r9 . A9),(X . A9)} is non empty finite V153() V154() V155() set
{(r9 . A9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(r9 . A9),(X . A9)},{(r9 . A9)}} is non empty finite V28() set
i1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
X . i1 is complex real ext-real Element of REAL
r9 . i1 is complex real ext-real Element of REAL
g . i1 is complex real ext-real Element of REAL
A . i1 is complex real ext-real Element of REAL
[:(Seg G),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg G),REAL:] is non empty non trivial non finite set
i1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i1 . A9 is complex real ext-real Element of REAL
Y1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
i1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i1 . Y1 is complex real ext-real Element of REAL
g . Y1 is complex real ext-real Element of REAL
i1 . Y1 is complex real ext-real Element of REAL
i2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i2 . Y1 is complex real ext-real Element of REAL
A . Y1 is complex real ext-real Element of REAL
i2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i2 . Y1 is complex real ext-real Element of REAL
[:(Seg G),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg G),REAL:] is non empty non trivial non finite set
i2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i2 . A9 is complex real ext-real Element of REAL
Y1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
i2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i2 . Y1 is complex real ext-real Element of REAL
A . Y1 is complex real ext-real Element of REAL
i2 . Y1 is complex real ext-real Element of REAL
i1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i1 . Y1 is complex real ext-real Element of REAL
g . Y1 is complex real ext-real Element of REAL
i1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
i1 . Y1 is complex real ext-real Element of REAL
i1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
g . i1 is complex real ext-real Element of REAL
r9 . i1 is complex real ext-real Element of REAL
A . i1 is complex real ext-real Element of REAL
X . i1 is complex real ext-real Element of REAL
(G,k) is Element of (G,k,G)
(G,k) is Element of (G,k,G)
i1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
X . i1 is complex real ext-real Element of REAL
r9 . i1 is complex real ext-real Element of REAL
i2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
r9 . i2 is complex real ext-real Element of REAL
X . i2 is complex real ext-real Element of REAL
A99 is set
B9 is set
0 + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,(0 + 1)) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
k is Element of (d,G,(0 + 1))
{k} is non empty trivial finite 1 -element Element of bool (d,G,(0 + 1))
bool (d,G,(0 + 1)) is non empty finite V28() set
(d,G,0,{k}) is finite Element of bool (d,G,0)
(d,G,0) is non empty finite Element of bool (bool (REAL d))
bool (d,G,0) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,0) : ( 0 + 1 <= d & not card ((d,G,0,b₁) /\ {k}) is even ) } is set
card (d,G,0,{k}) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
f is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
g is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,f,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A is complex real ext-real Element of REAL
f . A is complex real ext-real Element of REAL
[(f . A),(g . A)] is non empty non natural Element of [:REAL,REAL:]
{(f . A),(g . A)} is non empty finite V153() V154() V155() set
{(f . A)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(f . A),(g . A)},{(f . A)}} is non empty finite V28() set
(d,G,A) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A9 is complex real ext-real Element of REAL
(d,G,A9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
g . A99 is complex real ext-real Element of REAL
(d,G,A99) is non empty non trivial finite V153() V154() V155() Element of bool REAL
(d,f,f) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( f . b₂ <= b₁ . b₂ & b₁ . b₂ <= f . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not f . b₂ <= f . b₂ & ( b₁ . b₂ <= f . b₂ or f . b₂ <= b₁ . b₂ ) ) ) } is set
(d,g,g) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( g . b₂ <= b₁ . b₂ & b₁ . b₂ <= g . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not g . b₂ <= g . b₂ & ( b₁ . b₂ <= g . b₂ or g . b₂ <= b₁ . b₂ ) ) ) } is set
A9 is Element of (d,G,0)
A99 is Element of (d,G,0)
{f} is non empty trivial functional finite V28() 1 -element FinSequence-membered Element of bool (REAL d)
{g} is non empty trivial functional finite V28() 1 -element FinSequence-membered Element of bool (REAL d)
B9 is set
A9 is Element of (d,G,0)
B9 is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d
(d,B9,B9) is non empty functional FinSequence-membered Element of bool (REAL d)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite d -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL d : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d holds
( B9 . b₂ <= b₁ . b₂ & b₁ . b₂ <= B9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d st
( not B9 . b₂ <= B9 . b₂ & ( b₁ . b₂ <= B9 . b₂ or B9 . b₂ <= b₁ . b₂ ) ) ) } is set
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A1 is complex real ext-real Element of REAL
g . A1 is complex real ext-real Element of REAL
B9 . A is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A1 is complex real ext-real Element of REAL
g . A1 is complex real ext-real Element of REAL
B9 . A1 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A1 . A2 is complex real ext-real Element of REAL
B1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B1 . A2 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A1 . A2 is complex real ext-real Element of REAL
B1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B1 . A2 is complex real ext-real Element of REAL
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
f . A1 is complex real ext-real Element of REAL
g . A1 is complex real ext-real Element of REAL
B9 . A1 is complex real ext-real Element of REAL
B9 . A is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
B1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B1 . B2 is complex real ext-real Element of REAL
A2 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A2 . B2 is complex real ext-real Element of REAL
[:(Seg d),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg d),REAL:] is non empty non trivial non finite set
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg d
B1 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
B1 . B2 is complex real ext-real Element of REAL
A2 is Relation-like Seg d -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg d),REAL:]
A2 . B2 is complex real ext-real Element of REAL
A9 is set
A99 is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G) is finite Element of bool (d,G,d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,d) is non empty finite V28() set
(d,G,d) is finite Element of bool (d,G,d)
(d,G,d) ` is finite Element of bool (d,G,d)
(d,G,d) \ (d,G,d) is finite set
(d,G) ` is finite Element of bool (d,G,d)
(d,G,d) \ (d,G) is finite set
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(G,k,d) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,d) is non empty finite V28() set
f is finite Element of bool (G,k,d)
(G,k,d) is finite Element of bool (G,k,d)
(G,k,d,f,(G,k,d)) is finite Element of bool (G,k,d)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,d) is non empty finite V28() set
(d,G) is finite Element of bool (d,G,d)
k is finite Element of bool (d,G,d)
k ` is finite Element of bool (d,G,d)
(d,G,d) \ k is finite set
(d,G,d,k,(d,G)) is finite Element of bool (d,G,d)
k /\ (d,G,d) is finite Element of bool (bool (REAL d))
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,(d + 1)) is finite Element of bool (G,k,(d + 1))
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,(d + 1)) is non empty finite V28() set
(G,k,d,(G,k,(d + 1))) is finite Element of bool (G,k,d)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (G,k,d) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1))) is even ) } is set
