:: AOFA_000 semantic presentation

REAL is complex-membered ext-real-membered real-membered V107() V146() V147() V149() set
NAT is non empty non trivial ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() V144() V146() Element of bool REAL
bool REAL is non empty set
ExtREAL is ext-real-membered V149() set
[:NAT,ExtREAL:] is Relation-like V92() set
bool [:NAT,ExtREAL:] is non empty set
bool ExtREAL is non empty set
omega is non empty non trivial ordinal non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() V144() V146() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
COMPLEX is complex-membered V107() set
RAT is complex-membered ext-real-membered real-membered rational-membered V107() set
INT is complex-membered ext-real-membered real-membered rational-membered integer-membered V107() set
{} is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set
the empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set
1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
{{},1} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
K395() is non empty set
0 is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V89() V90() V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() Element of NAT
{0,1} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
NAT * is non empty functional FinSequence-membered FinSequenceSet of NAT
HFuncs NAT is non empty M21(NAT * , NAT )
K345((NAT *),NAT) is M21(NAT * , NAT )
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like M22(NAT * , NAT ,K345((NAT *),NAT)) : b₁ is homogeneous } is set
bool (HFuncs NAT) is non empty set
bool (HFuncs NAT) is non empty Element of bool (bool (HFuncs NAT))
bool (bool (HFuncs NAT)) is non empty set
[:NAT,(bool (HFuncs NAT)):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool (HFuncs NAT)):] is non empty non trivial non finite set
2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
K481() is L10()
the carrier of K481() is set
Trees is non empty constituted-Trees set
bool Trees is non empty set
FinTrees is non empty constituted-Trees constituted-FinTrees Element of bool Trees
PeanoNat is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
the carrier of PeanoNat is non empty set
FinTrees the carrier of PeanoNat is non empty functional constituted-DTrees DTree-set of the carrier of PeanoNat
TS PeanoNat is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of PeanoNat)
bool (FinTrees the carrier of PeanoNat) is non empty set
[:(TS PeanoNat),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(TS PeanoNat),NAT:] is non empty non trivial non finite set
[:NAT,(TS PeanoNat):] is non empty non trivial Relation-like non finite set
bool [:NAT,(TS PeanoNat):] is non empty non trivial non finite set
2 -tuples_on K395() is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of K395()
[:(2 -tuples_on K395()),K395():] is non empty Relation-like set
bool [:(2 -tuples_on K395()),K395():] is non empty set
3 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
3 -tuples_on K395() is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of K395()
[:(3 -tuples_on K395()),K395():] is non empty Relation-like set
bool [:(3 -tuples_on K395()),K395():] is non empty set
K671() is non empty Relation-like 2 -tuples_on K395() -defined K395() -valued Function-like total quasi_total V114() Element of bool [:(2 -tuples_on K395()),K395():]
[:NAT,REAL:] is Relation-like V91() V92() V93() set
bool [:NAT,REAL:] is non empty set
bool (bool REAL) is non empty set
bool RAT is non empty set
[:1,1:] is non empty Relation-like RAT -valued INT -valued finite V91() V92() V93() to-naturals set
bool [:1,1:] is non empty finite V44() set
[:[:1,1:],1:] is non empty Relation-like RAT -valued INT -valued finite V91() V92() V93() to-naturals set
bool [:[:1,1:],1:] is non empty finite V44() set
dom {} is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set
rng {} is empty trivial ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() with_non-empty_elements finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals V95() V96() V97() V98() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set
{{}} is non empty trivial functional finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Tree-like V188() set
4 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
Seg 4 is non empty finite 4 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 4 ) } is set
{1,2,3,4} is non empty finite set
card {} is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() set
+infty is non empty V80() ext-real positive non negative set
-infty is non empty V80() ext-real non positive negative set
A is Relation-like Function-like set
dom A is set
S is Relation-like Function-like set
dom S is set
T is Relation-like Function-like set
rng T is set
T * A is Relation-like Function-like set
T * S is Relation-like Function-like set
f is set
dom (T * A) is set
dom T is set
dom (T * S) is set
P is set
(T * A) . P is set
T . P is set
A . (T . P) is set
(T * S) . P is set
S . (T . P) is set
A is non empty set
S is non empty set
<*A,S*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*A*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,A] is non empty V22() set
{1,A} is non empty finite set
{1} is non empty trivial finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1,A},{1}} is non empty finite V44() set
{[1,A]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*A*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
1 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
rng <*A,S*> is non empty finite set
{A,S} is non empty finite set
A is Relation-like non-empty NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
S is Relation-like non-empty NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
A ^ S is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng (A ^ S) is finite set
rng A is with_non-empty_elements finite set
rng S is with_non-empty_elements finite set
(rng A) \/ (rng S) is finite set
A is set
S is non empty set
T is functional FinSequence-membered FinSequenceSet of A
f is Element of S
T --> f is set
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),S:] is non empty Relation-like set
bool [:(A *),S:] is non empty set
T --> f is Relation-like T -defined S -valued Function-like total quasi_total Element of bool [:T,S:]
[:T,S:] is Relation-like set
bool [:T,S:] is non empty set
dom (T --> f) is functional FinSequence-membered Element of bool T
bool T is non empty set
rng (T --> f) is Element of bool S
bool S is non empty set
{f} is non empty trivial finite 1 -element set
A is non empty set
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
S is Element of A
(A,A,(T -tuples_on A),S) is Relation-like A * -defined A -valued Function-like Element of bool [:(A *),A:]
dom (A,A,(T -tuples_on A),S) is functional FinSequence-membered Element of bool (A *)
bool (A *) is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
f -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
C is non empty Relation-like A * -defined A -valued Function-like homogeneous Element of bool [:(A *),A:]
arity C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is Relation-like A * -defined A -valued Function-like Element of bool [:(A *),A:]
A is non empty set
S is Element of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(T -tuples_on A),S) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total Element of bool [:(A *),A:]
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
arity (A,A,(T -tuples_on A),S) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom (A,A,(T -tuples_on A),S) is non empty functional FinSequence-membered with_common_domain Element of bool (A *)
bool (A *) is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
f -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
A is non empty set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
S is Element of A
(A,A,(T -tuples_on A),S) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total Element of bool [:(A *),A:]
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
arity (A,A,(T -tuples_on A),S) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty set
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
S is Element of A
(A,A,(T -tuples_on A),S) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total Element of bool [:(A *),A:]
f is Relation-like A * -defined A -valued Function-like homogeneous Element of bool [:(A *),A:]
A is non empty set
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
the Element of A is Element of A
2 -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
[:(2 -tuples_on A),A:] is non empty Relation-like set
bool [:(2 -tuples_on A),A:] is non empty set
T is non empty Relation-like 2 -tuples_on A -defined A -valued Function-like total quasi_total Element of bool [:(2 -tuples_on A),A:]
T is non empty Relation-like 2 -tuples_on A -defined A -valued Function-like total quasi_total Element of bool [:(2 -tuples_on A),A:]
rng T is non empty Element of bool A
bool A is non empty set
dom T is non empty functional FinSequence-membered Element of bool (2 -tuples_on A)
bool (2 -tuples_on A) is non empty set
f is non empty Relation-like A * -defined A -valued Function-like homogeneous Element of bool [:(A *),A:]
arity f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total Element of bool [:(A *),A:]
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
C is set
I is set
<*C,I*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
dom P is non empty functional FinSequence-membered with_common_domain Element of bool (A *)
bool (A *) is non empty set
s is set
<*I,s*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <*C,I*> is set
<*(P . <*C,I*>),s*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(P . <*C,I*>)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . <*C,I*>)] is non empty V22() set
{1,(P . <*C,I*>)} is non empty finite set
{{1,(P . <*C,I*>)},{1}} is non empty finite V44() set
{[1,(P . <*C,I*>)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . <*C,I*>)*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <*I,s*> is set
<*C,(P . <*I,s*>)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(P . <*I,s*>)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . <*I,s*>)] is non empty V22() set
{1,(P . <*I,s*>)} is non empty finite set
{{1,(P . <*I,s*>)},{1}} is non empty finite V44() set
{[1,(P . <*I,s*>)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*(P . <*I,s*>)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
IFEQ (C, the Element of A,I,C) is set
IFEQ (I, the Element of A,s,I) is set
IFEQ (C, the Element of A,(IFEQ (I, the Element of A,s,I)),C) is set
IFEQ (C, the Element of A,s,C) is set
P . <*(P . <*C,I*>),s*> is set
r is Element of A
i is Element of A
IFEQ (r, the Element of A,i,r) is Element of A
s is Element of A
<*(IFEQ (r, the Element of A,i,r)),s*> is non empty Relation-like NAT -defined A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of A
<*(IFEQ (r, the Element of A,i,r))*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(IFEQ (r, the Element of A,i,r))] is non empty V22() set
{1,(IFEQ (r, the Element of A,i,r))} is non empty finite set
{{1,(IFEQ (r, the Element of A,i,r))},{1}} is non empty finite V44() set
{[1,(IFEQ (r, the Element of A,i,r))]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(IFEQ (r, the Element of A,i,r))*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <*(IFEQ (r, the Element of A,i,r)),s*> is set
IFEQ ((IFEQ (C, the Element of A,I,C)), the Element of A,s,(IFEQ (C, the Element of A,I,C))) is set
IFEQ (i, the Element of A,s,i) is Element of A
<*r,(IFEQ (i, the Element of A,s,i))*> is non empty Relation-like NAT -defined A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(IFEQ (i, the Element of A,s,i))*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(IFEQ (i, the Element of A,s,i))] is non empty V22() set
{1,(IFEQ (i, the Element of A,s,i))} is non empty finite set
{{1,(IFEQ (i, the Element of A,s,i))},{1}} is non empty finite V44() set
{[1,(IFEQ (i, the Element of A,s,i))]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*(IFEQ (i, the Element of A,s,i))*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <*r,(IFEQ (i, the Element of A,s,i))*> is set
P . <*C,(P . <*I,s*>)*> is set
C is set
I is set
<*C,I*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
dom P is non empty functional FinSequence-membered with_common_domain set
<*I,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*I*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<* the Element of A,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<* the Element of A*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1, the Element of A] is non empty V22() set
{1, the Element of A} is non empty finite set
{{1, the Element of A},{1}} is non empty finite V44() set
{[1, the Element of A]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the Element of A*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <* the Element of A,C*> is set
<*C, the Element of A*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*C*> ^ <* the Element of A*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
P . <*C, the Element of A*> is set
s is Element of A
<* the Element of A,s*> is non empty Relation-like NAT -defined A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the Element of A*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
IFEQ ( the Element of A, the Element of A,s, the Element of A) is Element of A
<*s, the Element of A*> is non empty Relation-like NAT -defined A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of A
<*s*> ^ <* the Element of A*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
IFEQ (s, the Element of A, the Element of A,s) is Element of A
the Element of A is Element of A
0 -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(0 -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:(A *),A:]
arity (A,A,(0 -tuples_on A), the Element of A) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the Element of A is Element of A
3 -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(3 -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:(A *),A:]
arity (A,A,(3 -tuples_on A), the Element of A) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty set
FinTrees A is non empty functional constituted-DTrees DTree-set of A
S is Relation-like NAT -defined FinTrees A -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of FinTrees A
rng S is functional finite constituted-DTrees Element of bool (FinTrees A)
bool (FinTrees A) is non empty set
f is set
T is set
T -tree S is Relation-like Function-like DecoratedTree-like set
P is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
dom P is non empty finite Tree-like set
C is non empty finite Tree-like set
{ H₁(b₁) where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C : S₁[b₁] } is set
the Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C
len the Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is finite set
r is set
i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is non empty finite complex-membered ext-real-membered real-membered V144() V145() V146() V147() V148() set
max r is V79() V80() ext-real set
i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of C
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
s is set
S . s is Relation-like Function-like set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like V189() set
[1,r] is non empty V22() set
{1,r} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ i is non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (T -tree S) is non empty Tree-like set
len (<*r*> ^ i) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
1 + (len i) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
S is Relation-like Function-like set
A is Relation-like Function-like set
T is set
A | T is Relation-like Function-like set
S +* (A | T) is Relation-like Function-like set
A is Relation-like Function-like set
dom A is set
S is Relation-like Function-like set
T is set
f is set
(A,S,f) is Relation-like Function-like set
A | f is Relation-like Function-like set
S +* (A | f) is Relation-like Function-like set
(A,S,f) . T is set
A . T is set
dom (A | f) is set
(dom A) /\ f is set
(A | f) . T is set
S is Relation-like Function-like set
dom S is set
A is Relation-like Function-like set
T is set
f is set
(A,S,f) is Relation-like Function-like set
A | f is Relation-like Function-like set
S +* (A | f) is Relation-like Function-like set
(A,S,f) . T is set
S . T is set
dom A is set
(dom A) /\ f is set
dom (A | f) is set
A is non empty set
S is non empty set
Funcs (A,S) is non empty functional FUNCTION_DOMAIN of A,S
T is Relation-like A -defined S -valued Function-like total quasi_total Element of Funcs (A,S)
f is Relation-like A -defined S -valued Function-like total quasi_total Element of Funcs (A,S)
P is set
(T,f,P) is Relation-like Function-like set
T | P is Relation-like P -defined A -defined S -valued Function-like set
f +* (T | P) is Relation-like Function-like set
dom f is Element of bool A
bool A is non empty set
T | P is Relation-like A -defined P -defined A -defined S -valued Function-like Element of bool [:A,S:]
[:A,S:] is non empty Relation-like set
bool [:A,S:] is non empty set
dom (T | P) is Element of bool A
(dom f) \/ (dom (T | P)) is Element of bool A
dom (T,f,P) is set
rng (T,f,P) is set
rng f is Element of bool S
bool S is non empty set
rng (T | P) is Element of bool S
(rng f) \/ (rng (T | P)) is Element of bool S
A is non empty set
S is non empty set
Funcs (A,S) is non empty functional FUNCTION_DOMAIN of A,S
T is non empty set
Funcs (S,T) is non empty functional FUNCTION_DOMAIN of S,T
f is Relation-like A -defined S -valued Function-like total quasi_total Element of Funcs (A,S)
P is Relation-like S -defined T -valued Function-like total quasi_total Element of Funcs (S,T)
f * P is Relation-like A -defined T -valued Function-like set
Funcs (A,T) is non empty functional FUNCTION_DOMAIN of A,T
P * f is non empty Relation-like A -defined T -valued Function-like total quasi_total Element of bool [:A,T:]
[:A,T:] is non empty Relation-like set
bool [:A,T:] is non empty set
A is Relation-like Function-like set
S is set
{ ((iter (A,b₁)) . S) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : S in dom (iter (A,b₁)) } is set
A is Relation-like Function-like set
dom A is set
S is set
(A,S) is set
{ ((iter (A,b₁)) . S) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : S in dom (iter (A,b₁)) } is set
rng A is set
(dom A) \/ (rng A) is set
iter (A,0) is Relation-like Function-like set
field A is set
id (field A) is Relation-like field A -defined field A -valued Function-like one-to-one total quasi_total Element of bool [:(field A),(field A):]
[:(field A),(field A):] is Relation-like set
bool [:(field A),(field A):] is non empty set
(iter (A,0)) . S is set
dom (iter (A,0)) is set
A is Relation-like Function-like set
rng A is set
dom A is set
T is set
S is set
(A,S) is set
{ ((iter (A,b₁)) . S) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : S in dom (iter (A,b₁)) } is set
A . T is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,f) is Relation-like Function-like set
(iter (A,f)) . S is set
dom (iter (A,f)) is set
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,(f + 1)) is Relation-like Function-like set
(iter (A,f)) * A is Relation-like Function-like set
(iter (A,(f + 1))) . S is set
rng (iter (A,f)) is set
field A is set
(dom A) \/ (rng A) is set
dom (iter (A,(f + 1))) is set
A is Relation-like Function-like set
dom A is set
S is set
A . S is set
(A,S) is set
{ ((iter (A,b₁)) . S) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : S in dom (iter (A,b₁)) } is set
iter (A,1) is Relation-like Function-like set
A is Relation-like Function-like set
dom A is set
S is set
A . S is set
(A,(A . S)) is set
{ ((iter (A,b₁)) . (A . S)) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : A . S in dom (iter (A,b₁)) } is set
(A,S) is set
{ ((iter (A,b₁)) . S) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : S in dom (iter (A,b₁)) } is set
T is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,f) is Relation-like Function-like set
(iter (A,f)) . (A . S) is set
dom (iter (A,f)) is set
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,(f + 1)) is Relation-like Function-like set
A * (iter (A,f)) is Relation-like Function-like set
(iter (A,(f + 1))) . S is set
dom (iter (A,(f + 1))) is set
A is Relation-like Function-like set
rng A is set
dom A is set
field A is set
(dom A) \/ (rng A) is set
iter (A,0) is Relation-like Function-like set
id (dom A) is Relation-like dom A -defined dom A -valued Function-like one-to-one total quasi_total Element of bool [:(dom A),(dom A):]
[:(dom A),(dom A):] is Relation-like set
bool [:(dom A),(dom A):] is non empty set
dom (iter (A,0)) is set
f is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,f) is Relation-like Function-like set
dom (iter (A,f)) is set
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,(f + 1)) is Relation-like Function-like set
dom (iter (A,(f + 1))) is set
P is set
(iter (A,f)) . P is set
rng (iter (A,f)) is set
(iter (A,f)) * A is Relation-like Function-like set
S is set
T is set
f is set
P is set
(dom A) \ f is Element of bool (dom A)
bool (dom A) is non empty set
P is set
(A,P) is set
{ ((iter (A,b₁)) . P) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : P in dom (iter (A,b₁)) } is set
C is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . P is set
dom (iter (A,I)) is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . P is set
s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,r) is Relation-like Function-like set
(iter (A,r)) . P is set
P is Relation-like Function-like set
dom P is set
f --> T is Relation-like f -defined {T} -valued Function-like total quasi_total Element of bool [:f,{T}:]
{T} is non empty trivial finite 1 -element set
[:f,{T}:] is Relation-like set
bool [:f,{T}:] is non empty set
(f --> T) +* P is Relation-like Function-like set
C is Relation-like Function-like set
dom C is set
dom (f --> T) is Element of bool f
bool f is non empty set
f \/ ((dom A) \ f) is set
I is set
(A,I) is set
{ ((iter (A,b₁)) . I) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : I in dom (iter (A,b₁)) } is set
C . I is set
(f --> T) . I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . I is set
dom (iter (A,s)) is set
P . I is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,r) is Relation-like Function-like set
(iter (A,r)) . I is set
f is Relation-like Function-like set
dom f is set
P is Relation-like Function-like set
dom P is set
C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
f . C is set
P . C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . C is set
dom (iter (A,s)) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . C is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,r) is Relation-like Function-like set
(iter (A,r)) . C is set
f . C is set
P . C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
A is Relation-like Function-like set
rng A is set
dom A is set
field A is set
(dom A) \/ (rng A) is set
iter (A,0) is Relation-like Function-like set
id (dom A) is Relation-like dom A -defined dom A -valued Function-like one-to-one total quasi_total Element of bool [:(dom A),(dom A):]
[:(dom A),(dom A):] is Relation-like set
bool [:(dom A),(dom A):] is non empty set
dom (iter (A,0)) is set
f is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,f) is Relation-like Function-like set
dom (iter (A,f)) is set
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,(f + 1)) is Relation-like Function-like set
dom (iter (A,(f + 1))) is set
P is set
(iter (A,f)) . P is set
rng (iter (A,f)) is set
(iter (A,f)) * A is Relation-like Function-like set
S is set
T is Relation-like Function-like set
f is set
P is set
(dom A) \ f is Element of bool (dom A)
bool (dom A) is non empty set
P is set
(A,P) is set
{ ((iter (A,b₁)) . P) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : P in dom (iter (A,b₁)) } is set
C is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . P is set
dom (iter (A,I)) is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . P is set
s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,r) is Relation-like Function-like set
(iter (A,r)) . P is set
P is Relation-like Function-like set
dom P is set
f --> 0 is Relation-like f -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V91() V92() V93() to-naturals Element of bool [:f,NAT:]
[:f,NAT:] is Relation-like RAT -valued INT -valued V91() V92() V93() to-naturals set
bool [:f,NAT:] is non empty set
T | f is Relation-like Function-like set
(f --> 0) +* (T | f) is Relation-like Function-like set
((f --> 0) +* (T | f)) +* P is Relation-like Function-like set
C is Relation-like Function-like set
dom C is set
dom (f --> 0) is Element of bool f
bool f is non empty set
dom ((f --> 0) +* (T | f)) is set
dom (T | f) is set
f \/ (dom (T | f)) is set
f \/ ((dom A) \ f) is set
I is set
(A,I) is set
{ ((iter (A,b₁)) . I) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : I in dom (iter (A,b₁)) } is set
C . I is set
T . I is set
((f --> 0) +* (T | f)) . I is set
(T | f) . I is set
(f --> 0) . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom T is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . I is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,r) is Relation-like Function-like set
dom (iter (A,r)) is set
P . I is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,i) is Relation-like Function-like set
(iter (A,i)) . I is set
f is Relation-like Function-like set
dom f is set
P is Relation-like Function-like set
dom P is set
C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
f . C is set
T . C is set
P . C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . C is set
dom (iter (A,s)) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,s) is Relation-like Function-like set
(iter (A,s)) . C is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,r) is Relation-like Function-like set
(iter (A,r)) . C is set
f . C is set
P . C is set
(A,C) is set
{ ((iter (A,b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (A,b₁)) } is set
A is Relation-like Function-like set
rng A is set
dom A is set
S is Relation-like Function-like set
T is set
(A,T) is set
{ ((iter (A,b₁)) . T) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : T in dom (iter (A,b₁)) } is set
f is set
(A,f,S) is Relation-like Function-like set
(A,f,S) . T is set
P is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (A,C) is Relation-like Function-like set
(iter (A,C)) . T is set
dom (iter (A,C)) is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,C) is Relation-like Function-like set
(iter (A,C)) . T is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . T is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,I) is Relation-like Function-like set
(iter (A,I)) . T is set
A is Relation-like Function-like set
rng A is set
dom A is set
S is Relation-like Function-like set
A * S is Relation-like Function-like set
T is set
f is set
(A,f,S) is Relation-like Function-like set
(A,f,S) . T is set
A . T is set
(A,f,S) . (A . T) is set
(A,(A . T)) is set
{ ((iter (A,b₁)) . (A . T)) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : A . T in dom (iter (A,b₁)) } is set
(A,T) is set
{ ((iter (A,b₁)) . T) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : T in dom (iter (A,b₁)) } is set
S . (A . T) is set
S . T is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,P) is Relation-like Function-like set
(iter (A,P)) . T is set
field A is set
(dom A) \/ (rng A) is set
iter (A,0) is Relation-like Function-like set
id (dom A) is Relation-like dom A -defined dom A -valued Function-like one-to-one total quasi_total Element of bool [:(dom A),(dom A):]
[:(dom A),(dom A):] is Relation-like set
bool [:(dom A),(dom A):] is non empty set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + C is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,C) is Relation-like Function-like set
A * (iter (A,C)) is Relation-like Function-like set
(iter (A,C)) . (A . T) is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (A,(I + 1)) is Relation-like Function-like set
iter (A,I) is Relation-like Function-like set
A * (iter (A,I)) is Relation-like Function-like set
(iter (A,(I + 1))) . T is set
(iter (A,I)) . (A . T) is set
A is Relation-like Function-like set
rng A is set
dom A is set
S is Relation-like Function-like set
T is set
f is set
(A,f,S) is Relation-like Function-like set
(A,f,S) . T is set
field A is set
(dom A) \/ (rng A) is set
iter (A,0) is Relation-like Function-like set
id (dom A) is Relation-like dom A -defined dom A -valued Function-like one-to-one total quasi_total Element of bool [:(dom A),(dom A):]
[:(dom A),(dom A):] is Relation-like set
bool [:(dom A),(dom A):] is non empty set
(iter (A,0)) . T is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (A,P) is Relation-like Function-like set
(iter (A,P)) . T is set
A is non empty set
Funcs (A,A) is non empty functional FUNCTION_DOMAIN of A,A
S is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
T is set
f is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(S,T,f) is Relation-like Function-like set
dom S is Element of bool A
bool A is non empty set
rng S is Element of bool A
dom (S,T,f) is set
rng (S,T,f) is set
P is set
C is set
(S,T,f) . C is set
I is Element of A
(S,I) is set
{ ((iter (S,b₁)) . I) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : I in dom (iter (S,b₁)) } is set
f . I is Element of A
I is Element of A
(S,I) is set
{ ((iter (S,b₁)) . I) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : I in dom (iter (S,b₁)) } is set
(S,T,f) . I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (S,s) is Relation-like Function-like set
(iter (S,s)) . I is set
dom (iter (S,s)) is set
rng (iter (S,s)) is set
I is Element of A
(S,I) is set
{ ((iter (S,b₁)) . I) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : I in dom (iter (S,b₁)) } is set
A is non empty set
S is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom S is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
len S is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
Seg (len S) is non empty finite len S -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= len S ) } is set
the Element of A is Element of A
f is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
C -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(C -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total C -ary Element of bool [:(A *),A:]
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
f is Relation-like Function-like set
dom f is set
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng P is finite set
A * is non empty functional FinSequence-membered FinSequenceSet of A
PFuncs ((A *),A) is set
C is set
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
I is set
P . I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(r -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total r -ary Element of bool [:(A *),A:]
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
dom (A,A,(r -tuples_on A), the Element of A) is non empty functional FinSequence-membered with_common_domain Element of bool (A *)
bool (A *) is non empty set
rng (A,A,(r -tuples_on A), the Element of A) is non empty Element of bool A
bool A is non empty set
{ the Element of A} is non empty trivial finite 1 -element set
C is Relation-like NAT -defined PFuncs ((A *),A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs ((A *),A)
UAStr(# A,C #) is non empty strict UAStr
the carrier of UAStr(# A,C #) is non empty set
the carrier of UAStr(# A,C #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# A,C #)
[:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):] is non empty set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# A,C #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #))
PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) is set
dom the charact of UAStr(# A,C #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# A,C #) . s is set
r is Relation-like the carrier of UAStr(# A,C #) * -defined the carrier of UAStr(# A,C #) -valued Function-like Element of bool [:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):]
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(s -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total s -ary Element of bool [:(A *),A:]
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
the charact of UAStr(# A,C #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #))
the carrier of UAStr(# A,C #) is non empty set
the carrier of UAStr(# A,C #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# A,C #)
PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) is set
rng the charact of UAStr(# A,C #) is finite Element of bool (PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)))
bool (PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #))) is non empty set
dom C is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
s is set
C . s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(i -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total i -ary Element of bool [:(A *),A:]
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
the carrier of UAStr(# A,C #) is non empty set
the carrier of UAStr(# A,C #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# A,C #)
[:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):] is non empty set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# A,C #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #))
PFuncs (( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #)) is set
dom the charact of UAStr(# A,C #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# A,C #) . s is set
r is Relation-like the carrier of UAStr(# A,C #) * -defined the carrier of UAStr(# A,C #) -valued Function-like Element of bool [:( the carrier of UAStr(# A,C #) *), the carrier of UAStr(# A,C #):]
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(s -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total s -ary Element of bool [:(A *),A:]
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
s is non empty partial quasi_total non-empty UAStr
the carrier of s is non empty set
signature s is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the carrier of s * is non empty functional FinSequence-membered FinSequenceSet of the carrier of s
[:( the carrier of s *), the carrier of s:] is non empty Relation-like set
bool [:( the carrier of s *), the carrier of s:] is non empty set
i is non empty Relation-like the carrier of s * -defined the carrier of s -valued Function-like homogeneous Element of bool [:( the carrier of s *), the carrier of s:]
the charact of s is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of s *), the carrier of s) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of s *), the carrier of s)
PFuncs (( the carrier of s *), the carrier of s) is set
the charact of s . r is homogeneous set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
(A,A,(s -tuples_on A), the Element of A) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total s -ary Element of bool [:(A *),A:]
[:(A *),A:] is non empty Relation-like set
bool [:(A *),A:] is non empty set
the Relation-like NAT -defined A -valued Function-like finite s -element FinSequence-like FinSubsequence-like Element of s -tuples_on A is Relation-like NAT -defined A -valued Function-like finite s -element FinSequence-like FinSubsequence-like Element of s -tuples_on A
dom i is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of s *)
bool ( the carrier of s *) is non empty set
len the Relation-like NAT -defined A -valued Function-like finite s -element FinSequence-like FinSubsequence-like Element of s -tuples_on A is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
arity i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom S is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
len S is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
Seg (len S) is non empty finite len S -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= len S ) } is set
T is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(P -tuples_on NAT),f) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total P -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
T is Relation-like Function-like set
dom T is set
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng f is finite set
PFuncs ((NAT *),NAT) is set
P is set
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
C is set
f . C is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(s -tuples_on NAT),I) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total s -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
dom (NAT,NAT,(s -tuples_on NAT),I) is non empty functional FinSequence-membered with_common_domain Element of bool (NAT *)
bool (NAT *) is non empty set
{I} is non empty trivial finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
rng (NAT,NAT,(s -tuples_on NAT),I) is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V146() Element of bool RAT
P is Relation-like NAT -defined PFuncs ((NAT *),NAT) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs ((NAT *),NAT)
UAStr(# NAT,P #) is non empty strict UAStr
the carrier of UAStr(# NAT,P #) is non empty set
the carrier of UAStr(# NAT,P #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# NAT,P #)
[:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):] is non empty set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# NAT,P #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #))
PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) is set
dom the charact of UAStr(# NAT,P #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# NAT,P #) . I is set
s is Relation-like the carrier of UAStr(# NAT,P #) * -defined the carrier of UAStr(# NAT,P #) -valued Function-like Element of bool [:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):]
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(i -tuples_on NAT),r) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total i -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
the charact of UAStr(# NAT,P #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #))
the carrier of UAStr(# NAT,P #) is non empty set
the carrier of UAStr(# NAT,P #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# NAT,P #)
PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) is set
rng the charact of UAStr(# NAT,P #) is finite Element of bool (PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)))
bool (PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #))) is non empty set
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
I is set
P . I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(r -tuples_on NAT),s) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total r -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
the carrier of UAStr(# NAT,P #) is non empty set
the carrier of UAStr(# NAT,P #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# NAT,P #)
[:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):] is non empty set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# NAT,P #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #))
PFuncs (( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #)) is set
dom the charact of UAStr(# NAT,P #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# NAT,P #) . I is set
s is Relation-like the carrier of UAStr(# NAT,P #) * -defined the carrier of UAStr(# NAT,P #) -valued Function-like Element of bool [:( the carrier of UAStr(# NAT,P #) *), the carrier of UAStr(# NAT,P #):]
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(i -tuples_on NAT),r) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total i -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
I is non empty partial quasi_total non-empty UAStr
the carrier of I is non empty set
signature I is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
the charact of I is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of I *), the carrier of I) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of I *), the carrier of I)
the carrier of I * is non empty functional FinSequence-membered FinSequenceSet of the carrier of I
PFuncs (( the carrier of I *), the carrier of I) is set
len P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
[:( the carrier of I *), the carrier of I:] is non empty Relation-like set
bool [:( the carrier of I *), the carrier of I:] is non empty set
r is non empty Relation-like the carrier of I * -defined the carrier of I -valued Function-like homogeneous Element of bool [:( the carrier of I *), the carrier of I:]
the charact of I . s is homogeneous set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(s -tuples_on NAT),i) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total s -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
the Relation-like NAT -defined NAT -valued Function-like finite s -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of s -tuples_on NAT is Relation-like NAT -defined NAT -valued Function-like finite s -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of s -tuples_on NAT
dom r is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of I *)
bool ( the carrier of I *) is non empty set
len the Relation-like NAT -defined NAT -valued Function-like finite s -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals Element of s -tuples_on NAT is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
arity r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of I . s is homogeneous set
r -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(r -tuples_on NAT) --> s is non empty Relation-like r -tuples_on NAT -defined {s} -valued Function-like total quasi_total V91() V92() V93() to-naturals Element of bool [:(r -tuples_on NAT),{s}:]
{s} is non empty trivial finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
[:(r -tuples_on NAT),{s}:] is non empty Relation-like RAT -valued INT -valued V91() V92() V93() to-naturals set
bool [:(r -tuples_on NAT),{s}:] is non empty set
P . s is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
(NAT,NAT,(s -tuples_on NAT),i) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total s -ary Element of bool [:(NAT *),NAT:]
A is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
A . S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P is non empty set
T -tuples_on P is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of P
[:(T -tuples_on P),P:] is non empty Relation-like set
bool [:(T -tuples_on P),P:] is non empty set
C is non empty partial quasi_total non-empty UAStr
the carrier of C is non empty set
signature C is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
I is non empty Relation-like T -tuples_on P -defined P -valued Function-like total quasi_total Element of bool [:(T -tuples_on P),P:]
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
f -tuples_on P is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of P
P * is non empty functional FinSequence-membered FinSequenceSet of P
[:(P *),P:] is non empty Relation-like set
bool [:(P *),P:] is non empty set
the Relation-like NAT -defined P -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on P is Relation-like NAT -defined P -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on P
s is Relation-like P * -defined P -valued Function-like Element of bool [:(P *),P:]
dom s is functional FinSequence-membered Element of bool (P *)
bool (P *) is non empty set
len the Relation-like NAT -defined P -valued Function-like finite T -element FinSequence-like FinSubsequence-like Element of T -tuples_on P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is non empty Relation-like P * -defined P -valued Function-like homogeneous Element of bool [:(P *),P:]
arity i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of C is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of C *), the carrier of C) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of C *), the carrier of C)
the carrier of C * is non empty functional FinSequence-membered FinSequenceSet of the carrier of C
PFuncs (( the carrier of C *), the carrier of C) is set
the charact of C +* (S,i) is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
PFuncs ((P *),P) is set
{i} is non empty trivial functional finite 1 -element set
rng ( the charact of C +* (S,i)) is finite set
rng the charact of C is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of C *), the carrier of C))
bool (PFuncs (( the carrier of C *), the carrier of C)) is non empty set
(rng the charact of C) \/ {i} is non empty finite set
s is Relation-like NAT -defined PFuncs ((P *),P) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs ((P *),P)
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
dom the charact of C is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
UAStr(# P,s #) is non empty strict UAStr
the carrier of UAStr(# P,s #) is non empty set
the carrier of UAStr(# P,s #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# P,s #)
[:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):] is non empty set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# P,s #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #))
PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) is set
dom the charact of UAStr(# P,s #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# P,s #) . i is set
ri is Relation-like the carrier of UAStr(# P,s #) * -defined the carrier of UAStr(# P,s #) -valued Function-like Element of bool [:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):]
the charact of C . i is homogeneous set
the charact of UAStr(# P,s #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #))
the carrier of UAStr(# P,s #) is non empty set
the carrier of UAStr(# P,s #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# P,s #)
PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) is set
rng the charact of UAStr(# P,s #) is finite Element of bool (PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)))
bool (PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #))) is non empty set
i is set
s . i is set
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of C . ri is homogeneous set
the carrier of UAStr(# P,s #) is non empty set
the carrier of UAStr(# P,s #) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of UAStr(# P,s #)
[:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):] is non empty Relation-like set
bool [:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):] is non empty set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of UAStr(# P,s #) is Relation-like NAT -defined PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #))
PFuncs (( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #)) is set
dom the charact of UAStr(# P,s #) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
the charact of UAStr(# P,s #) . i is set
ri is Relation-like the carrier of UAStr(# P,s #) * -defined the carrier of UAStr(# P,s #) -valued Function-like Element of bool [:( the carrier of UAStr(# P,s #) *), the carrier of UAStr(# P,s #):]
the charact of C . i is homogeneous set
i is non empty partial quasi_total non-empty UAStr
the carrier of i is non empty set
signature i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
the charact of i is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of i *), the carrier of i) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of i *), the carrier of i)
the carrier of i * is non empty functional FinSequence-membered FinSequenceSet of the carrier of i
PFuncs (( the carrier of i *), the carrier of i) is set
the charact of i . S is homogeneous set
len A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of C is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
[:( the carrier of i *), the carrier of i:] is non empty Relation-like set
bool [:( the carrier of i *), the carrier of i:] is non empty set
r is non empty Relation-like the carrier of i * -defined the carrier of i -valued Function-like homogeneous Element of bool [:( the carrier of i *), the carrier of i:]
the charact of i . ri is homogeneous set
the charact of C . ri is homogeneous set
A . ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
arity r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
S is non empty disjoint_with_NAT set
FreeUnivAlgNSG (A,S) is non empty strict partial quasi_total non-empty free UAStr
DTConUA (A,S) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
dom A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(dom A) \/ S is non empty set
REL (A,S) is Relation-like (dom A) \/ S -defined ((dom A) \/ S) * -valued Element of bool [:((dom A) \/ S),(((dom A) \/ S) *):]
((dom A) \/ S) * is non empty functional FinSequence-membered FinSequenceSet of (dom A) \/ S
[:((dom A) \/ S),(((dom A) \/ S) *):] is non empty Relation-like set
bool [:((dom A) \/ S),(((dom A) \/ S) *):] is non empty set
DTConstrStr(# ((dom A) \/ S),(REL (A,S)) #) is non empty strict DTConstrStr
TS (DTConUA (A,S)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA (A,S)))
the carrier of (DTConUA (A,S)) is non empty set
FinTrees the carrier of (DTConUA (A,S)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA (A,S))
bool (FinTrees the carrier of (DTConUA (A,S))) is non empty set
FreeOpSeqNSG (A,S) is Relation-like NAT -defined PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S))))
(TS (DTConUA (A,S))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA (A,S))
PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S)))) is set
UAStr(# (TS (DTConUA (A,S))),(FreeOpSeqNSG (A,S)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG (A,S)) is non empty set
T is Element of the carrier of (FreeUnivAlgNSG (A,S))
f is Element of the carrier of (FreeUnivAlgNSG (A,S))
A is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
S is non empty disjoint_with_NAT set
FreeUnivAlgNSG (A,S) is non empty strict partial quasi_total non-empty free UAStr
DTConUA (A,S) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
dom A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(dom A) \/ S is non empty set