(G,k,d) is finite Element of bool (G,k,d)
f is Element of (G,k,d)
(G,k,d,f) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : f c= b<sub>1</sub> } is set
(G,k,d,f) /\ (G,k,(d + 1)) is finite Element of bool (G,k,(d + 1))
card ((G,k,d,f) /\ (G,k,(d + 1))) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
2 * 0 is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered even ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of NAT
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg (d + 1) is non empty finite d + 1 -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg (d + 1) -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d + 1)
((d + 1),G) is finite Element of bool ((d + 1),G,(d + 1))
((d + 1),G,(d + 1)) is non empty finite Element of bool (bool (REAL (d + 1)))
REAL (d + 1) is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL (d + 1)) is non empty set
bool (bool (REAL (d + 1))) is non empty set
bool ((d + 1),G,(d + 1)) is non empty finite V28() set
((d + 1),G,d,((d + 1),G)) is finite Element of bool ((d + 1),G,d)
((d + 1),G,d) is non empty finite Element of bool (bool (REAL (d + 1)))
bool ((d + 1),G,d) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of ((d + 1),G,d) : ( d + 1 <= d + 1 & not card (((d + 1),G,d,b<sub>1</sub>) /\ ((d + 1),G)) is even ) } is set
((d + 1),G,d) is finite Element of bool ((d + 1),G,d)
k is Element of ((d + 1),G,d)
((d + 1),G,d,k) is finite Element of bool ((d + 1),G,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of ((d + 1),G,(d + 1)) : k c= b<sub>1</sub> } is set
((d + 1),G,d,k) /\ ((d + 1),G) is finite Element of bool ((d + 1),G,(d + 1))
card (((d + 1),G,d,k) /\ ((d + 1),G)) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
2 * 1 is ordinal natural complex real finite cardinal V43() even ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,(d + 1)) is non empty finite V28() set
f is finite Element of bool (G,k,(d + 1))
g is finite Element of bool (G,k,(d + 1))
(G,k,(d + 1),f,g) is finite Element of bool (G,k,(d + 1))
(G,k,d,(G,k,(d + 1),f,g)) is finite Element of bool (G,k,d)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (G,k,d) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),f,g)) is even ) } is set
(G,k,d,f) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ f) is even ) } is set
(G,k,d,g) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ g) is even ) } is set
(G,k,d,(G,k,d,f),(G,k,d,g)) is finite Element of bool (G,k,d)
A is Element of (G,k,d)
(G,k,d,A) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : A c= b<sub>1</sub> } is set
f \+\ g is finite Element of bool (G,k,(d + 1))
(G,k,d,A) /\ (f \+\ g) is finite Element of bool (G,k,(d + 1))
(G,k,d,A) /\ f is finite Element of bool (G,k,(d + 1))
(G,k,d,A) /\ g is finite Element of bool (G,k,(d + 1))
((G,k,d,A) /\ f) \+\ ((G,k,d,A) /\ g) is finite Element of bool (G,k,(d + 1))
card ((G,k,d,A) /\ (f \+\ g)) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card ((G,k,d,A) /\ f) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card ((G,k,d,A) /\ g) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg (d + 1) is non empty finite d + 1 -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg (d + 1) -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d + 1)
((d + 1),G,(d + 1)) is non empty finite Element of bool (bool (REAL (d + 1)))
REAL (d + 1) is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL (d + 1)) is non empty set
bool (bool (REAL (d + 1))) is non empty set
bool ((d + 1),G,(d + 1)) is non empty finite V28() set
k is finite Element of bool ((d + 1),G,(d + 1))
k ` is finite Element of bool ((d + 1),G,(d + 1))
((d + 1),G,(d + 1)) \ k is finite set
((d + 1),G,d,(k `)) is finite Element of bool ((d + 1),G,d)
((d + 1),G,d) is non empty finite Element of bool (bool (REAL (d + 1)))
bool ((d + 1),G,d) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of ((d + 1),G,d) : ( d + 1 <= d + 1 & not card (((d + 1),G,d,b<sub>1</sub>) /\ (k `)) is even ) } is set
((d + 1),G,d,k) is finite Element of bool ((d + 1),G,d)
{ b₁ where b₁ is Element of ((d + 1),G,d) : ( d + 1 <= d + 1 & not card (((d + 1),G,d,b₁) /\ k) is even ) } is set
((d + 1),G) is finite Element of bool ((d + 1),G,(d + 1))
((d + 1),G,(d + 1),k,((d + 1),G)) is finite Element of bool ((d + 1),G,(d + 1))
((d + 1),G,d,((d + 1),G,(d + 1),k,((d + 1),G))) is finite Element of bool ((d + 1),G,d)
{ b₁ where b₁ is Element of ((d + 1),G,d) : ( d + 1 <= d + 1 & not card (((d + 1),G,d,b₁) /\ ((d + 1),G,(d + 1),k,((d + 1),G))) is even ) } is set
((d + 1),G,d,((d + 1),G)) is finite Element of bool ((d + 1),G,d)
{ b₁ where b₁ is Element of ((d + 1),G,d) : ( d + 1 <= d + 1 & not card (((d + 1),G,d,b₁) /\ ((d + 1),G)) is even ) } is set
((d + 1),G,d,((d + 1),G,d,k),((d + 1),G,d,((d + 1),G))) is finite Element of bool ((d + 1),G,d)
((d + 1),G,d) is finite Element of bool ((d + 1),G,d)
((d + 1),G,d,((d + 1),G,d,k),((d + 1),G,d)) is finite Element of bool ((d + 1),G,d)
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d + 1) + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,((d + 1) + 1)) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,((d + 1) + 1)) is non empty finite V28() set
(G,k,d) is finite Element of bool (G,k,d)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (G,k,d) is non empty finite V28() set
f is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),f) is finite Element of bool (G,k,(d + 1))
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
bool (G,k,(d + 1)) is non empty finite V28() set
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ f) is even ) } is set
(G,k,d,(G,k,(d + 1),f)) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),f)) is even ) } is set
g is Element of (G,k,((d + 1) + 1))
{g} is non empty trivial finite 1 -element Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),{g}) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ {g}) is even ) } is set
(G,k,d,(G,k,(d + 1),{g})) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),{g})) is even ) } is set
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is set
B9 is Element of (G,k,d)
(G,k,d,B9) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : B9 c= b₁ } is set
(G,k,d,B9) /\ (G,k,(d + 1),{g}) is finite Element of bool (G,k,(d + 1))
B9 is Element of (G,k,(d + 1))
card ((G,k,d,B9) /\ (G,k,(d + 1),{g})) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B9 is set
A1 is Element of (G,k,(d + 1))
the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
A9 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
bool (Seg G) is non empty finite V28() set
A2 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B2 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
r9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,B2,r9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( B2 . b₂ <= b₁ . b₂ & b₁ . b₂ <= r9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not B2 . b₂ <= r9 . b₂ & ( b₁ . b₂ <= r9 . b₂ or B2 . b₂ <= b₁ . b₂ ) ) ) } is set
B2 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
r9 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
X is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card X is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
X is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card X is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A2 \ X is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card (A2 \ X) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
1 + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + (1 + 1) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d + (1 + 1)) - d is complex real ext-real Element of REAL
i1 is set
i2 is set
i1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
{i1} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
X \/ {i1} is non empty finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
l1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . l1 is complex real ext-real Element of REAL
B2 . l1 is complex real ext-real Element of REAL
[:(Seg G),REAL:] is non empty non trivial Relation-like non finite V143() V144() V145() set