REL (A,S) is Relation-like (dom A) \/ S -defined ((dom A) \/ S) * -valued Element of bool [:((dom A) \/ S),(((dom A) \/ S) *):]
((dom A) \/ S) * is non empty functional FinSequence-membered FinSequenceSet of (dom A) \/ S
[:((dom A) \/ S),(((dom A) \/ S) *):] is non empty Relation-like set
bool [:((dom A) \/ S),(((dom A) \/ S) *):] is non empty set
DTConstrStr(# ((dom A) \/ S),(REL (A,S)) #) is non empty strict DTConstrStr
TS (DTConUA (A,S)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA (A,S)))
the carrier of (DTConUA (A,S)) is non empty set
FinTrees the carrier of (DTConUA (A,S)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA (A,S))
bool (FinTrees the carrier of (DTConUA (A,S))) is non empty set
FreeOpSeqNSG (A,S) is Relation-like NAT -defined PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S))))
(TS (DTConUA (A,S))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA (A,S))
PFuncs (((TS (DTConUA (A,S))) *),(TS (DTConUA (A,S)))) is set
UAStr(# (TS (DTConUA (A,S))),(FreeOpSeqNSG (A,S)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG (A,S)) is non empty set
T is Relation-like Function-like Element of the carrier of (FreeUnivAlgNSG (A,S))
T is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (A,S)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (FreeUnivAlgNSG (A,S))
A is non empty DTConstrStr
TS A is functional constituted-DTrees Element of bool (FinTrees the carrier of A)
the carrier of A is non empty set
FinTrees the carrier of A is non empty functional constituted-DTrees DTree-set of the carrier of A
bool (FinTrees the carrier of A) is non empty set
Terminals A is set
S is set
{S} is non empty trivial finite 1 -element set
(TS A) \ {S} is functional constituted-DTrees Element of bool (FinTrees the carrier of A)
f is Element of the carrier of A
root-tree f is Relation-like the carrier of A -valued Function-like DecoratedTree-like Element of FinTrees the carrier of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> f is non empty Relation-like elementary_tree 0 -defined {f} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{f}:]
{f} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{f}:] is non empty Relation-like finite set
bool [:(elementary_tree 0),{f}:] is non empty finite V44() set
T is functional constituted-DTrees Element of bool (FinTrees the carrier of A)
P is Relation-like NAT -defined FinTrees the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
rng P is functional finite constituted-DTrees Element of bool (FinTrees the carrier of A)
f is Element of the carrier of A
roots P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
C is Relation-like NAT -defined FinTrees the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of TS A
f -tree C is Relation-like the carrier of A -valued Function-like DecoratedTree-like Element of FinTrees the carrier of A
f -tree P is Relation-like the carrier of A -valued Function-like DecoratedTree-like Element of FinTrees the carrier of A
A is non empty disjoint_with_NAT set
S is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom S is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
FreeUnivAlgNSG (S,A) is non empty strict partial quasi_total non-empty free UAStr
DTConUA (S,A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom S) \/ A is non empty set
REL (S,A) is Relation-like (dom S) \/ A -defined ((dom S) \/ A) * -valued Element of bool [:((dom S) \/ A),(((dom S) \/ A) *):]
((dom S) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom S) \/ A
[:((dom S) \/ A),(((dom S) \/ A) *):] is non empty Relation-like set
bool [:((dom S) \/ A),(((dom S) \/ A) *):] is non empty set
DTConstrStr(# ((dom S) \/ A),(REL (S,A)) #) is non empty strict DTConstrStr
TS (DTConUA (S,A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA (S,A)))
the carrier of (DTConUA (S,A)) is non empty set
FinTrees the carrier of (DTConUA (S,A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA (S,A))
bool (FinTrees the carrier of (DTConUA (S,A))) is non empty set
FreeOpSeqNSG (S,A) is Relation-like NAT -defined PFuncs (((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A))))
(TS (DTConUA (S,A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA (S,A))
PFuncs (((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A)))) is set
UAStr(# (TS (DTConUA (S,A))),(FreeOpSeqNSG (S,A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG (S,A)) is non empty set
the charact of (FreeUnivAlgNSG (S,A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG (S,A)) *), the carrier of (FreeUnivAlgNSG (S,A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG (S,A)) *), the carrier of (FreeUnivAlgNSG (S,A)))
the carrier of (FreeUnivAlgNSG (S,A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG (S,A))
PFuncs (( the carrier of (FreeUnivAlgNSG (S,A)) *), the carrier of (FreeUnivAlgNSG (S,A))) is set
dom the charact of (FreeUnivAlgNSG (S,A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S . C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
In (C,(dom the charact of (FreeUnivAlgNSG (S,A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG (S,A))
Den ((In (C,(dom the charact of (FreeUnivAlgNSG (S,A))))),(FreeUnivAlgNSG (S,A))) is Relation-like the carrier of (FreeUnivAlgNSG (S,A)) * -defined the carrier of (FreeUnivAlgNSG (S,A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG (S,A))
rng the charact of (FreeUnivAlgNSG (S,A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG (S,A)) *), the carrier of (FreeUnivAlgNSG (S,A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG (S,A)) *), the carrier of (FreeUnivAlgNSG (S,A)))) is non empty set
the charact of (FreeUnivAlgNSG (S,A)) . (In (C,(dom the charact of (FreeUnivAlgNSG (S,A))))) is non empty homogeneous set
T is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
T /. C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
I is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (S,A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG (S,A))
len I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(Den ((In (C,(dom the charact of (FreeUnivAlgNSG (S,A))))),(FreeUnivAlgNSG (S,A)))) . I is set
C -tree I is Relation-like Function-like DecoratedTree-like set
(T /. C) -tuples_on (TS (DTConUA (S,A))) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of TS (DTConUA (S,A))
FreeOpNSG (C,S,A) is non empty Relation-like (TS (DTConUA (S,A))) * -defined TS (DTConUA (S,A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A))):]
[:((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A))):] is non empty Relation-like set
bool [:((TS (DTConUA (S,A))) *),(TS (DTConUA (S,A))):] is non empty set
dom (FreeOpNSG (C,S,A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA (S,A))) *)
bool ((TS (DTConUA (S,A))) *) is non empty set
len the charact of (FreeUnivAlgNSG (S,A)) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len S is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(FreeOpNSG (C,S,A)) . I is Relation-like Function-like set
Sym (C,S,A) is Element of the carrier of (DTConUA (S,A))
(Sym (C,S,A)) -tree I is Relation-like Function-like DecoratedTree-like set
A is non empty non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
bool the carrier of A is non empty Element of bool (bool the carrier of A)
bool (bool the carrier of A) is non empty set
[:NAT,(bool the carrier of A):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool the carrier of A):] is non empty non trivial non finite set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S is Element of bool the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= a₂ ) } is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= P ) } is set
P \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= P ) } is set
I is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . s is non empty set
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (s,A)) . r is set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng r is finite Element of bool the carrier of A
rng (Den (s,A)) is Element of bool the carrier of A
I is Element of bool the carrier of A
f is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
f . 0 is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
f . P is Element of bool the carrier of A
C is Element of bool the carrier of A
f . T is Element of bool the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
f . (I + 1) is Element of bool the carrier of A
f . I is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= f . I ) } is set
(f . I) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= f . I ) } is set
f is Element of bool the carrier of A
P is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= a₂ ) } is set
C is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
C . T is Element of bool the carrier of A
C . 0 is Element of bool the carrier of A
I is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
I . T is Element of bool the carrier of A
I . 0 is Element of bool the carrier of A
dom C is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V146() Element of bool NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C . (s + 1) is Element of bool the carrier of A
C . s is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= C . s ) } is set
(C . s) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= C . s ) } is set
dom I is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V146() Element of bool NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . (s + 1) is Element of bool the carrier of A
I . s is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= I . s ) } is set
(I . s) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= I . s ) } is set
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
(A,S,0) is Element of bool the carrier of A
bool the carrier of A is non empty Element of bool (bool the carrier of A)
bool (bool the carrier of A) is non empty set
[:NAT,(bool the carrier of A):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool the carrier of A):] is non empty non trivial non finite set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
T is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
T . 0 is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is Element of bool the carrier of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + 1)) is Element of bool the carrier of A
(A,S,T) is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
(A,S,T) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= a₂ ) } is set
bool the carrier of A is non empty Element of bool (bool the carrier of A)
bool (bool the carrier of A) is non empty set
[:NAT,(bool the carrier of A):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool the carrier of A):] is non empty non trivial non finite set
f is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
f . T is Element of bool the carrier of A
f . 0 is Element of bool the carrier of A
P is non empty Relation-like NAT -defined bool the carrier of A -valued Function-like total quasi_total Element of bool [:NAT,(bool the carrier of A):]
P . (T + 1) is Element of bool the carrier of A
P . 0 is Element of bool the carrier of A
dom f is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V146() Element of bool NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
f . (C + 1) is Element of bool the carrier of A
f . C is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= f . C ) } is set
(f . C) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= f . C ) } is set
dom P is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V146() Element of bool NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
P . (C + 1) is Element of bool the carrier of A
P . C is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= P . C ) } is set
(P . C) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= P . C ) } is set
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is Element of bool the carrier of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + 1)) is Element of bool the carrier of A
(A,S,T) is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
P is set
(A,S,T) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (C,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . C is non empty homogeneous set
I is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (C,A)) . I is set
dom (Den (C,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng I is finite Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,T) is Element of bool the carrier of A
(A,S,f) is Element of bool the carrier of A
T + 0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,(T + 0)) is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,(T + P)) is Element of bool the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= a₂ ) } is set
(T + P) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,((T + P) + 1)) is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,(T + P)) ) } is set
(A,S,(T + P)) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,(T + P)) ) } is set
P + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
T + (P + 1) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + (P + 1))) is Element of bool the carrier of A
T + (P + 1) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative set
(A,S,(T + (P + 1))) is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
T is Element of bool the carrier of A
(A,S,0) is Element of bool the carrier of A
(A,T,0) is Element of bool the carrier of A
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,f) is Element of bool the carrier of A
(A,T,f) is Element of bool the carrier of A
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(f + 1)) is Element of bool the carrier of A
(A,T,(f + 1)) is Element of bool the carrier of A
P is set
C is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . I is non empty homogeneous set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (I,A)) . s is set
dom (Den (I,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng s is finite Element of bool the carrier of A
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,f) is Element of bool the carrier of A
(A,T,f) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is Element of bool the carrier of A
(A,S,0) is Element of bool the carrier of A
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(0 + 1)) is Element of bool the carrier of A
T is set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + 1)) is Element of bool the carrier of A
(A,S,T) is Element of bool the carrier of A
(T + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,((T + 1) + 1)) is Element of bool the carrier of A
f is set
(T + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,((T + 1) + 1)) is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (P,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . P is non empty homogeneous set
C is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (P,A)) . C is set
dom (Den (P,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng C is finite Element of bool the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . I is non empty homogeneous set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (I,A)) . s is set
dom (Den (I,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng s is finite Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (P,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . P is non empty homogeneous set
C is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (P,A)) . C is set
dom (Den (P,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng C is finite Element of bool the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . I is non empty homogeneous set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (I,A)) . s is set
dom (Den (I,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng s is finite Element of bool the carrier of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + 1)) is Element of bool the carrier of A
(A,S,T) is Element of bool the carrier of A
f is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (P,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . P is non empty homogeneous set
C is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (P,A)) . C is set
dom (Den (P,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng C is finite Element of bool the carrier of A
F₂() is set
F₁() is non empty set
F₁() /\ F₂() is set
A is set
S is Element of F₁()
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
A is Relation-like Function-like set
dom A is set
rng A is set
S is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is Element of F₁()
A . f is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
{ (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
union { (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
bool the carrier of A is non empty Element of bool (bool the carrier of A)
bool (bool the carrier of A) is non empty set
union (bool the carrier of A) is Element of bool the carrier of A
P is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,C) is Element of bool the carrier of A
P is Element of bool the carrier of A
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
C is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
I is set
the charact of A . I is homogeneous set
dom C is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
arity C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity C) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of P
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
C . r is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . s is non empty homogeneous set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng r is finite Element of bool the carrier of A
i is Element of the carrier of A
s is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,s) is Element of bool the carrier of A
i is Element of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,s) is Element of bool the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,s) is Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,i) is Element of bool the carrier of A
s is set
(Den (s,A)) . r is set
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,i) ) } is set
(A,S,i) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,i) ) } is set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(i + 1)) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
T is Element of bool the carrier of A
S is Element of bool the carrier of A
{ (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
union { (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
f is set
P is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,C) is Element of bool the carrier of A
(A,S,0) is Element of bool the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,I) is Element of bool the carrier of A
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(I + 1)) is Element of bool the carrier of A
s is set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,I) ) } is set
(A,S,I) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,I) ) } is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (r,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . r is non empty homogeneous set
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (r,A)) . i is set
dom (Den (r,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng i is finite Element of bool the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
bool the carrier of A is non empty set
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
T is Element of the carrier of A
f is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng f is non empty Element of bool the carrier of A
f is set
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng P is non empty Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
Constants A is Element of bool the carrier of A
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ b₁ where b₁ is Element of the carrier of A : ex b₂ being non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A st
( arity b₂ = 0 & b₁ in rng b₂ ) } is set
S is Element of bool the carrier of A
T is set
f is Element of the carrier of A
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng P is non empty Element of bool the carrier of A
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng P is non empty Element of bool the carrier of A
dom P is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
C is set
P . C is set
0 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
I is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
len I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng I is finite Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
Constants A is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ b₁ where b₁ is Element of the carrier of A : ex b₂ being non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A st
( arity b₂ = 0 & b₁ in rng b₂ ) } is set
{} A is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() Element of bool the carrier of A
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
T is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of {} A
len T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
S . T is set
(arity S) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom S is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng S is non empty Element of bool the carrier of A
f is Element of the carrier of A
A is non empty partial quasi_total non-empty UAStr
Constants A is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ b₁ where b₁ is Element of the carrier of A : ex b₂ being non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A st
( arity b₂ = 0 & b₁ in rng b₂ ) } is set
S is GeneratorSet of A
{} A is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
{ (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
union { (A,S,b₁) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : verum } is set
f is Element of bool the carrier of A
(A,S,0) is Element of bool the carrier of A
P is Element of the carrier of A
C is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,I) is Element of bool the carrier of A
P is Element of bool the carrier of A
C is set
I is Element of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,s) is Element of bool the carrier of A
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,r) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
S is Element of bool the carrier of A
T is GeneratorSet of A
f is Element of the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,T,P) is Element of bool the carrier of A
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,T,C) is Element of bool the carrier of A
(A,S,C) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
S is GeneratorSet of A
T is Element of the carrier of A
(A,S,0) is Element of bool the carrier of A
bool the carrier of A is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,f) is Element of bool the carrier of A
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(f + 1)) is Element of bool the carrier of A
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,f) ) } is set
(A,S,f) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,f) ) } is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (P,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . P is non empty homogeneous set
C is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (P,A)) . C is set
dom (Den (P,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng C is finite Element of bool the carrier of A
rng (Den (P,A)) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
S is GeneratorSet of A
T is set
f is Element of the carrier of A
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng P is non empty Element of bool the carrier of A
A is non empty partial quasi_total non-empty free UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
S is free GeneratorSet of A
signature A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
T is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
f is non empty partial quasi_total non-empty UAStr
the carrier of f is non empty set
signature f is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
the charact of f is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of f *), the carrier of f) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of f *), the carrier of f)
the carrier of f * is non empty functional FinSequence-membered FinSequenceSet of the carrier of f
PFuncs (( the carrier of f *), the carrier of f) is set
[:S, the carrier of f:] is Relation-like set
bool [:S, the carrier of f:] is non empty set
S --> 0 is Relation-like S -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V91() V92() V93() to-naturals Element of bool [:S,NAT:]
[:S,NAT:] is Relation-like RAT -valued INT -valued V91() V92() V93() to-naturals set
bool [:S,NAT:] is non empty set
[: the carrier of A, the carrier of f:] is non empty Relation-like set
bool [: the carrier of A, the carrier of f:] is non empty set
P is Relation-like S -defined the carrier of f -valued Function-like total quasi_total Element of bool [:S, the carrier of f:]
C is non empty Relation-like the carrier of A -defined the carrier of f -valued Function-like total quasi_total Element of bool [: the carrier of A, the carrier of f:]
C | S is Relation-like the carrier of A -defined S -defined the carrier of A -defined the carrier of f -valued Function-like Element of bool [: the carrier of A, the carrier of f:]
len T is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of f is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom the charact of f is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
I is set
s is Element of the carrier of A
r is set
i is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng i is non empty Element of bool the carrier of A
s is set
the charact of A . s is homogeneous set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
T . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
Operations f is non empty PFuncsDomHQN of the carrier of f
rng the charact of f is non empty with_non-empty_elements finite set
the charact of f . s is homogeneous set
dom i is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
ri is set
i . ri is set
arity i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity i) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
C * r is Relation-like NAT -defined the carrier of f -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of f *
len (C * r) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
r -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
P . s is set
i . r is set
C . (i . r) is set
i is non empty Relation-like the carrier of f * -defined the carrier of f -valued Function-like homogeneous quasi_total Element of Operations f
i . (C * r) is set
(NAT,NAT,(r -tuples_on NAT),s) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total r -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
(NAT,NAT,(r -tuples_on NAT),s) . (C * r) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is set
s is Element of the carrier of A
r is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng r is non empty Element of bool the carrier of A
A is non empty partial quasi_total non-empty free UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
the free GeneratorSet of A is free GeneratorSet of A
T is GeneratorSet of A
A is non empty partial quasi_total non-empty free UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
the free GeneratorSet of A is free GeneratorSet of A
A is set
S is set
[:A,S:] is Relation-like set
T is set
f is set
P is set
[f,P] is non empty V22() set
{f,P} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,P},{f}} is non empty finite V44() set
A is non empty partial quasi_total non-empty free UAStr
the carrier of A is non empty set
S is GeneratorSet of A
T is non empty partial quasi_total non-empty UAStr
the carrier of T is non empty set
[: the carrier of A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of A, the carrier of T:] is non empty set
f is non empty Relation-like the carrier of A -defined the carrier of T -valued Function-like total quasi_total Element of bool [: the carrier of A, the carrier of T:]
P is non empty Relation-like the carrier of A -defined the carrier of T -valued Function-like total quasi_total Element of bool [: the carrier of A, the carrier of T:]
f | S is Relation-like the carrier of A -defined S -defined the carrier of A -defined the carrier of T -valued Function-like Element of bool [: the carrier of A, the carrier of T:]
P | S is Relation-like the carrier of A -defined S -defined the carrier of A -defined the carrier of T -valued Function-like Element of bool [: the carrier of A, the carrier of T:]
(A,S,0) is Element of bool the carrier of A
bool the carrier of A is non empty set
C is Element of the carrier of A
f . C is Element of the carrier of T
P . C is Element of the carrier of T
(f | S) . C is set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,C) is Element of bool the carrier of A
C + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(C + 1)) is Element of bool the carrier of A
I is Element of the carrier of A
f . I is Element of the carrier of T
P . I is Element of the carrier of T
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . s is non empty homogeneous set
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (s,A)) . r is set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng r is finite Element of bool the carrier of A
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
len the charact of T is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
Operations T is non empty PFuncsDomHQN of the carrier of T
rng the charact of T is non empty with_non-empty_elements finite set
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den (i,T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . i is non empty homogeneous set
f * r is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of T *
(Den (i,T)) . (f * r) is set
P * r is Relation-like NAT -defined the carrier of T -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of T *
(Den (i,T)) . (P * r) is set
dom f is non empty Element of bool the carrier of A
dom P is non empty Element of bool the carrier of A
s is set
f . s is set
P . s is set
C is Element of the carrier of A
f . C is Element of the carrier of T
P . C is Element of the carrier of T
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,S,I) is Element of bool the carrier of A
A is non empty partial quasi_total non-empty free UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
the free GeneratorSet of A is free GeneratorSet of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (T,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . T is non empty homogeneous set
dom (Den (T,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (f,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . f is non empty homogeneous set
dom (Den (f,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
C is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (T,A)) . P is set
(Den (f,A)) . C is set
signature A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
I is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
dom I is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
s is non empty partial quasi_total non-empty UAStr
the carrier of s is non empty set
signature s is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
the charact of s is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of s *), the carrier of s) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of s *), the carrier of s)
the carrier of s * is non empty functional FinSequence-membered FinSequenceSet of the carrier of s
PFuncs (( the carrier of s *), the carrier of s) is set
[: the free GeneratorSet of A, the carrier of s:] is Relation-like disjoint_with_NAT set
bool [: the free GeneratorSet of A, the carrier of s:] is non empty set
the free GeneratorSet of A --> 0 is Relation-like the free GeneratorSet of A -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total V91() V92() V93() to-naturals Element of bool [: the free GeneratorSet of A,NAT:]
[: the free GeneratorSet of A,NAT:] is Relation-like RAT -valued INT -valued V91() V92() V93() to-naturals disjoint_with_NAT set
bool [: the free GeneratorSet of A,NAT:] is non empty set
[: the carrier of A, the carrier of s:] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of A, the carrier of s:] is non empty set
r is Relation-like the free GeneratorSet of A -defined the carrier of s -valued Function-like total quasi_total Element of bool [: the free GeneratorSet of A, the carrier of s:]
i is non empty Relation-like the carrier of A -defined the carrier of s -valued Function-like total quasi_total Element of bool [: the carrier of A, the carrier of s:]
i | the free GeneratorSet of A is Relation-like the carrier of A -defined the free GeneratorSet of A -defined the carrier of A -defined the carrier of s -valued Function-like Element of bool [: the carrier of A, the carrier of s:]
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom the charact of s is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i * i is Relation-like NAT -defined the carrier of s -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of s *
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i * i is Relation-like NAT -defined the carrier of s -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of s *
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
J is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom J is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
arity J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity J) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
a is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom a is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
arity a is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity a) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i0 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
ri -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
gI2 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r -tuples_on NAT is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of s
Den (s,s) is Relation-like the carrier of s * -defined the carrier of s -valued Function-like Element of rng the charact of s
rng the charact of s is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of s *), the carrier of s))
bool (PFuncs (( the carrier of s *), the carrier of s)) is non empty set
the charact of s . s is non empty homogeneous set
Operations s is non empty PFuncsDomHQN of the carrier of s
rng the charact of s is non empty with_non-empty_elements finite set
(Den (T,A)) . i is set
i . ((Den (T,A)) . i) is set
(Den (s,s)) . i0 is set
(Den (f,A)) . i is set
i . ((Den (f,A)) . i) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of s
Den (s,s) is Relation-like the carrier of s * -defined the carrier of s -valued Function-like Element of rng the charact of s
the charact of s . s is non empty homogeneous set
(Den (s,s)) . gI2 is set
(NAT,NAT,(ri -tuples_on NAT),r) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total ri -ary Element of bool [:(NAT *),NAT:]
[:(NAT *),NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals disjoint_with_NAT set
bool [:(NAT *),NAT:] is non empty non trivial non finite set
(NAT,NAT,(r -tuples_on NAT),i) is non empty Relation-like NAT * -defined NAT -valued RAT -valued INT -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total r -ary Element of bool [:(NAT *),NAT:]
A is non empty partial quasi_total non-empty free UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
the free GeneratorSet of A is free GeneratorSet of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (T,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . T is non empty homogeneous set
dom (Den (T,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (f,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . f is non empty homogeneous set
dom (Den (f,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
C is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (T,A)) . P is set
(Den (f,A)) . C is set
signature A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
{ the carrier of A} is non empty trivial finite 1 -element set
the free GeneratorSet of A \/ { the carrier of A} is non empty set
i is Relation-like Function-like set
dom i is set
{0} is non empty trivial functional finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Tree-like V188() set
r is non empty set
[:{0},r:] is non empty Relation-like disjoint_with_NAT set
I is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
FreeUnivAlgNSG (I,[:{0},r:]) is non empty strict partial quasi_total non-empty free UAStr
DTConUA (I,[:{0},r:]) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
dom I is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(dom I) \/ [:{0},r:] is non empty set
REL (I,[:{0},r:]) is Relation-like (dom I) \/ [:{0},r:] -defined ((dom I) \/ [:{0},r:]) * -valued Element of bool [:((dom I) \/ [:{0},r:]),(((dom I) \/ [:{0},r:]) *):]
((dom I) \/ [:{0},r:]) * is non empty functional FinSequence-membered FinSequenceSet of (dom I) \/ [:{0},r:]
[:((dom I) \/ [:{0},r:]),(((dom I) \/ [:{0},r:]) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom I) \/ [:{0},r:]),(((dom I) \/ [:{0},r:]) *):] is non empty set
DTConstrStr(# ((dom I) \/ [:{0},r:]),(REL (I,[:{0},r:])) #) is non empty strict DTConstrStr
TS (DTConUA (I,[:{0},r:])) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA (I,[:{0},r:])))
the carrier of (DTConUA (I,[:{0},r:])) is non empty set
FinTrees the carrier of (DTConUA (I,[:{0},r:])) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA (I,[:{0},r:]))
bool (FinTrees the carrier of (DTConUA (I,[:{0},r:]))) is non empty set
FreeOpSeqNSG (I,[:{0},r:]) is Relation-like NAT -defined PFuncs (((TS (DTConUA (I,[:{0},r:]))) *),(TS (DTConUA (I,[:{0},r:])))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA (I,[:{0},r:]))) *),(TS (DTConUA (I,[:{0},r:]))))
(TS (DTConUA (I,[:{0},r:]))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA (I,[:{0},r:]))
PFuncs (((TS (DTConUA (I,[:{0},r:]))) *),(TS (DTConUA (I,[:{0},r:])))) is set
UAStr(# (TS (DTConUA (I,[:{0},r:]))),(FreeOpSeqNSG (I,[:{0},r:])) #) is non empty strict UAStr
signature (FreeUnivAlgNSG (I,[:{0},r:])) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
Terminals (DTConUA (I,[:{0},r:])) is non empty Element of bool the carrier of (DTConUA (I,[:{0},r:]))
bool the carrier of (DTConUA (I,[:{0},r:])) is non empty set
rng i is set
FreeGenSetNSG (I,[:{0},r:]) is free GeneratorSet of FreeUnivAlgNSG (I,[:{0},r:])
{ (root-tree b₁) where b₁ is Element of the carrier of (DTConUA (I,[:{0},r:])) : b₁ in Terminals (DTConUA (I,[:{0},r:])) } is set
r is set
i is set
i . i is set
ri is Element of the carrier of A
[0,ri] is non empty V22() set
{0,ri} is non empty finite set
{{0,ri},{0}} is non empty finite V44() set
r is Element of the carrier of (DTConUA (I,[:{0},r:]))
root-tree r is Relation-like the carrier of (DTConUA (I,[:{0},r:])) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA (I,[:{0},r:]))
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> r is non empty Relation-like elementary_tree 0 -defined {r} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{r}:]
{r} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{r}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{r}:] is non empty finite V44() set
the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty set
[: the free GeneratorSet of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):] is Relation-like disjoint_with_NAT set
bool [: the free GeneratorSet of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):] is non empty set
[: the carrier of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):] is non empty set
r is Relation-like the free GeneratorSet of A -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like total quasi_total Element of bool [: the free GeneratorSet of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):]
i is non empty Relation-like the carrier of A -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like total quasi_total Element of bool [: the carrier of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):]
i | the free GeneratorSet of A is Relation-like the carrier of A -defined the free GeneratorSet of A -defined the carrier of A -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like Element of bool [: the carrier of A, the carrier of (FreeUnivAlgNSG (I,[:{0},r:])):]
(A, the free GeneratorSet of A,0) is Element of bool the carrier of A
bool the carrier of A is non empty set
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
the charact of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *), the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *), the carrier of (FreeUnivAlgNSG (I,[:{0},r:])))
the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))
PFuncs (( the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *), the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))) is set
len the charact of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom the charact of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (ri,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . ri is non empty homogeneous set
dom (Den (ri,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i * i is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
Operations (FreeUnivAlgNSG (I,[:{0},r:])) is non empty PFuncsDomHQN of the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))
rng the charact of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty with_non-empty_elements finite set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG (I,[:{0},r:]))
Den (J,(FreeUnivAlgNSG (I,[:{0},r:]))) is Relation-like the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) * -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG (I,[:{0},r:]))
rng the charact of (FreeUnivAlgNSG (I,[:{0},r:])) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *), the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *), the carrier of (FreeUnivAlgNSG (I,[:{0},r:])))) is non empty set
the charact of (FreeUnivAlgNSG (I,[:{0},r:])) . J is non empty homogeneous set
I . J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i0 is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom i0 is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
arity i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity i0) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
b is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i * r is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
len (i * r) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i0 . i is set
i . (i0 . i) is set
a is non empty Relation-like the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) * -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like homogeneous quasi_total Element of Operations (FreeUnivAlgNSG (I,[:{0},r:]))
i is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG (I,[:{0},r:]))
a . i is set
In (J,(dom the charact of (FreeUnivAlgNSG (I,[:{0},r:])))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG (I,[:{0},r:]))
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(Den (ri,A)) . r is set
i . ((Den (ri,A)) . r) is set
ri -tree (i * r) is Relation-like Function-like DecoratedTree-like set
ri is set
r is set
i . ri is set
i . r is set
r . ri is set
r . r is set
[0,ri] is non empty V22() set
{0,ri} is non empty finite set
{{0,ri},{0}} is non empty finite V44() set
root-tree [0,ri] is Relation-like Function-like DecoratedTree-like set
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> [0,ri] is non empty Relation-like elementary_tree 0 -defined {[0,ri]} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{[0,ri]}:]
{[0,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
[:(elementary_tree 0),{[0,ri]}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{[0,ri]}:] is non empty finite V44() set
[0,r] is non empty V22() set
{0,r} is non empty finite set
{{0,r},{0}} is non empty finite V44() set
root-tree [0,r] is Relation-like Function-like DecoratedTree-like set
(elementary_tree 0) --> [0,r] is non empty Relation-like elementary_tree 0 -defined {[0,r]} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{[0,r]}:]
{[0,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
[:(elementary_tree 0),{[0,r]}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{[0,r]}:] is non empty finite V44() set
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (ri,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . ri is non empty homogeneous set
dom (Den (ri,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
(Den (ri,A)) . r is set
i . ((Den (ri,A)) . r) is set
i * r is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
ri -tree (i * r) is Relation-like Function-like DecoratedTree-like set
i is set
i . i is set
r . i is set
[0,i] is non empty V22() set
{0,i} is non empty finite set
{{0,i},{0}} is non empty finite V44() set
root-tree [0,i] is Relation-like Function-like DecoratedTree-like set
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> [0,i] is non empty Relation-like elementary_tree 0 -defined {[0,i]} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{[0,i]}:]
{[0,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
[:(elementary_tree 0),{[0,i]}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{[0,i]}:] is non empty finite V44() set
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A, the free GeneratorSet of A,ri) is Element of bool the carrier of A
ri + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A, the free GeneratorSet of A,(ri + 1)) is Element of bool the carrier of A
r is set
i is set
i . r is set
i . i is set
r . r is set
r . i is set
[0,r] is non empty V22() set
{0,r} is non empty finite set
{{0,r},{0}} is non empty finite V44() set
root-tree [0,r] is Relation-like Function-like DecoratedTree-like set
(elementary_tree 0) --> [0,r] is non empty Relation-like elementary_tree 0 -defined {[0,r]} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{[0,r]}:]
{[0,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
[:(elementary_tree 0),{[0,r]}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{[0,r]}:] is non empty finite V44() set
[0,i] is non empty V22() set
{0,i} is non empty finite set
{{0,i},{0}} is non empty finite V44() set
root-tree [0,i] is Relation-like Function-like DecoratedTree-like set
(elementary_tree 0) --> [0,i] is non empty Relation-like elementary_tree 0 -defined {[0,i]} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{[0,i]}:]
{[0,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
[:(elementary_tree 0),{[0,i]}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{[0,i]}:] is non empty finite V44() set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i is non empty homogeneous set
i0 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (i,A)) . i0 is set
dom (Den (i,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng i0 is finite Element of bool the carrier of A
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (gI2,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . gI2 is non empty homogeneous set
J is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (gI2,A)) . J is set
dom (Den (gI2,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng J is finite Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i is non empty homogeneous set
i0 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (i,A)) . i0 is set
dom (Den (i,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng i0 is finite Element of bool the carrier of A
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (gI2,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . gI2 is non empty homogeneous set
J is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (gI2,A)) . J is set
dom (Den (gI2,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng J is finite Element of bool the carrier of A
a is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (a,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . a is non empty homogeneous set
b is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (a,A)) . b is set
dom (Den (a,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng b is finite Element of bool the carrier of A
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (gI2,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . gI2 is non empty homogeneous set
J is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (gI2,A)) . J is set
dom (Den (gI2,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng J is finite Element of bool the carrier of A
i * i0 is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
i -tree (i * i0) is Relation-like Function-like DecoratedTree-like set
i * J is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
gI2 -tree (i * J) is Relation-like Function-like DecoratedTree-like set
len i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom i0 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
Seg (len i0) is finite len i0 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= len i0 ) } is set
dom J is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
a is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 . a is set
J . a is set
i . (i0 . a) is set
(i * i0) . a is set
i . (J . a) is set
(i * J) . a is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i is non empty homogeneous set
i0 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (i,A)) . i0 is set
dom (Den (i,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng i0 is finite Element of bool the carrier of A
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (gI2,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . gI2 is non empty homogeneous set
J is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (gI2,A)) . J is set
dom (Den (gI2,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng J is finite Element of bool the carrier of A
ri is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i * ri is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
T -tree (i * ri) is Relation-like Function-like DecoratedTree-like set
i . ((Den (T,A)) . P) is set