bool [:(Seg G),REAL:] is non empty non trivial non finite set
l1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . r1 is complex real ext-real Element of REAL
r9 . r1 is complex real ext-real Element of REAL
r1 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
l1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
r1 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
l1 . B1 is complex real ext-real Element of REAL
r1 . B1 is complex real ext-real Element of REAL
A . B1 is complex real ext-real Element of REAL
A9 . B1 is complex real ext-real Element of REAL
B2 . B1 is complex real ext-real Element of REAL
r9 . B1 is complex real ext-real Element of REAL
card (X \/ {i1}) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card {i1} is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card X) + (card {i1}) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
r1 . B1 is complex real ext-real Element of REAL
l1 . B1 is complex real ext-real Element of REAL
[(l1 . B1),(r1 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(l1 . B1),(r1 . B1)} is non empty finite V153() V154() V155() set
{(l1 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(l1 . B1),(r1 . B1)},{(l1 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A . B1 is complex real ext-real Element of REAL
A9 . B1 is complex real ext-real Element of REAL
B2 . B1 is complex real ext-real Element of REAL
r9 . B1 is complex real ext-real Element of REAL
(G,l1,r1) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( l1 . b₂ <= b₁ . b₂ & b₁ . b₂ <= r1 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not l1 . b₂ <= r1 . b₂ & ( b₁ . b₂ <= r1 . b₂ or l1 . b₂ <= b₁ . b₂ ) ) ) } is set
i2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
{i2} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
X \/ {i2} is non empty finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
l2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . l2 is complex real ext-real Element of REAL
B2 . l2 is complex real ext-real Element of REAL
l2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . r2 is complex real ext-real Element of REAL
r9 . r2 is complex real ext-real Element of REAL
r2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
card (X \/ {i2}) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card {i2} is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(card X) + (card {i2}) is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
r2 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
l2 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
r2 . B2 is complex real ext-real Element of REAL
l2 . B2 is complex real ext-real Element of REAL
[(l2 . B2),(r2 . B2)] is non empty non natural Element of [:REAL,REAL:]
{(l2 . B2),(r2 . B2)} is non empty finite V153() V154() V155() set
{(l2 . B2)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(l2 . B2),(r2 . B2)},{(l2 . B2)}} is non empty finite V28() set
(G,k,B2) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A . B2 is complex real ext-real Element of REAL
A9 . B2 is complex real ext-real Element of REAL
B2 . B2 is complex real ext-real Element of REAL
r9 . B2 is complex real ext-real Element of REAL
(G,l2,r2) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( l2 . b₂ <= b₁ . b₂ & b₁ . b₂ <= r2 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not l2 . b₂ <= r2 . b₂ & ( b₁ . b₂ <= r2 . b₂ or l2 . b₂ <= b₁ . b₂ ) ) ) } is set
B1 is Element of (G,k,(d + 1))
B2 is Element of (G,k,(d + 1))
B is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
l1 . B is complex real ext-real Element of REAL
B2 . B is complex real ext-real Element of REAL
r9 . B is complex real ext-real Element of REAL
r1 . B is complex real ext-real Element of REAL
A . B is complex real ext-real Element of REAL
A9 . B is complex real ext-real Element of REAL
B is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
l2 . B is complex real ext-real Element of REAL
B2 . B is complex real ext-real Element of REAL
r9 . B is complex real ext-real Element of REAL
r2 . B is complex real ext-real Element of REAL
A . B is complex real ext-real Element of REAL
A9 . B is complex real ext-real Element of REAL
l1 . i1 is complex real ext-real Element of REAL
A . i1 is complex real ext-real Element of REAL
r1 . i1 is complex real ext-real Element of REAL
A9 . i1 is complex real ext-real Element of REAL
l2 . i1 is complex real ext-real Element of REAL
B2 . i1 is complex real ext-real Element of REAL
r2 . i1 is complex real ext-real Element of REAL
r9 . i1 is complex real ext-real Element of REAL
B is set
B is Element of (G,k,(d + 1))
l99 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
r99 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,l99,r99) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( l99 . b₂ <= b₁ . b₂ & b₁ . b₂ <= r99 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not l99 . b₂ <= r99 . b₂ & ( b₁ . b₂ <= r99 . b₂ or l99 . b₂ <= b₁ . b₂ ) ) ) } is set
l99 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
r99 . the ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G is complex real ext-real Element of REAL
Y is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card Y is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
Y is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card Y is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
Y \ X is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card (Y \ X) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(d + 1) - d is complex real ext-real Element of REAL
i9 is set
{i9} is non empty trivial finite 1 -element set
i9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
X \/ Y is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
{i9} is non empty trivial finite V28() 1 -element V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
X \/ {i9} is non empty finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
i is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
l99 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
l99 . i is complex real ext-real Element of REAL
A . i is complex real ext-real Element of REAL
l2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
l2 . i is complex real ext-real Element of REAL
r99 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r99 . i is complex real ext-real Element of REAL
A9 . i is complex real ext-real Element of REAL
r2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r2 . i is complex real ext-real Element of REAL
B2 . i is complex real ext-real Element of REAL
r9 . i is complex real ext-real Element of REAL
i is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
l99 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
l99 . i is complex real ext-real Element of REAL
A . i is complex real ext-real Element of REAL
l2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
l2 . i is complex real ext-real Element of REAL
r99 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r99 . i is complex real ext-real Element of REAL
A9 . i is complex real ext-real Element of REAL
r2 is Relation-like Seg G -defined REAL -valued Function-like finite quasi_total V143() V144() V145() Element of bool [:(Seg G),REAL:]
r2 . i is complex real ext-real Element of REAL
B2 . i is complex real ext-real Element of REAL
r9 . i is complex real ext-real Element of REAL
2 * 1 is ordinal natural complex real finite cardinal V43() even ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
i1 is set
i2 is set
g is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),g) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ g) is even ) } is set
(G,k,d,(G,k,(d + 1),g)) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),g)) is even ) } is set
A is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),A) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ A) is even ) } is set
(G,k,d,(G,k,(d + 1),A)) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),A)) is even ) } is set
(G,k,((d + 1) + 1),g,A) is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),(G,k,((d + 1) + 1),g,A)) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ (G,k,((d + 1) + 1),g,A)) is even ) } is set