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i * r is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding Element of the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) *
f -tree (i * r) is Relation-like Function-like DecoratedTree-like set
P * i is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite set
C * i is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG (I,[:{0},r:])) -valued Function-like finite set
len ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I . f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom ri is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
Seg (len ri) is finite len ri -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= len ri ) } is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
ri . i is set
rng ri is finite Element of bool the carrier of A
r . i is set
rng r is finite Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A, the free GeneratorSet of A,i) is Element of bool the carrier of A
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A, the free GeneratorSet of A,i0) is Element of bool the carrier of A
max (i,i0) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
gI2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A, the free GeneratorSet of A,gI2) is Element of bool the carrier of A
i . (ri . i) is set
(i * ri) . i is set
i . (r . i) is set
(i * r) . i is set
A is non empty partial quasi_total non-empty free UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom S is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
T is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
dom T is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
f is set
the charact of A . f is homogeneous set
P is set
the charact of A . P is homogeneous set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (C,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . C is non empty homogeneous set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . I is non empty homogeneous set
s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
S . s is set
T . r is set
A is non empty partial quasi_total non-empty free UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (S,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . S is non empty homogeneous set
dom (Den (S,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
T is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
rng T is finite set
(Den (S,A)) . T is set
f is set
P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
rng P is finite Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is free GeneratorSet of A
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
C is Element of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,(A),s) is Element of bool the carrier of A
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,(A),r) is Element of bool the carrier of A
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . s is non empty homogeneous set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
i is Element of the carrier of A
rng s is finite Element of bool the carrier of A
(Den (s,A)) . s is set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . s is non empty homogeneous set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
i is Element of the carrier of A
rng s is finite Element of bool the carrier of A
(Den (s,A)) . s is set
r is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
rng r is non empty Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,(A),ri) is Element of bool the carrier of A
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i is non empty homogeneous set
dom (Den (i,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
r is Element of the carrier of A
rng i is finite Element of bool the carrier of A
(Den (i,A)) . i is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (r,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . r is non empty homogeneous set
i is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (r,A)) . i is set
dom (Den (r,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng i is finite Element of bool the carrier of A
gI2 is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i0,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i0 is non empty homogeneous set
dom (Den (i0,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
i is Element of the carrier of A
rng gI2 is finite Element of bool the carrier of A
(Den (i0,A)) . gI2 is set
A is non empty partial quasi_total non-empty free UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
S is GeneratorSet of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (T,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . T is non empty homogeneous set
dom (Den (T,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (T,A)) . f is set
rng f is finite set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
P + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(P + 1)) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A,S,P) is Element of bool the carrier of A
C is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(I + 1)) is Element of bool the carrier of A
(A,S,I) is Element of bool the carrier of A
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den (T,A)) . s is set
rng s is finite Element of bool the carrier of A
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den (T,A)) . s is set
rng s is finite Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(A,S,s) is Element of bool the carrier of A
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den (T,A)) . s is set
rng s is finite Element of bool the carrier of A
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (i,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . i is non empty homogeneous set
s is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (i,A)) . s is set
dom (Den (i,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng s is finite Element of bool the carrier of A
A is non empty partial quasi_total non-empty free UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(A) is free GeneratorSet of A
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
bool the carrier of A is non empty set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (T,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . T is non empty homogeneous set
dom (Den (T,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (f,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . f is non empty homogeneous set
P is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom (Den (f,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
C is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
C . P is set
rng C is non empty Element of bool the carrier of A
(Den (f,A)) . P is set
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (T,A)) . f is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
P + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,(A),(P + 1)) is Element of bool the carrier of A
rng f is finite set
(A,(A),P) is Element of bool the carrier of A
A is non empty set
A * is non empty functional FinSequence-membered FinSequenceSet of A
[:(A *),A:] is non empty Relation-like disjoint_with_NAT set
bool [:(A *),A:] is non empty set
0 -tuples_on A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of A
S is Element of A
(A,A,(0 -tuples_on A),S) is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:(A *),A:]
T is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 2 -ary () () Element of bool [:(A *),A:]
<*(A,A,(0 -tuples_on A),S),T*> is non empty Relation-like non-empty NAT -defined bool [:(A *),A:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:(A *),A:]
<*(A,A,(0 -tuples_on A),S)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,(A,A,(0 -tuples_on A),S)] is non empty V22() set
{1,(A,A,(0 -tuples_on A),S)} is non empty finite set
{{1,(A,A,(0 -tuples_on A),S)},{1}} is non empty finite V44() set
{[1,(A,A,(0 -tuples_on A),S)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(A,A,(0 -tuples_on A),S)*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
f is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:(A *),A:]
P is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:(A *),A:]
<*f,P*> is non empty Relation-like non-empty NAT -defined bool [:(A *),A:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:(A *),A:]
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<*(A,A,(0 -tuples_on A),S),T*> ^ <*f,P*> is non empty Relation-like non-empty NAT -defined bool [:(A *),A:] -valued Function-like finite 2 + 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:(A *),A:]
2 + 2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is non empty Relation-like A * -defined A -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:(A *),A:]
<*s,T*> is non empty Relation-like non-empty NAT -defined bool [:(A *),A:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:(A *),A:]
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<*s,T*> ^ <*f,P*> is non empty Relation-like non-empty NAT -defined bool [:(A *),A:] -valued Function-like finite 2 + 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:(A *),A:]
PFuncs ((A *),A) is set
i is set
rng (<*s,T*> ^ <*f,P*>) is non empty with_non-empty_elements finite set
rng (<*s,T*> ^ <*f,P*>) is non empty with_non-empty_elements finite Element of bool (bool [:(A *),A:])
bool (bool [:(A *),A:]) is non empty set
rng <*s,T*> is non empty with_non-empty_elements finite Element of bool (bool [:(A *),A:])
rng <*f,P*> is non empty with_non-empty_elements finite Element of bool (bool [:(A *),A:])
(rng <*s,T*>) \/ (rng <*f,P*>) is non empty finite Element of bool (bool [:(A *),A:])
{s,T} is non empty functional finite set
{f,P} is non empty functional finite set
i is non empty Relation-like non-empty NAT -defined PFuncs ((A *),A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs ((A *),A)
UAStr(# A,i #) is non empty strict UAStr
s is non empty strict non-empty UAStr
the carrier of s is non empty set
the charact of s is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of s *), the carrier of s) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of s *), the carrier of s)
the carrier of s * is non empty functional FinSequence-membered FinSequenceSet of the carrier of s
PFuncs (( the carrier of s *), the carrier of s) is set
len <*s,T*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len <*f,P*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
2 + 2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom i is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
[:( the carrier of s *), the carrier of s:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of s *), the carrier of s:] is non empty set
s is non empty Relation-like the carrier of s * -defined the carrier of s -valued Function-like homogeneous 0 -ary Element of bool [:( the carrier of s *), the carrier of s:]
r is non empty Relation-like the carrier of s * -defined the carrier of s -valued Function-like homogeneous quasi_total 2 -ary () Element of bool [:( the carrier of s *), the carrier of s:]
<*s,r*> is non empty Relation-like non-empty NAT -defined bool [:( the carrier of s *), the carrier of s:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:( the carrier of s *), the carrier of s:]
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<*s,r*> . 1 is set
<*s,r*> . 2 is set
<*f,P*> . 1 is set
<*f,P*> . 2 is set
dom the charact of s is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
the charact of s . 1 is set
dom <*s,r*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
the charact of s . 2 is set
In (1,(dom the charact of s)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of s
Den ((In (1,(dom the charact of s))),s) is Relation-like the carrier of s * -defined the carrier of s -valued Function-like Element of rng the charact of s
rng the charact of s is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of s *), the carrier of s))
bool (PFuncs (( the carrier of s *), the carrier of s)) is non empty set
the charact of s . (In (1,(dom the charact of s))) is non empty set
(Den ((In (1,(dom the charact of s))),s)) . {} is set
the charact of s . 3 is set
dom <*f,P*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
2 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (2 + 1) is set
i . (2 + 2) is set
the charact of s . 4 is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of s . i is set
ri is Relation-like the carrier of s * -defined the carrier of s -valued Function-like Element of bool [:( the carrier of s *), the carrier of s:]
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
the charact of s . i is set
ri is Relation-like the carrier of s * -defined the carrier of s -valued Function-like Element of bool [:( the carrier of s *), the carrier of s:]
r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
dom ri is functional FinSequence-membered set
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
{{}} * is non empty functional FinSequence-membered FinSequenceSet of {{}}
[:({{}} *),{{}}:] is non empty Relation-like RAT -valued INT -valued V91() V92() V93() to-naturals disjoint_with_NAT set
bool [:({{}} *),{{}}:] is non empty set
the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:] is non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]
the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}} is ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}
T is set
arity the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:] is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:] is non empty functional FinSequence-membered with_common_domain Element of bool ({{}} *)
bool ({{}} *) is non empty set
2 -tuples_on {{}} is non empty functional finite FinSequence-membered with_common_domain product-like FinSequenceSet of {{}}
<* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}, the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}*> is non empty Relation-like NAT -defined {{}} -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals V189() FinSequence of {{}}
<* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like V189() set
[1, the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}] is non empty V22() set
{1, the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1, the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}},{1}} is non empty finite V44() set
{[1, the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}*> ^ <* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}},T*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:] is non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]
the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:] is non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]
0 -tuples_on {{}} is non empty functional finite FinSequence-membered with_common_domain product-like FinSequenceSet of {{}}
f is ordinal natural Relation-like NAT -defined NAT -valued Function-like V22() finite cardinal FinSequence-like FinSubsequence-like V79() V80() ext-real non negative V89() V90() V91() V92() V93() to-naturals Element of {{}}
({{}},{{}},(0 -tuples_on {{}}),f) is non empty Relation-like {{}} * -defined RAT -valued INT -valued {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 0 -ary Element of bool [:({{}} *),{{}}:]
<*({{}},{{}},(0 -tuples_on {{}}),f), the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]*> is non empty Relation-like non-empty NAT -defined bool [:({{}} *),{{}}:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:({{}} *),{{}}:]
<*({{}},{{}},(0 -tuples_on {{}}),f)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1,({{}},{{}},(0 -tuples_on {{}}),f)] is non empty V22() set
{1,({{}},{{}},(0 -tuples_on {{}}),f)} is non empty finite set
{{1,({{}},{{}},(0 -tuples_on {{}}),f)},{1}} is non empty finite V44() set
{[1,({{}},{{}},(0 -tuples_on {{}}),f)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]] is non empty V22() set
{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]} is non empty finite set
{{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]},{1}} is non empty finite V44() set
{[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*({{}},{{}},(0 -tuples_on {{}}),f)*> ^ <* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:], the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]*> is non empty Relation-like non-empty NAT -defined bool [:({{}} *),{{}}:] -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:({{}} *),{{}}:]
<* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]] is non empty V22() set
{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]} is non empty finite set
{{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]},{1}} is non empty finite V44() set
{[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like set
[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]] is non empty V22() set
{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]} is non empty finite set
{{1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]},{1}} is non empty finite V44() set
{[1, the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:]*> ^ <* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like set
<*({{}},{{}},(0 -tuples_on {{}}),f), the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary () () Element of bool [:({{}} *),{{}}:]*> ^ <* the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 3 -ary Element of bool [:({{}} *),{{}}:], the non empty Relation-like {{}} * -defined {{}} -valued Function-like V91() V92() V93() to-naturals homogeneous quasi_total 2 -ary Element of bool [:({{}} *),{{}}:]*> is non empty Relation-like non-empty NAT -defined bool [:({{}} *),{{}}:] -valued Function-like finite 2 + 2 -element Function-yielding FinSequence-like FinSubsequence-like FinSequence of bool [:({{}} *),{{}}:]
2 + 2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I is non empty strict non-empty UAStr
the carrier of I is non empty set
the charact of I is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of I *), the carrier of I) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of I *), the carrier of I)
the carrier of I * is non empty functional FinSequence-membered FinSequenceSet of the carrier of I
PFuncs (( the carrier of I *), the carrier of I) is set
A is non empty non-empty () UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty set
dom (Den ((In (1,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 1 is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom S is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
0 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty non-empty () UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 1 is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:( the carrier of A *), the carrier of A:]
dom S is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
A is non empty partial quasi_total non-empty () UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(A) is Element of the carrier of A
(Den ((In (1,(dom the charact of A))),A)) . {} is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng S is non empty Element of bool the carrier of A
bool the carrier of A is non empty set
dom (Den ((In (1,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
<*> the carrier of A is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined the carrier of A -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of the carrier of A
[:NAT, the carrier of A:] is non empty non trivial Relation-like non finite disjoint_with_NAT set
len (<*> the carrier of A) is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V89() V90() V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() Element of NAT
A is non empty non-empty () UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty set
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 2 is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty non-empty () UAStr
the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 2 is set
f is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
dom f is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty non-empty () () () UAStr
the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
S is Element of the carrier of A
(A,(A),S) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty set
<*(A),S*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*(A)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A)] is non empty V22() set
{1,(A)} is non empty finite set
{{1,(A)},{1}} is non empty finite V44() set
{[1,(A)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(A)*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*(A),S*> is set
(A,S,(A)) is Element of the carrier of A
<*S,(A)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> ^ <*(A)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,(A)*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 2 is set
T is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom T is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
<*S,S*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
A is non empty non-empty () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
f is Element of the carrier of A
(A,(A,S,T),f) is Element of the carrier of A
<*(A,S,T),f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*(A,S,T)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A,S,T)] is non empty V22() set
{1,(A,S,T)} is non empty finite set
{{1,(A,S,T)},{1}} is non empty finite V44() set
{[1,(A,S,T)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(A,S,T)*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*(A,S,T),f*> is set
(A,T,f) is Element of the carrier of A
<*T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,f*> is set
(A,S,(A,T,f)) is Element of the carrier of A
<*S,(A,T,f)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*(A,T,f)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A,T,f)] is non empty V22() set
{1,(A,T,f)} is non empty finite set
{{1,(A,T,f)},{1}} is non empty finite V44() set
{[1,(A,T,f)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*(A,T,f)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,(A,T,f)*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 2 is set
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary () Element of bool [:( the carrier of A *), the carrier of A:]
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom P is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty non-empty () UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty set
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 3 is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty non-empty () UAStr
the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(1 + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 3 is set
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom P is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty non-empty () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
(A,S,T,(A)) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty set
<*S,T,(A)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*(A)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A)] is non empty V22() set
{1,(A)} is non empty finite set
{{1,(A)},{1}} is non empty finite V44() set
{[1,(A)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*(A)*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,(A)*> is set
A is non empty non-empty () UAStr
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (4,(dom the charact of A))) is non empty set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 4 is set
S is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty non-empty () UAStr
the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (4,(dom the charact of A))) is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 4 is set
f is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom f is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*f,P*> is set
(A,T,(A,f,P)) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*T,(A,f,P)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(A,f,P)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A,f,P)] is non empty V22() set
{1,(A,f,P)} is non empty finite set
{{1,(A,f,P)},{1}} is non empty finite V44() set
{[1,(A,f,P)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*(A,f,P)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*T,(A,f,P)*> is set
(A,S,(A,T,(A,f,P))) is Element of the carrier of A
<*S,(A,T,(A,f,P))*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(A,T,(A,f,P))*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A,T,(A,f,P))] is non empty V22() set
{1,(A,T,(A,f,P))} is non empty finite set
{{1,(A,T,(A,f,P))},{1}} is non empty finite V44() set
{[1,(A,T,(A,f,P))]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*(A,T,(A,f,P))*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,(A,T,(A,f,P))*> is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
bool the carrier of A is non empty set
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
(A) is Element of the carrier of A
the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
(A,S,T,f) is Element of the carrier of A
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 3 is homogeneous set
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
C is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity C) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 4 is homogeneous set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity P) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
T is set
(Den ((In (3,(dom the charact of A))),A)) . T is set
[:( the carrier of A *), the carrier of A:] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of A *), the carrier of A:] is non empty set
the charact of A . 3 is homogeneous set
f is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity f) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
P is set
C is set
I is set
<*P,C,I*> is non empty Relation-like NAT -defined Function-like finite 3 -element FinSequence-like FinSubsequence-like set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*P*> ^ <*C*>) ^ <*I*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
s is Element of the carrier of A
r is Element of the carrier of A
i is Element of the carrier of A
(A,s,r,i) is Element of the carrier of A
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*s,r,i*> is set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
T is set
(Den ((In (4,(dom the charact of A))),A)) . T is set
the charact of A . 4 is homogeneous set
f is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of A *), the carrier of A:]
arity f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(arity f) -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
P is set
C is set
<*P,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
I is Element of the carrier of A
s is Element of the carrier of A
(A,I,s) is Element of the carrier of A
<*I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*I,s*> is set
<*0,2*> is non empty Relation-like NAT -defined NAT -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals V189() FinSequence of NAT
<*0*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like V189() set
[1,0] is non empty V22() set
{1,0} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1,0},{1}} is non empty finite V44() set
{[1,0]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like V189() set
[1,2] is non empty V22() set
{1,2} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1,2},{1}} is non empty finite V44() set
{[1,2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*0*> ^ <*2*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*3,2*> is non empty Relation-like non-empty NAT -defined NAT -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals V189() FinSequence of NAT
<*3*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like V189() set
[1,3] is non empty V22() set
{1,3} is non empty finite V44() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{1,3},{1}} is non empty finite V44() set
{[1,3]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*3*> ^ <*2*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*0,2*> ^ <*3,2*> is non empty Relation-like NAT -defined NAT -valued Function-like finite 2 + 2 -element FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
2 + 2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
() is non empty Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len () is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom () is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
() . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . 3 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . 4 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len <*0,2*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len <*3,2*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom <*0,2*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
dom <*3,2*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
<*3,2*> . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
<*3,2*> . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
2 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
<*0,2*> . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
<*0,2*> . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty partial non-empty UAStr
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite FinSequence-like FinSubsequence-like homogeneous FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
signature A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
len <*0,2*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
len <*3,2*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
2 + 2 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
len the charact of (FreeUnivAlgNSG ((),A)) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom <*0,2*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
dom <*3,2*> is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
the charact of (FreeUnivAlgNSG ((),A)) . 1 is homogeneous set
[:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty set
FreeOpNSG (1,(),A) is non empty Relation-like (TS (DTConUA ((),A))) * -defined TS (DTConUA ((),A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):]
[:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty set
P is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
FreeOpNSG (1,P,A) is non empty Relation-like (TS (DTConUA (P,A))) * -defined TS (DTConUA (P,A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):]
DTConUA (P,A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
dom P is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(dom P) \/ A is non empty set
REL (P,A) is Relation-like (dom P) \/ A -defined ((dom P) \/ A) * -valued Element of bool [:((dom P) \/ A),(((dom P) \/ A) *):]
((dom P) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom P) \/ A
[:((dom P) \/ A),(((dom P) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom P) \/ A),(((dom P) \/ A) *):] is non empty set
DTConstrStr(# ((dom P) \/ A),(REL (P,A)) #) is non empty strict DTConstrStr
TS (DTConUA (P,A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA (P,A)))
the carrier of (DTConUA (P,A)) is non empty set
FinTrees the carrier of (DTConUA (P,A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA (P,A))
bool (FinTrees the carrier of (DTConUA (P,A))) is non empty set
(TS (DTConUA (P,A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA (P,A))
[:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):] is non empty set
<*0,2*> . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
<*0,2*> . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P /. 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (1,P,A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA (P,A))) *)
bool ((TS (DTConUA (P,A))) *) is non empty set
0 -tuples_on the carrier of (FreeUnivAlgNSG ((),A)) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
f is non empty set
f * is non empty functional FinSequence-membered FinSequenceSet of f
[:(f *),f:] is non empty Relation-like disjoint_with_NAT set
bool [:(f *),f:] is non empty set
I is non empty Relation-like f * -defined f -valued Function-like homogeneous quasi_total Element of bool [:(f *),f:]
arity I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of (FreeUnivAlgNSG ((),A)) . 2 is homogeneous set
FreeOpNSG (2,P,A) is non empty Relation-like (TS (DTConUA (P,A))) * -defined TS (DTConUA (P,A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):]
P . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P /. 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (2,P,A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA (P,A))) *)
2 -tuples_on the carrier of (FreeUnivAlgNSG ((),A)) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
r is non empty Relation-like f * -defined f -valued Function-like homogeneous quasi_total Element of bool [:(f *),f:]
arity r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of (FreeUnivAlgNSG ((),A)) . 3 is homogeneous set
FreeOpNSG (3,P,A) is non empty Relation-like (TS (DTConUA (P,A))) * -defined TS (DTConUA (P,A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):]
<*3,2*> . 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
<*3,2*> . 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
2 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
P . (2 + 1) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P /. 3 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (3,P,A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA (P,A))) *)
3 -tuples_on the carrier of (FreeUnivAlgNSG ((),A)) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
s is non empty Relation-like f * -defined f -valued Function-like homogeneous quasi_total Element of bool [:(f *),f:]
arity s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
the charact of (FreeUnivAlgNSG ((),A)) . 4 is homogeneous set
FreeOpNSG (4,P,A) is non empty Relation-like (TS (DTConUA (P,A))) * -defined TS (DTConUA (P,A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA (P,A))) *),(TS (DTConUA (P,A))):]
P . (2 + 2) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P /. 4 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (4,P,A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA (P,A))) *)
r is non empty Relation-like f * -defined f -valued Function-like homogeneous quasi_total Element of bool [:(f *),f:]
arity r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
f is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
Terminals (DTConUA ((),A)) is non empty Element of bool the carrier of (DTConUA ((),A))
bool the carrier of (DTConUA ((),A)) is non empty set
P is Element of the carrier of (DTConUA ((),A))
root-tree P is Relation-like the carrier of (DTConUA ((),A)) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA ((),A))
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> P is non empty Relation-like elementary_tree 0 -defined {P} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{P}:]
{P} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{P}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{P}:] is non empty finite V44() set
P is Element of the carrier of (DTConUA ((),A))
C is Relation-like NAT -defined FinTrees the carrier of (DTConUA ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of TS (DTConUA ((),A))
roots C is Relation-like NAT -defined the carrier of (DTConUA ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of (DTConUA ((),A))
P -tree C is Relation-like the carrier of (DTConUA ((),A)) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA ((),A))
NonTerminals (DTConUA ((),A)) is non empty Element of bool the carrier of (DTConUA ((),A))
{ b₁ where b₁ is Element of the carrier of (DTConUA ((),A)) : ex b₂ being Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set st b₁ ==> b₂ } is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
roots s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
[I,(roots s)] is non empty V22() set
{I,(roots s)} is non empty finite V44() set
{I} is non empty trivial finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() V188() set
{{I,(roots s)},{I}} is non empty finite V44() set
the Rules of (DTConUA ((),A)) is Relation-like the carrier of (DTConUA ((),A)) -defined the carrier of (DTConUA ((),A)) * -valued Element of bool [: the carrier of (DTConUA ((),A)),( the carrier of (DTConUA ((),A)) *):]
the carrier of (DTConUA ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (DTConUA ((),A))
[: the carrier of (DTConUA ((),A)),( the carrier of (DTConUA ((),A)) *):] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of (DTConUA ((),A)),( the carrier of (DTConUA ((),A)) *):] is non empty set
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
dom (roots s) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
card (dom (roots s)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
len (roots s) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
1 -tree {} is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
((FreeUnivAlgNSG ((),A))) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
(Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . {} is set
[:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . 1 is homogeneous set
P is non empty Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like homogeneous quasi_total 0 -ary Element of bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):]
FreeOpNSG (1,(),A) is non empty Relation-like (TS (DTConUA ((),A))) * -defined TS (DTConUA ((),A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):]
[:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty set
T is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
T /. 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (1,(),A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA ((),A))) *)
bool ((TS (DTConUA ((),A))) *) is non empty set
(T /. 1) -tuples_on (TS (DTConUA ((),A))) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of TS (DTConUA ((),A))
<*> (TS (DTConUA ((),A))) is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined TS (DTConUA ((),A)) -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of TS (DTConUA ((),A))
[:NAT,(TS (DTConUA ((),A))):] is non empty non trivial Relation-like non finite disjoint_with_NAT set
P . {} is set
Sym (1,(),A) is Element of the carrier of (DTConUA ((),A))
(Sym (1,(),A)) -tree {} is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
1 -tree P is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
<*> (TS (DTConUA ((),A))) is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined TS (DTConUA ((),A)) -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of TS (DTConUA ((),A))
[:NAT,(TS (DTConUA ((),A))):] is non empty non trivial Relation-like non finite disjoint_with_NAT set
I -tree (<*> (TS (DTConUA ((),A)))) is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),P,C) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*P,C*> is set
2 -tree (P,C) is Relation-like Function-like DecoratedTree-like set
<*P,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
2 -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
[:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . 2 is homogeneous set
I is non empty Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):]
FreeOpNSG (2,(),A) is non empty Relation-like (TS (DTConUA ((),A))) * -defined TS (DTConUA ((),A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):]
[:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty set
T is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
T /. 2 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (2,(),A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA ((),A))) *)
bool ((TS (DTConUA ((),A))) *) is non empty set
2 -tuples_on (TS (DTConUA ((),A))) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of TS (DTConUA ((),A))
I . <*P,C*> is set
Sym (2,(),A) is Element of the carrier of (DTConUA ((),A))
(Sym (2,(),A)) -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
2 -tree P is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
<*> (TS (DTConUA ((),A))) is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined TS (DTConUA ((),A)) -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of TS (DTConUA ((),A))
[:NAT,(TS (DTConUA ((),A))):] is non empty non trivial Relation-like non finite disjoint_with_NAT set
I -tree (<*> (TS (DTConUA ((),A)))) is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P . 1 is set
P . 2 is set
<*(P . 1),(P . 2)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(P . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 1)] is non empty V22() set
{1,(P . 1)} is non empty finite set
{{1,(P . 1)},{1}} is non empty finite V44() set
{[1,(P . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 2)] is non empty V22() set
{1,(P . 2)} is non empty finite set
{{1,(P . 2)},{1}} is non empty finite V44() set
{[1,(P . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 1)*> ^ <*(P . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
rng P is finite Element of bool the carrier of (FreeUnivAlgNSG ((),A))
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
{(P . 1),(P . 2)} is non empty finite set
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
i is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*r,i*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),P,C) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*P,C*> is set
rng <*P,C*> is non empty finite Element of bool the carrier of (FreeUnivAlgNSG ((),A))
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
{P,C} is non empty functional finite set
2 -tree (P,C) is Relation-like Function-like DecoratedTree-like set
<*P,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
2 -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
2 -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
f is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),f,P) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*f,P*> is set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),C,I) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*C,I*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*C,I*> is set
2 -tree (f,P) is Relation-like Function-like DecoratedTree-like set
<*f,P*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
2 -tree <*f,P*> is Relation-like Function-like DecoratedTree-like set
2 -tree (C,I) is Relation-like Function-like DecoratedTree-like set
<*C,I*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
2 -tree <*C,I*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),P,C,I) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*P,C,I*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*P*> ^ <*C*>) ^ <*I*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*P,C,I*> is set
3 -tree <*P,C,I*> is Relation-like Function-like DecoratedTree-like set
[:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . 3 is homogeneous set
s is non empty Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like homogeneous quasi_total 3 -ary Element of bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):]
FreeOpNSG (3,(),A) is non empty Relation-like (TS (DTConUA ((),A))) * -defined TS (DTConUA ((),A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):]
[:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty set
T is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
T /. 3 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (3,(),A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA ((),A))) *)
bool ((TS (DTConUA ((),A))) *) is non empty set
3 -tuples_on (TS (DTConUA ((),A))) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of TS (DTConUA ((),A))
s . <*P,C,I*> is set
Sym (3,(),A) is Element of the carrier of (DTConUA ((),A))
(Sym (3,(),A)) -tree <*P,C,I*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
3 -tree P is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
<*> (TS (DTConUA ((),A))) is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined TS (DTConUA ((),A)) -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of TS (DTConUA ((),A))
[:NAT,(TS (DTConUA ((),A))):] is non empty non trivial Relation-like non finite disjoint_with_NAT set
I -tree (<*> (TS (DTConUA ((),A)))) is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P . 1 is set
P . 2 is set
P . 3 is set
<*(P . 1),(P . 2),(P . 3)*> is non empty Relation-like NAT -defined Function-like finite 3 -element FinSequence-like FinSubsequence-like set
<*(P . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 1)] is non empty V22() set
{1,(P . 1)} is non empty finite set
{{1,(P . 1)},{1}} is non empty finite V44() set
{[1,(P . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 2)] is non empty V22() set
{1,(P . 2)} is non empty finite set
{{1,(P . 2)},{1}} is non empty finite V44() set
{[1,(P . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 1)*> ^ <*(P . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*(P . 3)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 3)] is non empty V22() set
{1,(P . 3)} is non empty finite set
{{1,(P . 3)},{1}} is non empty finite V44() set
{[1,(P . 3)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*(P . 1)*> ^ <*(P . 2)*>) ^ <*(P . 3)*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
rng P is finite Element of bool the carrier of (FreeUnivAlgNSG ((),A))
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
{(P . 1),(P . 2),(P . 3)} is non empty finite set
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
i is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*r,i,s*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
f is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),f,C,I) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*f,C,I*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*C*>) ^ <*I*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*f,C,I*> is set
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),P,s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*P,s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*P*> ^ <*s*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*P,s,r*> is set
3 -tree <*f,C,I*> is Relation-like Function-like DecoratedTree-like set
3 -tree <*P,s,r*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),P,C) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*P,C*> is set
4 -tree (P,C) is Relation-like Function-like DecoratedTree-like set
<*P,C*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
4 -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
[:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty Relation-like disjoint_with_NAT set
bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):] is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . 4 is homogeneous set
I is non empty Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like homogeneous quasi_total 2 -ary Element of bool [:( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)):]
FreeOpNSG (4,(),A) is non empty Relation-like (TS (DTConUA ((),A))) * -defined TS (DTConUA ((),A)) -valued Function-like homogeneous quasi_total Element of bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):]
[:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty Relation-like disjoint_with_NAT set
bool [:((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))):] is non empty set
T is non empty Relation-like NAT -defined omega -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of omega
T /. 4 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of omega
dom (FreeOpNSG (4,(),A)) is non empty functional FinSequence-membered with_common_domain Element of bool ((TS (DTConUA ((),A))) *)
bool ((TS (DTConUA ((),A))) *) is non empty set
2 -tuples_on (TS (DTConUA ((),A))) is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of TS (DTConUA ((),A))
I . <*P,C*> is set
Sym (4,(),A) is Element of the carrier of (DTConUA ((),A))
(Sym (4,(),A)) -tree <*P,C*> is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
P is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
4 -tree P is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
<*> (TS (DTConUA ((),A))) is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined TS (DTConUA ((),A)) -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of TS (DTConUA ((),A))
[:NAT,(TS (DTConUA ((),A))):] is non empty non trivial Relation-like non finite disjoint_with_NAT set
I -tree (<*> (TS (DTConUA ((),A)))) is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P . 1 is set
P . 2 is set
<*(P . 1),(P . 2)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(P . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 1)] is non empty V22() set
{1,(P . 1)} is non empty finite set
{{1,(P . 1)},{1}} is non empty finite V44() set
{[1,(P . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(P . 2)] is non empty V22() set
{1,(P . 2)} is non empty finite set
{{1,(P . 2)},{1}} is non empty finite V44() set
{[1,(P . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(P . 1)*> ^ <*(P . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
rng P is finite Element of bool the carrier of (FreeUnivAlgNSG ((),A))
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
{(P . 1),(P . 2)} is non empty finite set
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
i is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*r,i*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
((FreeUnivAlgNSG ((),A))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
((FreeUnivAlgNSG ((),A))) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
(Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . {} is set
{((FreeUnivAlgNSG ((),A)))} is non empty trivial functional finite 1 -element set
the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))} is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (3,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (4,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
(( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
{ ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is set
((( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ { ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
f is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
P is Element of A
root-tree P is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> P is non empty Relation-like elementary_tree 0 -defined {P} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{P}:]