(G,k,d,(G,k,(d + 1),(G,k,((d + 1) + 1),g,A))) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),(G,k,((d + 1) + 1),g,A))) is even ) } is set
(G,k,(d + 1),(G,k,(d + 1),g),(G,k,(d + 1),A)) is finite Element of bool (G,k,(d + 1))
(G,k,d,(G,k,(d + 1),(G,k,(d + 1),g),(G,k,(d + 1),A))) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),(G,k,(d + 1),g),(G,k,(d + 1),A))) is even ) } is set
(G,k,d,(G,k,d),(G,k,d)) is finite Element of bool (G,k,d)
(G,k,((d + 1) + 1)) is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),(G,k,((d + 1) + 1))) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ (G,k,((d + 1) + 1))) is even ) } is set
(G,k,d,(G,k,(d + 1),(G,k,((d + 1) + 1)))) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),(G,k,((d + 1) + 1)))) is even ) } is set
(G,k,(d + 1)) is finite Element of bool (G,k,(d + 1))
(G,k,d,(G,k,(d + 1))) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1))) is even ) } is set
g is Element of (G,k,((d + 1) + 1))
{g} is non empty trivial finite 1 -element Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),{g}) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ {g}) is even ) } is set
(G,k,d,(G,k,(d + 1),{g})) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),{g})) is even ) } is set
bool (Seg G) is non empty finite V28() set
A is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A,A9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A . b₂ <= b₁ . b₂ & b₁ . b₂ <= A9 . b₂ ) or ex b₂ being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A . b₂ <= A9 . b₂ & ( b₁ . b₂ <= A9 . b₂ or A . b₂ <= b₁ . b₂ ) ) ) } is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
B9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(G,k) is Element of (G,k,G)
(G,k,G) is non empty finite Element of bool (bool (REAL G))
{g} ` is finite Element of bool (G,k,((d + 1) + 1))
(G,k,((d + 1) + 1)) \ {g} is finite set
B9 is Element of (G,k,((d + 1) + 1))
{B9} is non empty trivial finite 1 -element Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),{B9}) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ {B9}) is even ) } is set
(G,k,d,(G,k,(d + 1),{B9})) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),{B9})) is even ) } is set
A9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
B9 is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G
(G,A9,B9) is non empty functional FinSequence-membered Element of bool (REAL G)
{ b<sub>1</sub> where b<sub>1</sub> is Relation-like NAT -defined REAL -valued Function-like finite G -element FinSequence-like FinSubsequence-like V143() V144() V145() Element of REAL G : ( for b<sub>2</sub> being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G holds
( A9 . b<sub>2</sub> <= b<sub>1</sub> . b<sub>2</sub> & b<sub>1</sub> . b<sub>2</sub> <= B9 . b<sub>2</sub> ) or ex b<sub>2</sub> being ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G st
( not A9 . b<sub>2</sub> <= B9 . b<sub>2</sub> & ( b<sub>1</sub> . b<sub>2</sub> <= B9 . b<sub>2</sub> or A9 . b<sub>2</sub> <= b<sub>1</sub> . b<sub>2</sub> ) ) ) } is set
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . A1 is complex real ext-real Element of REAL
B9 . A1 is complex real ext-real Element of REAL
B1 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . A1 is complex real ext-real Element of REAL
B9 . A1 is complex real ext-real Element of REAL
B1 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A9 . B1 is complex real ext-real Element of REAL
B9 . B1 is complex real ext-real Element of REAL
[(A9 . B1),(B9 . B1)] is non empty non natural Element of [:REAL,REAL:]
{(A9 . B1),(B9 . B1)} is non empty finite V153() V154() V155() set
{(A9 . B1)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A9 . B1),(B9 . B1)},{(A9 . B1)}} is non empty finite V28() set
(G,k,B1) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A2 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A2 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
B9 is finite Element of bool (G,k,((d + 1) + 1))
A9 is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),B9) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ B9) is even ) } is set
(G,k,d,(G,k,(d + 1),B9)) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),B9)) is even ) } is set
(G,k,(d + 1),A9) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ A9) is even ) } is set
(G,k,d,(G,k,(d + 1),A9)) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),A9)) is even ) } is set
(G,k,((d + 1) + 1),B9,A9) is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),(G,k,((d + 1) + 1),B9,A9)) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ (G,k,((d + 1) + 1),B9,A9)) is even ) } is set
(G,k,d,(G,k,(d + 1),(G,k,((d + 1) + 1),B9,A9))) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),(G,k,((d + 1) + 1),B9,A9))) is even ) } is set
(G,k,(d + 1),({g} `)) is finite Element of bool (G,k,(d + 1))
{ b₁ where b₁ is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b₁) /\ ({g} `)) is even ) } is set
(G,k,d,(G,k,(d + 1),({g} `))) is finite Element of bool (G,k,d)
{ b₁ where b₁ is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b₁) /\ (G,k,(d + 1),({g} `))) is even ) } is set
A99 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . A99 is complex real ext-real Element of REAL
A9 . A99 is complex real ext-real Element of REAL
B9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() Element of Seg G
A . B9 is complex real ext-real Element of REAL
A9 . B9 is complex real ext-real Element of REAL
[(A . B9),(A9 . B9)] is non empty non natural Element of [:REAL,REAL:]
{(A . B9),(A9 . B9)} is non empty finite V153() V154() V155() set
{(A . B9)} is non empty trivial finite 1 -element V153() V154() V155() set
{{(A . B9),(A9 . B9)},{(A . B9)}} is non empty finite V28() set
(G,k,B9) is non empty non trivial finite V153() V154() V155() Element of bool REAL
A9 is finite V153() V154() V155() V156() V157() V158() Element of bool (Seg G)
card A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
g is Element of (G,k,((d + 1) + 1))
{g} is non empty trivial finite 1 -element Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),{g}) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ {g}) is even ) } is set
(G,k,d,(G,k,(d + 1),{g})) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),{g})) is even ) } is set
g is finite Element of bool (G,k,((d + 1) + 1))
A is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),g) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ g) is even ) } is set
(G,k,d,(G,k,(d + 1),g)) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),g)) is even ) } is set
(G,k,(d + 1),A) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ A) is even ) } is set
(G,k,d,(G,k,(d + 1),A)) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),A)) is even ) } is set
(G,k,((d + 1) + 1),g,A) is finite Element of bool (G,k,((d + 1) + 1))
(G,k,(d + 1),(G,k,((d + 1) + 1),g,A)) is finite Element of bool (G,k,(d + 1))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,(d + 1)) : ( (d + 1) + 1 <= G & not card ((G,k,(d + 1),b<sub>1</sub>) /\ (G,k,((d + 1) + 1),g,A)) is even ) } is set
(G,k,d,(G,k,(d + 1),(G,k,((d + 1) + 1),g,A))) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ (G,k,(d + 1),(G,k,((d + 1) + 1),g,A))) is even ) } is set
(G,k,(d + 1)) is finite Element of bool (G,k,(d + 1))
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
(d,G,k) is finite Element of bool (d,G,k)
card (d,G,k) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
f is ordinal natural complex real finite cardinal ext-real non negative set
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f is ordinal natural complex real finite cardinal ext-real non negative set
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is finite Element of bool (d,G,k)
A is finite Element of bool (d,G,k)
card A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(A9 + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,(A9 + 1)) is non empty finite V28() set
A99 is finite Element of bool (d,G,(A9 + 1))
B9 is finite Element of bool (d,G,(A9 + 1))
(d,G,A9,B9) is finite Element of bool (d,G,A9)
(d,G,A9) is non empty finite Element of bool (bool (REAL d))
bool (d,G,A9) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,A9) : ( A9 + 1 <= d & not card ((d,G,A9,b<sub>1</sub>) /\ B9) is even ) } is set
(d,G,A9) is finite Element of bool (d,G,A9)
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(A9 + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,(A9 + 1)) is non empty finite V28() set
A99 is finite Element of bool (d,G,(A9 + 1))
(d,G,A9,A99) is finite Element of bool (d,G,A9)
(d,G,A9) is non empty finite Element of bool (bool (REAL d))
bool (d,G,A9) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,A9) : ( A9 + 1 <= d & not card ((d,G,A9,b<sub>1</sub>) /\ A99) is even ) } is set
(d,G,A9) is finite Element of bool (d,G,A9)
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
d + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,(d + 1)) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,(d + 1)) is non empty finite V28() set
(G,k,d) is finite Element of bool (G,k,d)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
bool (G,k,d) is non empty finite V28() set
f is finite Element of bool (G,k,(d + 1))
(G,k,d,f) is finite Element of bool (G,k,d)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,d) : ( d + 1 <= G & not card ((G,k,d,b<sub>1</sub>) /\ f) is even ) } is set
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(G,k,(g + 1)) is non empty finite Element of bool (bool (REAL G))
bool (G,k,(g + 1)) is non empty finite V28() set
A is finite Element of bool (G,k,(g + 1))
(G,k,g,A) is finite Element of bool (G,k,g)
(G,k,g) is non empty finite Element of bool (bool (REAL G))
bool (G,k,g) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,g) : ( g + 1 <= G & not card ((G,k,g,b<sub>1</sub>) /\ A) is even ) } is set
(G,k,g) is finite Element of bool (G,k,g)
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
(G,k,d) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,d) is non empty finite V28() set
f is finite Element of bool (G,k,d)
g is ordinal natural complex real finite cardinal ext-real non negative set
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(G,k,(A + 1)) is non empty finite Element of bool (bool (REAL G))
bool (G,k,(A + 1)) is non empty finite V28() set
A9 is finite Element of bool (G,k,(A + 1))
(G,k,A,A9) is finite Element of bool (G,k,A)
(G,k,A) is non empty finite Element of bool (bool (REAL G))
bool (G,k,A) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (G,k,A) : ( A + 1 <= G & not card ((G,k,A,b<sub>1</sub>) /\ A9) is even ) } is set
(G,k,A) is finite Element of bool (G,k,A)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,0) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,0) is non empty finite V28() set
k is finite Element of bool (d,G,0)
card k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(f + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,(f + 1)) is non empty finite V28() set
g is finite Element of bool (d,G,(f + 1))
(d,G,f,g) is finite Element of bool (d,G,f)
(d,G,f) is non empty finite Element of bool (bool (REAL d))
bool (d,G,f) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,f) : ( f + 1 <= d & not card ((d,G,f,b<sub>1</sub>) /\ g) is even ) } is set
(d,G,f) is finite Element of bool (d,G,f)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
f is finite (d,G,k + 1)
(d,G,k,f) is finite Element of bool (d,G,k)
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b<sub>1</sub>) /\ f) is even ) } is set
(d,G,k) is finite Element of bool (d,G,k)
g is finite Element of bool (d,G,k)
A is finite Element of bool (d,G,k)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is finite Element of bool (d,G,k)
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
card {} is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of omega
2 * 0 is empty trivial ordinal natural complex real Relation-like non-empty empty-yielding RAT -valued functional finite finite-yielding V28() cardinal {} -element V43() FinSequence-like FinSequence-membered even ext-real non positive non negative V64() V143() V144() V145() V146() V153() V154() V155() V156() V157() V158() V159() Element of NAT
f is ordinal natural complex real finite cardinal ext-real non negative set
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f is ordinal natural complex real finite cardinal ext-real non negative set
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(g + 1)) is finite Element of bool (d,G,(g + 1))
(d,G,(g + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,(g + 1)) is non empty finite V28() set
(d,G,g,(d,G,(g + 1))) is finite Element of bool (d,G,g)
(d,G,g) is non empty finite Element of bool (bool (REAL d))
bool (d,G,g) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,g) : ( g + 1 <= d & not card ((d,G,g,b<sub>1</sub>) /\ (d,G,(g + 1))) is even ) } is set
(d,G,g) is finite Element of bool (d,G,g)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G) is finite Element of bool (d,G,d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,d) is non empty finite V28() set
k is ordinal natural complex real finite cardinal ext-real non negative set
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg (f + 1) is non empty finite f + 1 -element V153() V154() V155() V156() V157() V158() Element of bool NAT
g is Relation-like Seg (f + 1) -defined bool REAL -valued Function-like finite finite-yielding quasi_total (f + 1)
((f + 1),g) is finite Element of bool ((f + 1),g,(f + 1))
((f + 1),g,(f + 1)) is non empty finite Element of bool (bool (REAL (f + 1)))
REAL (f + 1) is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL (f + 1)) is non empty set
bool (bool (REAL (f + 1))) is non empty set
bool ((f + 1),g,(f + 1)) is non empty finite V28() set
((f + 1),g,f,((f + 1),g)) is finite Element of bool ((f + 1),g,f)
((f + 1),g,f) is non empty finite Element of bool (bool (REAL (f + 1)))
bool ((f + 1),g,f) is non empty finite V28() set
{ b<sub>1</sub> where b<sub>1</sub> is Element of ((f + 1),g,f) : ( f + 1 <= f + 1 & not card (((f + 1),g,f,b<sub>1</sub>) /\ ((f + 1),g)) is even ) } is set
((f + 1),g,f) is finite (f + 1,g,f)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
f is finite (d,G,k)
g is finite (d,G,k)
f \+\ g is finite set
card f is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
card g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
(d,G,k,f,g) is finite Element of bool (d,G,k)
(d,G,k) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,k) is non empty finite V28() set
card (d,G,k,f,g) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
A is ordinal natural complex real finite cardinal ext-real non negative set
A + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A is ordinal natural complex real finite cardinal ext-real non negative set
A + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A9 + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A99 is finite (d,G,A9 + 1)
(d,G,A9,A99) is finite Element of bool (d,G,A9)
(d,G,A9) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,A9) is non empty finite V28() set
(d,G,(A9 + 1)) is non empty finite Element of bool (bool (REAL d))
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,A9) : ( A9 + 1 <= d & not card ((d,G,A9,b<sub>1</sub>) /\ A99) is even ) } is set
(d,G,A9) is finite Element of bool (d,G,A9)
(d,G,A9) is finite (d,G,A9)
B9 is finite (d,G,A9 + 1)
(d,G,A9,B9) is finite Element of bool (d,G,A9)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,A9) : ( A9 + 1 <= d & not card ((d,G,A9,b<sub>1</sub>) /\ B9) is even ) } is set
(d,G,(A9 + 1),A99,B9) is finite Element of bool (d,G,(A9 + 1))
bool (d,G,(A9 + 1)) is non empty finite V28() set
(d,G,A9,(d,G,(A9 + 1),A99,B9)) is finite Element of bool (d,G,A9)
{ b<sub>1</sub> where b<sub>1</sub> is Element of (d,G,A9) : ( A9 + 1 <= d & not card ((d,G,A9,b<sub>1</sub>) /\ (d,G,(A9 + 1),A99,B9)) is even ) } is set
(d,G,A9,(d,G,A9),(d,G,A9)) is finite Element of bool (d,G,A9)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
(d,G,d) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
k is finite (d,G,d)
k ` is finite Element of bool (d,G,d)
bool (d,G,d) is non empty finite V28() set