{P} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{P}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{P}:] is non empty finite V44() set
P is Element of A
root-tree P is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> P is non empty Relation-like elementary_tree 0 -defined {P} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{P}:]
{P} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{P}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{P}:] is non empty finite V44() set
P -tree {} is Relation-like Function-like DecoratedTree-like set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
P -tree C is Relation-like Function-like DecoratedTree-like set
len C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
P -tree C is Relation-like Function-like DecoratedTree-like set
len C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*I,s*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
P -tree (I,s) is Relation-like Function-like DecoratedTree-like set
<*I,s*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
P -tree <*I,s*> is Relation-like Function-like DecoratedTree-like set
((FreeUnivAlgNSG ((),A)),I,s) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*I,s*> is set
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*I,s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*I*> ^ <*s*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
((FreeUnivAlgNSG ((),A)),I,s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*I,s,r*> is set
I is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*I,s*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
P -tree (I,s) is Relation-like Function-like DecoratedTree-like set
<*I,s*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
P -tree <*I,s*> is Relation-like Function-like DecoratedTree-like set
((FreeUnivAlgNSG ((),A)),I,s) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
(Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*I,s*> is set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
f is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
P is Element of A
P -tree f is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
I -tree s is Relation-like Function-like DecoratedTree-like set
len s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
I is Element of A
root-tree I is Relation-like A -valued Function-like DecoratedTree-like Element of FinTrees A
FinTrees A is non empty functional constituted-DTrees DTree-set of A
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I -tree {} is Relation-like Function-like DecoratedTree-like set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
((FreeUnivAlgNSG ((),A))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
((FreeUnivAlgNSG ((),A))) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
(Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . {} is set
{((FreeUnivAlgNSG ((),A)))} is non empty trivial functional finite 1 -element set
the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))} is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (3,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (4,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
(( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
{ ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is set
((( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ { ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
FreeGenSetNSG ((),A) is free GeneratorSet of FreeUnivAlgNSG ((),A)
Terminals (DTConUA ((),A)) is non empty Element of bool the carrier of (DTConUA ((),A))
bool the carrier of (DTConUA ((),A)) is non empty set
{ (root-tree b₁) where b₁ is Element of the carrier of (DTConUA ((),A)) : b₁ in Terminals (DTConUA ((),A)) } is set
card A is non empty ordinal cardinal set
card (FreeGenSetNSG ((),A)) is ordinal cardinal set
P is set
C is Element of A
C -tree {} is Relation-like Function-like DecoratedTree-like set
I is Element of the carrier of (DTConUA ((),A))
root-tree I is Relation-like the carrier of (DTConUA ((),A)) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA ((),A))
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
P is set
I is Element of the carrier of (DTConUA ((),A))
root-tree I is Relation-like the carrier of (DTConUA ((),A)) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA ((),A))
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
I -tree {} is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r*> is set
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
In (2,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
<*s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (2,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r*> is set
2 -tree (s,r) is Relation-like Function-like DecoratedTree-like set
<*s,r*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
2 -tree <*s,r*> is Relation-like Function-like DecoratedTree-like set
2 -tree <*s,r*> is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
i is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r,i) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r,i*> is set
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
i is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r,i) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 3 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r,i*> is set
3 -tree <*s,r,i*> is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r*> is set
s is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
r is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
((FreeUnivAlgNSG ((),A)),s,r) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
<*s,r*> is non empty Relation-like NAT -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding FinSequence of the carrier of (FreeUnivAlgNSG ((),A))
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element Function-yielding FinSequence-like FinSubsequence-like FuncSeq-like DTree-yielding set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
(Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . <*s,r*> is set
4 -tree (s,r) is Relation-like Function-like DecoratedTree-like set
<*s,r*> is non empty Relation-like NAT -defined Function-like finite 2 -element Function-yielding FinSequence-like FinSubsequence-like DTree-yielding set
4 -tree <*s,r*> is Relation-like Function-like DecoratedTree-like set
4 -tree <*s,r*> is Relation-like Function-like DecoratedTree-like set
C is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
P is Relation-like Function-like set
dom P is set
rng P is set
C is set
I is set
P . I is set
root-tree I is Relation-like Function-like DecoratedTree-like set
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
C is set
I is Element of the carrier of (DTConUA ((),A))
root-tree I is Relation-like the carrier of (DTConUA ((),A)) -valued Function-like DecoratedTree-like Element of FinTrees the carrier of (DTConUA ((),A))
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> I is non empty Relation-like elementary_tree 0 -defined {I} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{I}:]
{I} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{I}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{I}:] is non empty finite V44() set
s is Element of A
P . s is set
C is set
I is set
P . C is set
P . I is set
s is Element of A
root-tree s is Relation-like Function-like DecoratedTree-like set
elementary_tree 0 is non empty finite Tree-like set
(elementary_tree 0) --> s is non empty Relation-like elementary_tree 0 -defined {s} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{s}:]
{s} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{s}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{s}:] is non empty finite V44() set
r is Element of A
root-tree r is Relation-like Function-like DecoratedTree-like set
(elementary_tree 0) --> r is non empty Relation-like elementary_tree 0 -defined {r} -valued Function-like total quasi_total finite DecoratedTree-like Element of bool [:(elementary_tree 0),{r}:]
{r} is non empty trivial finite 1 -element set
[:(elementary_tree 0),{r}:] is non empty Relation-like finite disjoint_with_NAT set
bool [:(elementary_tree 0),{r}:] is non empty finite V44() set
{0} is non empty trivial functional finite V44() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Tree-like V188() set
[:NAT,{0}:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals disjoint_with_NAT set
A is non empty non trivial Relation-like RAT -valued INT -valued non finite V91() V92() V93() to-naturals disjoint_with_NAT set
S is set
T is set
f is set
[T,f] is non empty V22() set
{T,f} is non empty finite set
{T} is non empty trivial finite 1 -element set
{{T,f},{T}} is non empty finite V44() set
A is non empty non trivial non finite disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
card A is non empty non trivial ordinal non finite cardinal set
FreeGenSetNSG ((),A) is free GeneratorSet of FreeUnivAlgNSG ((),A)
Terminals (DTConUA ((),A)) is non empty Element of bool the carrier of (DTConUA ((),A))
bool the carrier of (DTConUA ((),A)) is non empty set
{ (root-tree b₁) where b₁ is Element of the carrier of (DTConUA ((),A)) : b₁ in Terminals (DTConUA ((),A)) } is set
card (FreeGenSetNSG ((),A)) is ordinal cardinal set
((FreeUnivAlgNSG ((),A))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
bool the carrier of (FreeUnivAlgNSG ((),A)) is non empty set
((FreeUnivAlgNSG ((),A))) is Relation-like Function-like DecoratedTree-like Element of the carrier of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))
the carrier of (FreeUnivAlgNSG ((),A)) * is non empty functional FinSequence-membered FinSequenceSet of the carrier of (FreeUnivAlgNSG ((),A))
PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))) is set
dom the charact of (FreeUnivAlgNSG ((),A)) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
rng the charact of (FreeUnivAlgNSG ((),A)) is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A))))
bool (PFuncs (( the carrier of (FreeUnivAlgNSG ((),A)) *), the carrier of (FreeUnivAlgNSG ((),A)))) is non empty set
the charact of (FreeUnivAlgNSG ((),A)) . (In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
(Den ((In (1,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) . {} is set
{((FreeUnivAlgNSG ((),A)))} is non empty trivial functional finite 1 -element set
the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))} is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (3,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
In (4,(dom the charact of (FreeUnivAlgNSG ((),A)))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of (FreeUnivAlgNSG ((),A))
Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))) is Relation-like the carrier of (FreeUnivAlgNSG ((),A)) * -defined the carrier of (FreeUnivAlgNSG ((),A)) -valued Function-like Element of rng the charact of (FreeUnivAlgNSG ((),A))
the charact of (FreeUnivAlgNSG ((),A)) . (In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
(( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A))))) is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
{ ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is set
((( the carrier of (FreeUnivAlgNSG ((),A)) \ {((FreeUnivAlgNSG ((),A)))}) \ (rng (Den ((In (3,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ (rng (Den ((In (4,(dom the charact of (FreeUnivAlgNSG ((),A))))),(FreeUnivAlgNSG ((),A)))))) \ { ((FreeUnivAlgNSG ((),A)),b₁,b₂) where b₁, b₂ is Element of the carrier of (FreeUnivAlgNSG ((),A)) : ( not b₁ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) & not b₂ = ((FreeUnivAlgNSG ((),A)),b₁,b₂) ) } is Element of bool the carrier of (FreeUnivAlgNSG ((),A))
card ((FreeUnivAlgNSG ((),A))) is ordinal cardinal set
A is non empty disjoint_with_NAT set
FreeUnivAlgNSG ((),A) is non empty strict partial quasi_total non-empty free () () () () UAStr
DTConUA ((),A) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ A is non empty set
REL ((),A) is Relation-like (dom ()) \/ A -defined ((dom ()) \/ A) * -valued Element of bool [:((dom ()) \/ A),(((dom ()) \/ A) *):]
((dom ()) \/ A) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ A
[:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ A),(((dom ()) \/ A) *):] is non empty set
DTConstrStr(# ((dom ()) \/ A),(REL ((),A)) #) is non empty strict DTConstrStr
TS (DTConUA ((),A)) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((),A)))
the carrier of (DTConUA ((),A)) is non empty set
FinTrees the carrier of (DTConUA ((),A)) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((),A))
bool (FinTrees the carrier of (DTConUA ((),A))) is non empty set
FreeOpSeqNSG ((),A) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A))))
(TS (DTConUA ((),A))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((),A))
PFuncs (((TS (DTConUA ((),A))) *),(TS (DTConUA ((),A)))) is set
UAStr(# (TS (DTConUA ((),A))),(FreeOpSeqNSG ((),A)) #) is non empty strict UAStr
signature (FreeUnivAlgNSG ((),A)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
A is non empty partial quasi_total non-empty () () () () UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
(A) is Element of bool the carrier of A
(A) is Element of the carrier of A
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is set
dom (Den ((In (1,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
T is Element of the carrier of A
rng (Den ((In (1,(dom the charact of A))),A)) is Element of bool the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*f,P*> is set
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*f,P*> is set
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
rng (Den ((In (2,(dom the charact of A))),A)) is Element of bool the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
C is Element of the carrier of A
(A,f,P,C) is Element of the carrier of A
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*f,P,C*> is set
f is Element of the carrier of A
P is Element of the carrier of A
C is Element of the carrier of A
(A,f,P,C) is Element of the carrier of A
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*f,P,C*> is set
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*f,P*> is set
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*f,P*> is set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
T is Element of the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
T is Element of the carrier of A
f is Element of the carrier of A
(A,T,f) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,f*> is set
S is Element of the carrier of A
(A,S,T,f) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
(A,S,T) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
dom (Den ((In (1,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
P is Element of the carrier of A
C is Element of the carrier of A
(A,P,C) is Element of the carrier of A
<*P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*P,C*> is set
f is Element of the carrier of A
(A,f,P,C) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*f,P,C*> is set
(A,f,P) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den ((In (4,(dom the charact of A))),A)) . <*f,P*> is set
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng <*S,T*> is non empty finite Element of bool the carrier of A
bool the carrier of A is non empty set
{S,T} is non empty finite set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
(A,S,T,f) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
P is Element of the carrier of A
C is Element of the carrier of A
(A,P,C) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*P,C*> is set
I is Element of the carrier of A
(A,P,C,I) is Element of the carrier of A
<*P,C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*P*> ^ <*C*>) ^ <*I*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*P,C,I*> is set
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
rng <*S,T,f*> is non empty finite Element of bool the carrier of A
bool the carrier of A is non empty set
{S,T,f} is non empty finite set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*f,P*> is set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
rng <*S,T*> is non empty finite Element of bool the carrier of A
bool the carrier of A is non empty set
{S,T} is non empty finite set
A is non empty partial quasi_total non-empty () () () () UAStr
S is non empty partial quasi_total non-empty free UAStr
the free GeneratorSet of S is free GeneratorSet of S
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } is set
the carrier of A \ (union { (rng b₁) where b₁ is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A : verum } ) is Element of bool the carrier of A
(A) is Element of bool the carrier of A
(A) is Element of the carrier of A
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
(A,f,P) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*f,P*> is set
I is Element of the carrier of A
C is Element of the carrier of A
(A,C,I) is Element of the carrier of A
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*C,I*> is set
s is Element of the carrier of A
r is Element of the carrier of A
(A,s,r) is Element of the carrier of A
<*s,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*s,r*> is set
f is Element of the carrier of A
P is Element of the carrier of A
C is Element of the carrier of A
(A,f,P,C) is Element of the carrier of A
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*f,P,C*> is set
I is Element of the carrier of A
s is Element of the carrier of A
(A,I,s) is Element of the carrier of A
<*I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*I,s*> is set
r is Element of the carrier of A
i is Element of the carrier of A
(A,r,i) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
s is Element of the carrier of A
s is Element of the carrier of A
r is Element of the carrier of A
(A,s,s,r) is Element of the carrier of A
<*s,s,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*s*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*s,s,r*> is set
i is Element of the carrier of A
ri is Element of the carrier of A
(A,i,ri) is Element of the carrier of A
<*i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,ri*> is set
r is Element of the carrier of A
i is Element of the carrier of A
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*r,i*> is set
i is Element of the carrier of A
i0 is Element of the carrier of A
gI2 is Element of the carrier of A
(A,i,i0,gI2) is Element of the carrier of A
<*i,i0,gI2*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i0*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i0] is non empty V22() set
{1,i0} is non empty finite set
{{1,i0},{1}} is non empty finite V44() set
{[1,i0]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*i0*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*i0*>) ^ <*gI2*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,i0,gI2*> is set
J is Element of the carrier of A
a is Element of the carrier of A
(A,J,a) is Element of the carrier of A
<*J,a*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> ^ <*a*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*J,a*> is set
the non empty non trivial non finite disjoint_with_NAT set is non empty non trivial non finite disjoint_with_NAT set
FreeUnivAlgNSG ((), the non empty non trivial non finite disjoint_with_NAT set ) is non empty strict partial quasi_total non-empty free () () () () () () () () UAStr
DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ) is non empty strict with_terminals with_nonterminals with_useful_nonterminals DTConstrStr
(dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set is non empty set
REL ((), the non empty non trivial non finite disjoint_with_NAT set ) is Relation-like (dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set -defined ((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ) * -valued Element of bool [:((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ),(((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ) *):]
((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ) * is non empty functional FinSequence-membered FinSequenceSet of (dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set
[:((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ),(((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ) *):] is non empty Relation-like disjoint_with_NAT set
bool [:((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ),(((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ) *):] is non empty set
DTConstrStr(# ((dom ()) \/ the non empty non trivial non finite disjoint_with_NAT set ),(REL ((), the non empty non trivial non finite disjoint_with_NAT set )) #) is non empty strict DTConstrStr
TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )) is non empty functional constituted-DTrees Element of bool (FinTrees the carrier of (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )))
the carrier of (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )) is non empty set
FinTrees the carrier of (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )) is non empty functional constituted-DTrees DTree-set of the carrier of (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))
bool (FinTrees the carrier of (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))) is non empty set
FreeOpSeqNSG ((), the non empty non trivial non finite disjoint_with_NAT set ) is Relation-like NAT -defined PFuncs (((TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))) *),(TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )))) -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of PFuncs (((TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))) *),(TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))))
(TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))) * is non empty functional FinSequence-membered FinSequenceSet of TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))
PFuncs (((TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))) *),(TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set )))) is set
UAStr(# (TS (DTConUA ((), the non empty non trivial non finite disjoint_with_NAT set ))),(FreeOpSeqNSG ((), the non empty non trivial non finite disjoint_with_NAT set )) #) is non empty strict UAStr
A is non empty partial quasi_total non-empty () () () () () UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
S is Element of bool the carrier of A
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(T + 1)) is Element of bool the carrier of A
(A,S,T) is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
(A,S,T) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,S,T) ) } is set
<*> the carrier of A is empty proper ordinal natural Relation-like non-empty empty-yielding NAT -defined the carrier of A -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() FinSequence of the carrier of A
[:NAT, the carrier of A:] is non empty non trivial Relation-like non finite disjoint_with_NAT set
P is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
rng P is finite Element of bool the carrier of A
dom (Den ((In (1,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
C is Element of the carrier of A
I is Element of the carrier of A
s is Element of the carrier of A
(A,I,s) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*I,s*> is set
(A,C,I,s) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*C,I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(<*C*> ^ <*I*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*C,I,s*> is set
(A,C,I) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den ((In (4,(dom the charact of A))),A)) . <*C,I*> is set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
rng <*I,s*> is non empty finite Element of bool the carrier of A
{I,s} is non empty finite set
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
rng <*C,I,s*> is non empty finite Element of bool the carrier of A
{C,I,s} is non empty finite set
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
rng <*C,I*> is non empty finite Element of bool the carrier of A
{C,I} is non empty finite set
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
A is non empty partial quasi_total non-empty () () () () () UAStr
(A) is Element of bool the carrier of A
the carrier of A is non empty set
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
T is set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,(A),(f + 1)) is Element of bool the carrier of A
(A,(A),f) is Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,(A),f) ) } is set
(A,(A),f) \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= (A,(A),f) ) } is set
P is Element of the carrier of A
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (C,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . C is non empty homogeneous set
I is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
(Den (C,A)) . I is set
dom (Den (C,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
rng I is finite Element of bool the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
rng the charact of A is non empty with_non-empty_elements finite set