(d,G,d) \ k is finite set
f is ordinal natural complex real finite cardinal ext-real non negative set
f + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg (g + 1) is non empty finite g + 1 -element V153() V154() V155() V156() V157() V158() Element of bool NAT
A is Relation-like Seg (g + 1) -defined bool REAL -valued Function-like finite finite-yielding quasi_total (g + 1)
A9 is finite (g + 1,A,g + 1)
((g + 1),A,g,A9) is finite Element of bool ((g + 1),A,g)
((g + 1),A,g) is non empty finite Element of bool (bool (REAL (g + 1)))
REAL (g + 1) is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL (g + 1)) is non empty set
bool (bool (REAL (g + 1))) is non empty set
bool ((g + 1),A,g) is non empty finite V28() set
((g + 1),A,(g + 1)) is non empty finite Element of bool (bool (REAL (g + 1)))
{ b₁ where b₁ is Element of ((g + 1),A,g) : ( g + 1 <= g + 1 & not card (((g + 1),A,g,b₁) /\ A9) is even ) } is set
((g + 1),A,g) is finite Element of bool ((g + 1),A,g)
((g + 1),A,g) is finite (g + 1,A,g)
A9 ` is finite Element of bool ((g + 1),A,(g + 1))
bool ((g + 1),A,(g + 1)) is non empty finite V28() set
((g + 1),A,(g + 1)) \ A9 is finite set
((g + 1),A,g,(A9 `)) is finite Element of bool ((g + 1),A,g)
{ b₁ where b₁ is Element of ((g + 1),A,g) : ( g + 1 <= g + 1 & not card (((g + 1),A,g,b₁) /\ (A9 `)) is even ) } is set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,(k + 1)) is non empty finite V28() set
f is finite Element of bool (d,G,(k + 1))
(d,G,k,f) is finite Element of bool (d,G,k)
(d,G,k) is non empty finite Element of bool (bool (REAL d))
bool (d,G,k) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ f) is even ) } is set
(d,G,(k + 1)) is finite (d,G,k + 1)
(d,G,k,(d,G,(k + 1))) is finite Element of bool (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ (d,G,(k + 1))) is even ) } is set
(d,G,k) is finite Element of bool (d,G,k)
(d,G,k) is finite (d,G,k)
(d,G,(0 + 1)) is non empty finite Element of bool (bool (REAL d))
g is Element of (d,G,(0 + 1))
{g} is non empty trivial finite 1 -element Element of bool (d,G,(0 + 1))
bool (d,G,(0 + 1)) is non empty finite V28() set
(d,G,0,{g}) is finite Element of bool (d,G,0)
(d,G,0) is non empty finite Element of bool (bool (REAL d))
bool (d,G,0) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,0) : ( 0 + 1 <= d & not card ((d,G,0,b₁) /\ {g}) is even ) } is set
card (d,G,0,{g}) is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of omega
2 * 1 is ordinal natural complex real finite cardinal V43() even ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g is Element of (d,G,(k + 1))
{g} is non empty trivial finite 1 -element Element of bool (d,G,(k + 1))
(d,G,k,{g}) is finite Element of bool (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ {g}) is even ) } is set
g is finite Element of bool (d,G,(k + 1))
A is finite Element of bool (d,G,(k + 1))
(d,G,k,g) is finite Element of bool (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ g) is even ) } is set
(d,G,k,A) is finite Element of bool (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ A) is even ) } is set
(d,G,(k + 1),g,A) is finite Element of bool (d,G,(k + 1))
(d,G,k,(d,G,(k + 1),g,A)) is finite Element of bool (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ (d,G,(k + 1),g,A)) is even ) } is set
A9 is finite (d,G,k)
A99 is finite (d,G,k)
(d,G,k,A9,A99) is finite (d,G,k)
g is ordinal natural complex real finite cardinal ext-real non negative set
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
g is ordinal natural complex real finite cardinal ext-real non negative set
g + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
A + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(A + 1) + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,((A + 1) + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,((A + 1) + 1)) is non empty finite V28() set
A9 is finite Element of bool (d,G,((A + 1) + 1))
(d,G,(A + 1),A9) is finite Element of bool (d,G,(A + 1))
(d,G,(A + 1)) is non empty finite Element of bool (bool (REAL d))
bool (d,G,(A + 1)) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,(A + 1)) : ( (A + 1) + 1 <= d & not card ((d,G,(A + 1),b₁) /\ A9) is even ) } is set
(d,G,A,(d,G,(A + 1),A9)) is finite Element of bool (d,G,A)
(d,G,A) is non empty finite Element of bool (bool (REAL d))
bool (d,G,A) is non empty finite V28() set
{ b₁ where b₁ is Element of (d,G,A) : ( A + 1 <= d & not card ((d,G,A,b₁) /\ (d,G,(A + 1),A9)) is even ) } is set
(d,G,A) is finite (d,G,A)
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,k) is non empty finite Element of bool (bool (REAL d))
bool (bool (REAL d)) is non empty set
((bool (REAL d)),(d,G,k)) is non empty finite V28() Element of bool (bool (bool (REAL d)))
bool (bool (bool (REAL d))) is non empty set
(d,G,k) is finite (d,G,k)
bool (d,G,k) is non empty finite V28() set
[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):] is non empty Relation-like finite set
[:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):] is non empty Relation-like finite set
bool [:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):] is non empty finite V28() set
f is Relation-like [:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):] -defined ((bool (REAL d)),(d,G,k)) -valued Function-like finite quasi_total Element of bool [:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):]
addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is strict addLoopStr
the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is set
A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A9 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A + A9 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Relation-like [: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] -defined the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) -valued Function-like quasi_total Element of bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):]
[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
[:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is non empty set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A,A9) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A,A9] is non empty non natural set
{A,A9} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,A9},{A}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A,A9] is set
A99 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
(A + A9) + A99 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . ((A + A9),A99) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[(A + A9),A99] is non empty non natural set
{(A + A9),A99} is non empty finite set
{(A + A9)} is non empty trivial finite 1 -element set
{{(A + A9),A99},{(A + A9)}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [(A + A9),A99] is set
A9 + A99 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A9,A99) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A9,A99] is non empty non natural set
{A9,A99} is non empty finite set
{A9} is non empty trivial finite 1 -element set
{{A9,A99},{A9}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A9,A99] is set
A + (A9 + A99) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A,(A9 + A99)) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A,(A9 + A99)] is non empty non natural set
{A,(A9 + A99)} is non empty finite set
{{A,(A9 + A99)},{A}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A,(A9 + A99)] is set
B9 is finite Element of bool (d,G,k)
A9 is finite Element of bool (d,G,k)
(d,G,k,B9,A9) is finite Element of bool (d,G,k)
f . ((d,G,k,B9,A9),A99) is set
[(d,G,k,B9,A9),A99] is non empty non natural set
{(d,G,k,B9,A9),A99} is non empty finite set
{(d,G,k,B9,A9)} is non empty trivial finite V28() 1 -element set
{{(d,G,k,B9,A9),A99},{(d,G,k,B9,A9)}} is non empty finite V28() set
f . [(d,G,k,B9,A9),A99] is set
B9 is finite Element of bool (d,G,k)
(d,G,k,(d,G,k,B9,A9),B9) is finite Element of bool (d,G,k)
(d,G,k,A9,B9) is finite Element of bool (d,G,k)
(d,G,k,B9,(d,G,k,A9,B9)) is finite Element of bool (d,G,k)
f . (A,(d,G,k,A9,B9)) is set
[A,(d,G,k,A9,B9)] is non empty non natural set
{A,(d,G,k,A9,B9)} is non empty finite set
{{A,(d,G,k,A9,B9)},{A}} is non empty finite V28() set
f . [A,(d,G,k,A9,B9)] is set
the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is set
A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is zero Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the ZeroF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A + (0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Relation-like [: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] -defined the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) -valued Function-like quasi_total Element of bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):]
[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
[:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is non empty set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A,(0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #))) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A,(0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #))] is non empty non natural set
{A,(0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #))} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,(0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #))},{A}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A,(0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #))] is set
A9 is finite Element of bool (d,G,k)
(d,G,k,A9,(d,G,k)) is finite Element of bool (d,G,k)
the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is set
A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A + A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Relation-like [: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] -defined the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) -valued Function-like quasi_total Element of bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):]
[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
[:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is non empty set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A,A) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A,A] is non empty non natural set
{A,A} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,A},{A}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A,A] is set
0. addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is zero Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the ZeroF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A9 is finite Element of bool (d,G,k)
(d,G,k,A9,A9) is finite Element of bool (d,G,k)
the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is set
A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A9 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
A + A9 is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) is Relation-like [: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] -defined the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) -valued Function-like quasi_total Element of bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):]
[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
[:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is Relation-like set
bool [:[: the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #), the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):], the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #):] is non empty set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A,A9) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A,A9] is non empty non natural set
{A,A9} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,A9},{A}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A,A9] is set
A9 + A is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . (A9,A) is Element of the carrier of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #)
[A9,A] is non empty non natural set
{A9,A} is non empty finite set
{A9} is non empty trivial finite 1 -element set
{{A9,A},{A9}} is non empty finite V28() set
the addF of addLoopStr(# ((bool (REAL d)),(d,G,k)),f,(d,G,k) #) . [A9,A] is set
A99 is finite Element of bool (d,G,k)
B9 is finite Element of bool (d,G,k)
(d,G,k,A99,B9) is finite Element of bool (d,G,k)
A is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of A is non empty set
0. A is zero Element of the carrier of A
the ZeroF of A is Element of the carrier of A
A9 is Element of the carrier of A
A99 is Element of the carrier of A
A9 + A99 is Element of the carrier of A
the addF of A is Relation-like [: the carrier of A, the carrier of A:] -defined the carrier of A -valued Function-like quasi_total Element of bool [:[: the carrier of A, the carrier of A:], the carrier of A:]
[: the carrier of A, the carrier of A:] is non empty Relation-like set
[:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty Relation-like set
bool [:[: the carrier of A, the carrier of A:], the carrier of A:] is non empty set
the addF of A . (A9,A99) is Element of the carrier of A
[A9,A99] is non empty non natural set
{A9,A99} is non empty finite set
{A9} is non empty trivial finite 1 -element set
{{A9,A99},{A9}} is non empty finite V28() set
the addF of A . [A9,A99] is set
B9 is finite Element of bool (d,G,k)
A9 is finite Element of bool (d,G,k)
(d,G,k,B9,A9) is finite Element of bool (d,G,k)
f is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of f is non empty set
0. f is zero Element of the carrier of f
the ZeroF of f is Element of the carrier of f
g is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of g is non empty set
0. g is zero Element of the carrier of g
the ZeroF of g is Element of the carrier of g
[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):] is non empty Relation-like finite set
[:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):] is non empty Relation-like finite set
bool [:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):] is non empty finite V28() set
the addF of f is Relation-like [: the carrier of f, the carrier of f:] -defined the carrier of f -valued Function-like quasi_total Element of bool [:[: the carrier of f, the carrier of f:], the carrier of f:]
[: the carrier of f, the carrier of f:] is non empty Relation-like set
[:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty Relation-like set
bool [:[: the carrier of f, the carrier of f:], the carrier of f:] is non empty set
the addF of g is Relation-like [: the carrier of g, the carrier of g:] -defined the carrier of g -valued Function-like quasi_total Element of bool [:[: the carrier of g, the carrier of g:], the carrier of g:]
[: the carrier of g, the carrier of g:] is non empty Relation-like set
[:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty Relation-like set
bool [:[: the carrier of g, the carrier of g:], the carrier of g:] is non empty set
B9 is Element of [:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):]
A9 is finite Element of bool (d,G,k)
B9 is finite Element of bool (d,G,k)
[A9,B9] is non empty non natural Element of [:(bool (d,G,k)),(bool (d,G,k)):]
[:(bool (d,G,k)),(bool (d,G,k)):] is non empty Relation-like finite set
{A9,B9} is non empty finite V28() set
{A9} is non empty trivial finite V28() 1 -element set
{{A9,B9},{A9}} is non empty finite V28() set
A9 is Relation-like [:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):] -defined ((bool (REAL d)),(d,G,k)) -valued Function-like finite quasi_total Element of bool [:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):]
A9 . B9 is finite Element of ((bool (REAL d)),(d,G,k))
A1 is Element of the carrier of f
B1 is Element of the carrier of f
A1 + B1 is Element of the carrier of f
the addF of f . (A1,B1) is Element of the carrier of f
[A1,B1] is non empty non natural set
{A1,B1} is non empty finite set
{A1} is non empty trivial finite 1 -element set
{{A1,B1},{A1}} is non empty finite V28() set
the addF of f . [A1,B1] is set
(d,G,k,A9,B9) is finite Element of bool (d,G,k)
A2 is Element of the carrier of g
B2 is Element of the carrier of g
A2 + B2 is Element of the carrier of g
the addF of g . (A2,B2) is Element of the carrier of g
[A2,B2] is non empty non natural set
{A2,B2} is non empty finite set
{A2} is non empty trivial finite 1 -element set
{{A2,B2},{A2}} is non empty finite V28() set
the addF of g . [A2,B2] is set
A99 is Relation-like [:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):] -defined ((bool (REAL d)),(d,G,k)) -valued Function-like finite quasi_total Element of bool [:[:((bool (REAL d)),(d,G,k)),((bool (REAL d)),(d,G,k)):],((bool (REAL d)),(d,G,k)):]
A99 . B9 is finite Element of ((bool (REAL d)),(d,G,k))
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg G is non empty finite G -element V153() V154() V155() V156() V157() V158() Element of bool NAT
A is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg A is non empty finite A -element V153() V154() V155() V156() V157() V158() Element of bool NAT
f is set
k is Relation-like Seg G -defined bool REAL -valued Function-like finite finite-yielding quasi_total (G)
d is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(G,k,d) is non empty finite Element of bool (bool (REAL G))
REAL G is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL G) is non empty set
bool (bool (REAL G)) is non empty set
bool (G,k,d) is non empty finite V28() set
(G,k,d) is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of (G,k,d) is non empty set
A99 is set
A9 is Relation-like Seg A -defined bool REAL -valued Function-like finite finite-yielding quasi_total (A)
g is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(A,A9,g) is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of (A,A9,g) is non empty set
(A,A9,g) is non empty finite Element of bool (bool (REAL A))
REAL A is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL A) is non empty set
bool (bool (REAL A)) is non empty set
bool (A,A9,g) is non empty finite V28() set
d is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
Seg d is non empty finite d -element V153() V154() V155() V156() V157() V158() Element of bool NAT
G is Relation-like Seg d -defined bool REAL -valued Function-like finite finite-yielding quasi_total (d)
k is ordinal natural complex real finite cardinal V43() ext-real non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
k + 1 is non empty ordinal natural complex real finite cardinal V43() ext-real positive non negative V64() V153() V154() V155() V156() V157() V158() Element of NAT
(d,G,(k + 1)) is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of (d,G,(k + 1)) is non empty set
(d,G,k) is non empty strict right_complementable Abelian add-associative right_zeroed addLoopStr
the carrier of (d,G,k) is non empty set
[: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):] is non empty Relation-like set
bool [: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):] is non empty set
(d,G,(k + 1)) is non empty finite Element of bool (bool (REAL d))
REAL d is non empty functional FinSequence-membered FinSequenceSet of REAL
bool (REAL d) is non empty set
bool (bool (REAL d)) is non empty set
bool (d,G,(k + 1)) is non empty finite V28() set
((bool (REAL d)),(d,G,(k + 1))) is non empty finite V28() Element of bool (bool (bool (REAL d)))
bool (bool (bool (REAL d))) is non empty set
(d,G,k) is non empty finite Element of bool (bool (REAL d))
((bool (REAL d)),(d,G,k)) is non empty finite V28() Element of bool (bool (bool (REAL d)))
[:((bool (REAL d)),(d,G,(k + 1))),((bool (REAL d)),(d,G,k)):] is non empty Relation-like finite set
bool [:((bool (REAL d)),(d,G,(k + 1))),((bool (REAL d)),(d,G,k)):] is non empty finite V28() set
f is Relation-like ((bool (REAL d)),(d,G,(k + 1))) -defined ((bool (REAL d)),(d,G,k)) -valued Function-like finite quasi_total Element of bool [:((bool (REAL d)),(d,G,(k + 1))),((bool (REAL d)),(d,G,k)):]
A is Element of the carrier of (d,G,(k + 1))
A9 is Element of the carrier of (d,G,(k + 1))
A + A9 is Element of the carrier of (d,G,(k + 1))
the addF of (d,G,(k + 1)) is Relation-like [: the carrier of (d,G,(k + 1)), the carrier of (d,G,(k + 1)):] -defined the carrier of (d,G,(k + 1)) -valued Function-like quasi_total Element of bool [:[: the carrier of (d,G,(k + 1)), the carrier of (d,G,(k + 1)):], the carrier of (d,G,(k + 1)):]
[: the carrier of (d,G,(k + 1)), the carrier of (d,G,(k + 1)):] is non empty Relation-like set
[:[: the carrier of (d,G,(k + 1)), the carrier of (d,G,(k + 1)):], the carrier of (d,G,(k + 1)):] is non empty Relation-like set
bool [:[: the carrier of (d,G,(k + 1)), the carrier of (d,G,(k + 1)):], the carrier of (d,G,(k + 1)):] is non empty set
the addF of (d,G,(k + 1)) . (A,A9) is Element of the carrier of (d,G,(k + 1))
[A,A9] is non empty non natural set
{A,A9} is non empty finite set
{A} is non empty trivial finite 1 -element set
{{A,A9},{A}} is non empty finite V28() set
the addF of (d,G,(k + 1)) . [A,A9] is set
f . (A + A9) is set
A99 is finite Element of bool (d,G,(k + 1))
B9 is finite Element of bool (d,G,(k + 1))
(d,G,(k + 1),A99,B9) is finite Element of bool (d,G,(k + 1))
f . (d,G,(k + 1),A99,B9) is finite Element of ((bool (REAL d)),(d,G,k))
(d,G,k,(d,G,(k + 1),A99,B9)) is finite (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ (d,G,(k + 1),A99,B9)) is even ) } is set
(d,G,k,A99) is finite (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ A99) is even ) } is set
(d,G,k,B9) is finite (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ B9) is even ) } is set
(d,G,k,(d,G,k,A99),(d,G,k,B9)) is finite (d,G,k)
f . B9 is finite Element of ((bool (REAL d)),(d,G,k))
(d,G,k,(d,G,k,A99),(f . B9)) is finite Element of bool (d,G,k)
bool (d,G,k) is non empty finite V28() set
f . A99 is finite Element of ((bool (REAL d)),(d,G,k))
(d,G,k,(f . A99),(f . B9)) is finite Element of bool (d,G,k)
g is Relation-like the carrier of (d,G,(k + 1)) -defined the carrier of (d,G,k) -valued Function-like quasi_total Element of bool [: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):]
g . A is Element of the carrier of (d,G,k)
g . A9 is Element of the carrier of (d,G,k)
(g . A) + (g . A9) is Element of the carrier of (d,G,k)
the addF of (d,G,k) is Relation-like [: the carrier of (d,G,k), the carrier of (d,G,k):] -defined the carrier of (d,G,k) -valued Function-like quasi_total Element of bool [:[: the carrier of (d,G,k), the carrier of (d,G,k):], the carrier of (d,G,k):]
[: the carrier of (d,G,k), the carrier of (d,G,k):] is non empty Relation-like set
[:[: the carrier of (d,G,k), the carrier of (d,G,k):], the carrier of (d,G,k):] is non empty Relation-like set
bool [:[: the carrier of (d,G,k), the carrier of (d,G,k):], the carrier of (d,G,k):] is non empty set
the addF of (d,G,k) . ((g . A),(g . A9)) is Element of the carrier of (d,G,k)
[(g . A),(g . A9)] is non empty non natural set
{(g . A),(g . A9)} is non empty finite set
{(g . A)} is non empty trivial finite 1 -element set
{{(g . A),(g . A9)},{(g . A)}} is non empty finite V28() set
the addF of (d,G,k) . [(g . A),(g . A9)] is set
A is Relation-like the carrier of (d,G,(k + 1)) -defined the carrier of (d,G,k) -valued Function-like quasi_total additive Element of bool [: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):]
A9 is Element of the carrier of (d,G,(k + 1))
A99 is finite Element of bool (d,G,(k + 1))
A . A9 is Element of the carrier of (d,G,k)
(d,G,k,A99) is finite (d,G,k)
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ A99) is even ) } is set
f is Relation-like the carrier of (d,G,(k + 1)) -defined the carrier of (d,G,k) -valued Function-like quasi_total additive Element of bool [: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):]
g is Relation-like the carrier of (d,G,(k + 1)) -defined the carrier of (d,G,k) -valued Function-like quasi_total additive Element of bool [: the carrier of (d,G,(k + 1)), the carrier of (d,G,k):]
A is Element of the carrier of (d,G,(k + 1))
A9 is Element of the carrier of (d,G,(k + 1))
f . A9 is Element of the carrier of (d,G,k)
A99 is finite Element of bool (d,G,(k + 1))
(d,G,k,A99) is finite (d,G,k)
(d,G,k) is non empty finite Element of bool (bool (REAL d))
{ b₁ where b₁ is Element of (d,G,k) : ( k + 1 <= d & not card ((d,G,k,b₁) /\ A99) is even ) } is set
g . A9 is Element of the carrier of (d,G,k)
f . A is Element of the carrier of (d,G,k)
g . A is Element of the carrier of (d,G,k)