signature A is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like V91() V92() V93() to-naturals FinSequence of NAT
len (signature A) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
len the charact of A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom (signature A) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
len I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(signature A) . s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
() . C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
Seg 2 is non empty finite 2 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 2 ) } is set
Seg 3 is non empty finite 3 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= 3 ) } is set
I . 1 is set
I . 2 is set
<*(I . 1),(I . 2)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(I . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 1)] is non empty V22() set
{1,(I . 1)} is non empty finite set
{{1,(I . 1)},{1}} is non empty finite V44() set
{[1,(I . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 2)] is non empty V22() set
{1,(I . 2)} is non empty finite set
{{1,(I . 2)},{1}} is non empty finite V44() set
{[1,(I . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 1)*> ^ <*(I . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
dom I is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
i is Element of the carrier of A
s is Element of the carrier of A
(A,i,s) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*i,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,s*> is set
I . 1 is set
I . 2 is set
I . 3 is set
<*(I . 1),(I . 2),(I . 3)*> is non empty Relation-like NAT -defined Function-like finite 3 -element FinSequence-like FinSubsequence-like set
<*(I . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 1)] is non empty V22() set
{1,(I . 1)} is non empty finite set
{{1,(I . 1)},{1}} is non empty finite V44() set
{[1,(I . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 2)] is non empty V22() set
{1,(I . 2)} is non empty finite set
{{1,(I . 2)},{1}} is non empty finite V44() set
{[1,(I . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 1)*> ^ <*(I . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*(I . 3)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 3)] is non empty V22() set
{1,(I . 3)} is non empty finite set
{{1,(I . 3)},{1}} is non empty finite V44() set
{[1,(I . 3)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*(I . 1)*> ^ <*(I . 2)*>) ^ <*(I . 3)*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
dom I is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
i is Element of the carrier of A
s is Element of the carrier of A
s is Element of the carrier of A
(A,i,s,s) is Element of the carrier of A
<*i,s,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*s*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,s,s*> is set
I . 1 is set
I . 2 is set
<*(I . 1),(I . 2)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*(I . 1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 1)] is non empty V22() set
{1,(I . 1)} is non empty finite set
{{1,(I . 1)},{1}} is non empty finite V44() set
{[1,(I . 1)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(I . 2)] is non empty V22() set
{1,(I . 2)} is non empty finite set
{{1,(I . 2)},{1}} is non empty finite V44() set
{[1,(I . 2)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*(I . 1)*> ^ <*(I . 2)*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
dom I is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
i is Element of the carrier of A
s is Element of the carrier of A
(A,i,s) is Element of the carrier of A
<*i,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*i,s*> is set
A is non empty partial quasi_total non-empty UAStr
the carrier of A is non empty set
bool the carrier of A is non empty set
Constants A is Element of bool the carrier of A
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
Operations A is non empty PFuncsDomHQN of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
PFuncs (( the carrier of A *), the carrier of A) is set
rng the charact of A is non empty with_non-empty_elements finite set
{ b₁ where b₁ is Element of the carrier of A : ex b₂ being non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A st
( arity b₂ = 0 & b₁ in rng b₂ ) } is set
S is Element of bool the carrier of A
(A,S,1) is Element of bool the carrier of A
T is set
f is Element of the carrier of A
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng P is non empty Element of bool the carrier of A
P is non empty Relation-like the carrier of A * -defined the carrier of A -valued Function-like homogeneous quasi_total Element of Operations A
arity P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
rng P is non empty Element of bool the carrier of A
dom P is non empty functional FinSequence-membered with_common_domain Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
C is set
P . C is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
I is set
the charact of A . I is homogeneous set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . s is non empty homogeneous set
0 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
r is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A *
rng r is finite Element of bool the carrier of A
{ ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= S ) } is set
(A,S,0) is Element of bool the carrier of A
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,S,(0 + 1)) is Element of bool the carrier of A
S \/ { ((Den (b₁,A)) . b₂) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A, b₂ is Relation-like NAT -defined the carrier of A -valued Function-like finite FinSequence-like FinSubsequence-like Element of the carrier of A * : ( b₂ in dom (Den (b₁,A)) & rng b₂ c= S ) } is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
T is Element of the carrier of A
F₁() is non empty partial quasi_total non-empty () () () () () () UAStr
the carrier of F₁() is non empty set
(F₁()) is Element of bool the carrier of F₁()
bool the carrier of F₁() is non empty set
(F₁()) is Element of the carrier of F₁()
the charact of F₁() is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of F₁() *), the carrier of F₁()) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of F₁() *), the carrier of F₁())
the carrier of F₁() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F₁()
PFuncs (( the carrier of F₁() *), the carrier of F₁()) is set
dom the charact of F₁() is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (1,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
rng the charact of F₁() is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of F₁() *), the carrier of F₁()))
bool (PFuncs (( the carrier of F₁() *), the carrier of F₁())) is non empty set
the charact of F₁() . (In (1,(dom the charact of F₁()))) is non empty homogeneous set
(Den ((In (1,(dom the charact of F₁()))),F₁())) . {} is set
{(F₁())} is non empty trivial finite 1 -element set
the carrier of F₁() \ {(F₁())} is Element of bool the carrier of F₁()
In (3,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (3,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (3,(dom the charact of F₁()))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of F₁()))),F₁())) is Element of bool the carrier of F₁()
( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁()))) is Element of bool the carrier of F₁()
In (4,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (4,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (4,(dom the charact of F₁()))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of F₁()))),F₁())) is Element of bool the carrier of F₁()
(( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁())))) \ (rng (Den ((In (4,(dom the charact of F₁()))),F₁()))) is Element of bool the carrier of F₁()
{ (F₁(),b₁,b₂) where b₁, b₂ is Element of the carrier of F₁() : ( not b₁ = (F₁(),b₁,b₂) & not b₂ = (F₁(),b₁,b₂) ) } is set
((( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁())))) \ (rng (Den ((In (4,(dom the charact of F₁()))),F₁())))) \ { (F₁(),b₁,b₂) where b₁, b₂ is Element of the carrier of F₁() : ( not b₁ = (F₁(),b₁,b₂) & not b₂ = (F₁(),b₁,b₂) ) } is Element of bool the carrier of F₁()
F₂() is Element of the carrier of F₁()
(F₁(),(F₁()),0) is Element of bool the carrier of F₁()
S is Element of the carrier of F₁()
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),S) is Element of bool the carrier of F₁()
S + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),(S + 1)) is Element of bool the carrier of F₁()
T is Element of the carrier of F₁()
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
(F₁(),f,P) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*f,P*> is set
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
(F₁(),f,P) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*f,P*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*f,P*> is set
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
C is Element of the carrier of F₁()
(F₁(),f,P,C) is Element of the carrier of F₁()
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*f,P,C*> is set
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
C is Element of the carrier of F₁()
(F₁(),f,P,C) is Element of the carrier of F₁()
<*f,P,C*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*f*> ^ <*P*>) ^ <*C*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*f,P,C*> is set
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
(F₁(),f,P) is Element of the carrier of F₁()
<*f,P*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*f,P*> is set
f is Element of the carrier of F₁()
P is Element of the carrier of F₁()
(F₁(),f,P) is Element of the carrier of F₁()
<*f,P*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*f,P*> is set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),S) is Element of bool the carrier of F₁()
S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
[:[:S, the carrier of A:],S:] is non empty Relation-like disjoint_with_NAT set
bool [:[:S, the carrier of A:],S:] is non empty set
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
[:[:S, the carrier of A:],S:] is non empty Relation-like disjoint_with_NAT set
bool [:[:S, the carrier of A:],S:] is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
(A,S,T,(A)) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,(A)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*(A)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A)] is non empty V22() set
{1,(A)} is non empty finite set
{{1,(A)},{1}} is non empty finite V44() set
{[1,(A)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*(A)*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,(A)*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
[:[:f, the carrier of A:],f:] is non empty Relation-like disjoint_with_NAT set
bool [:[:f, the carrier of A:],f:] is non empty set
P is Element of bool f
C is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total Element of bool [:[:f, the carrier of A:],f:]
I is Element of f
C . (I,S) is Element of f
[I,S] is non empty V22() set
{I,S} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,S},{I}} is non empty finite V44() set
C . [I,S] is set
C . (I,(A,S,T)) is Element of f
[I,(A,S,T)] is non empty V22() set
{I,(A,S,T)} is non empty finite set
{{I,(A,S,T)},{I}} is non empty finite V44() set
C . [I,(A,S,T)] is set
C . ((C . (I,S)),(A)) is Element of f
[(C . (I,S)),(A)] is non empty V22() set
{(C . (I,S)),(A)} is non empty finite set
{(C . (I,S))} is non empty trivial finite 1 -element set
{{(C . (I,S)),(A)},{(C . (I,S))}} is non empty finite V44() set
C . [(C . (I,S)),(A)] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
pr1 (S, the carrier of A) is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total Element of bool [:[:S, the carrier of A:],S:]
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
[:[:S, the carrier of A:],S:] is non empty Relation-like disjoint_with_NAT set
bool [:[:S, the carrier of A:],S:] is non empty set
T is Element of bool S
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
P is Element of S
(pr1 (S, the carrier of A)) . (P,(A)) is Element of S
[P,(A)] is non empty V22() set
{P,(A)} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,(A)},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,(A)] is set
P is Element of S
C is Element of the carrier of A
I is Element of the carrier of A
(A,C,I) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*C,I*> is set
(pr1 (S, the carrier of A)) . (P,(A,C,I)) is Element of S
[P,(A,C,I)] is non empty V22() set
{P,(A,C,I)} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,(A,C,I)},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,(A,C,I)] is set
(pr1 (S, the carrier of A)) . (P,C) is Element of S
[P,C] is non empty V22() set
{P,C} is non empty finite set
{{P,C},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,C] is set
(pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I) is Element of S
[((pr1 (S, the carrier of A)) . (P,C)),I] is non empty V22() set
{((pr1 (S, the carrier of A)) . (P,C)),I} is non empty finite set
{((pr1 (S, the carrier of A)) . (P,C))} is non empty trivial finite 1 -element set
{{((pr1 (S, the carrier of A)) . (P,C)),I},{((pr1 (S, the carrier of A)) . (P,C))}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [((pr1 (S, the carrier of A)) . (P,C)),I] is set
P is Element of S
C is Element of the carrier of A
I is Element of the carrier of A
s is Element of the carrier of A
(A,C,I,s) is Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*C,I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*C*> ^ <*I*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*C,I,s*> is set
(pr1 (S, the carrier of A)) . (P,(A,C,I,s)) is Element of S
[P,(A,C,I,s)] is non empty V22() set
{P,(A,C,I,s)} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,(A,C,I,s)},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,(A,C,I,s)] is set
(pr1 (S, the carrier of A)) . (P,C) is Element of S
[P,C] is non empty V22() set
{P,C} is non empty finite set
{{P,C},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,C] is set
(pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I) is Element of S
[((pr1 (S, the carrier of A)) . (P,C)),I] is non empty V22() set
{((pr1 (S, the carrier of A)) . (P,C)),I} is non empty finite set
{((pr1 (S, the carrier of A)) . (P,C))} is non empty trivial finite 1 -element set
{{((pr1 (S, the carrier of A)) . (P,C)),I},{((pr1 (S, the carrier of A)) . (P,C))}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [((pr1 (S, the carrier of A)) . (P,C)),I] is set
(pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),s) is Element of S
[((pr1 (S, the carrier of A)) . (P,C)),s] is non empty V22() set
{((pr1 (S, the carrier of A)) . (P,C)),s} is non empty finite set
{{((pr1 (S, the carrier of A)) . (P,C)),s},{((pr1 (S, the carrier of A)) . (P,C))}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [((pr1 (S, the carrier of A)) . (P,C)),s] is set
P is Element of S
C is Element of the carrier of A
(pr1 (S, the carrier of A)) . (P,C) is Element of S
[P,C] is non empty V22() set
{P,C} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,C},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,C] is set
I is Element of the carrier of A
(A,C,I) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*C,I*> is set
(pr1 (S, the carrier of A)) . (P,(A,C,I)) is Element of S
[P,(A,C,I)] is non empty V22() set
{P,(A,C,I)} is non empty finite set
{{P,(A,C,I)},{P}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [P,(A,C,I)] is set
(pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I) is Element of S
[((pr1 (S, the carrier of A)) . (P,C)),I] is non empty V22() set
{((pr1 (S, the carrier of A)) . (P,C)),I} is non empty finite set
{((pr1 (S, the carrier of A)) . (P,C))} is non empty trivial finite 1 -element set
{{((pr1 (S, the carrier of A)) . (P,C)),I},{((pr1 (S, the carrier of A)) . (P,C))}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [((pr1 (S, the carrier of A)) . (P,C)),I] is set
(pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I)),(A,C,I)) is Element of S
[((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I)),(A,C,I)] is non empty V22() set
{((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I)),(A,C,I)} is non empty finite set
{((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I))} is non empty trivial finite 1 -element set
{{((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I)),(A,C,I)},{((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I))}} is non empty finite V44() set
(pr1 (S, the carrier of A)) . [((pr1 (S, the carrier of A)) . (((pr1 (S, the carrier of A)) . (P,C)),I)),(A,C,I)] is set
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
[:[:S, the carrier of A:],S:] is non empty Relation-like disjoint_with_NAT set
bool [:[:S, the carrier of A:],S:] is non empty set
T is Element of bool S
pr1 (S, the carrier of A) is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total Element of bool [:[:S, the carrier of A:],S:]
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S,T)
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S,T)
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool T
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool T
S is Element of the carrier of A
f is Element of T
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,P)
+infty is non empty V80() ext-real positive non negative Element of ExtREAL
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
I . 1 is set
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . (len I) is set
(len I) - 1 is V79() V80() ext-real set
s is ext-real Element of ExtREAL
r is ext-real Element of ExtREAL
i is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len i) - 1 is V79() V80() ext-real set
i . 1 is set
i . (len i) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i . s is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (s + 1) is set
C . ((i . s),S) is set
[(i . s),S] is non empty V22() set
{(i . s),S} is non empty finite set
{(i . s)} is non empty trivial finite 1 -element set
{{(i . s),S},{(i . s)}} is non empty finite V44() set
C . [(i . s),S] is set
I is ext-real Element of ExtREAL
r is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len r) - 1 is V79() V80() ext-real set
r . 1 is set
r . (len r) is set
s is ext-real Element of ExtREAL
i is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len i) - 1 is V79() V80() ext-real set
i . 1 is set
i . (len i) is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (0 + 1) is set
i . (0 + 1) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (s + 1) is set
i . (s + 1) is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(s + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((s + 1) + 1) is set
i . ((s + 1) + 1) is set
(s + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((s + 1) + 1) is set
r . (s + 1) is set
C . ((r . (s + 1)),S) is set
[(r . (s + 1)),S] is non empty V22() set
{(r . (s + 1)),S} is non empty finite set
{(r . (s + 1))} is non empty trivial finite 1 -element set
{{(r . (s + 1)),S},{(r . (s + 1))}} is non empty finite V44() set
C . [(r . (s + 1)),S] is set
i . ((s + 1) + 1) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i . (len r) is set
s is ext-real Element of ExtREAL
i is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len i) - 1 is V79() V80() ext-real set
i . 1 is set
i . (len i) is set
r is ext-real Element of ExtREAL
s is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len s) - 1 is V79() V80() ext-real set
s . 1 is set
s . (len s) is set
s is ext-real Element of ExtREAL
r is ext-real Element of ExtREAL
i is ext-real Element of ExtREAL
ri is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len ri is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len ri) - 1 is V79() V80() ext-real set
ri . 1 is set
ri . (len ri) is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool T
P is Element of T
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,f)
(A,S,T,P,f,C) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len I) - 1 is V79() V80() ext-real set
I . 1 is set
I . (len I) is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool T
P is Element of T
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,f)
(A,S,T,P,f,C) is ext-real Element of ExtREAL
<*P*> is non empty trivial Relation-like NAT -defined T -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of T
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> . 1 is set
len <*P*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
<*P*> . s is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
<*P*> . (s + 1) is set
C . ((<*P*> . s),S) is set
[(<*P*> . s),S] is non empty V22() set
{(<*P*> . s),S} is non empty finite set
{(<*P*> . s)} is non empty trivial finite 1 -element set
{{(<*P*> . s),S},{(<*P*> . s)}} is non empty finite V44() set
C . [(<*P*> . s),S] is set
1 - 1 is V79() V80() ext-real set
0. is empty ordinal natural Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V22() finite finite-yielding V44() cardinal {} -element Function-yielding FinSequence-like FinSubsequence-like FinSequence-membered V79() V80() ext-real non positive non negative with_common_domain V91() V92() V93() to-naturals complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V107() FuncSeq-like V146() V147() V148() V149() constituted-Trees constituted-FinTrees constituted-DTrees Tree-yielding FinTree-yielding DTree-yielding V188() Element of ExtREAL
<*P*> . (len <*P*>) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
<*P*> . s is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
<*P*> . (s + 1) is set
C . ((<*P*> . s),S) is set
[(<*P*> . s),S] is non empty V22() set
{(<*P*> . s),S} is non empty finite set
{(<*P*> . s)} is non empty trivial finite 1 -element set
{{(<*P*> . s),S},{(<*P*> . s)}} is non empty finite V44() set
C . [(<*P*> . s),S] is set
1. is ext-real Element of ExtREAL
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool T
P is Element of T
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,f)
C . (P,S) is Element of T
[P,S] is non empty V22() set
{P,S} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,S},{P}} is non empty finite V44() set
C . [P,S] is set
(A,S,T,P,f,C) is ext-real Element of ExtREAL
(A,S,T,(C . (P,S)),f,C) is ext-real Element of ExtREAL
1. + (A,S,T,(C . (P,S)),f,C) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
I . 1 is set
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . (len I) is set
s is Element of T
<*s*> is non empty trivial Relation-like NAT -defined T -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of T
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
r is Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
<*s*> ^ r is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
1 + (len r) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . 1 is set
i . (len i) is set
len <*s*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom i is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
1 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . (1 + 1) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i . s is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (s + 1) is set
C . ((i . s),S) is set
[(i . s),S] is non empty V22() set
{(i . s),S} is non empty finite set
{(i . s)} is non empty trivial finite 1 -element set
{{(i . s),S},{(i . s)}} is non empty finite V44() set
C . [(i . s),S] is set
I . (s + 1) is set
(s + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . ((s + 1) + 1) is set
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
I . 1 is set
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I . (len I) is set
<*P*> is non empty trivial Relation-like NAT -defined T -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of T
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
s is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
len <*P*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
1 + (len I) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom I is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s . r is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (r + 1) is set
C . ((s . r),S) is set
[(s . r),S] is non empty V22() set
{(s . r),S} is non empty finite set
{(s . r)} is non empty trivial finite 1 -element set
{{(s . r),S},{(s . r)}} is non empty finite V44() set
C . [(s . r),S] is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (i + 1) is set
I . i is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . ((i + 1) + 1) is set
I . (i + 1) is set
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len I) - 1 is V79() V80() ext-real set
I . 1 is set
I . (len I) is set
<*P*> is non empty trivial Relation-like NAT -defined T -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like FinSequence of T
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len <*P*> is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . 1 is set
len (<*P*> ^ I) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
1 + (len I) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
1 + ((len I) - 1) is V79() V80() ext-real set
(len (<*P*> ^ I)) - 1 is V79() V80() ext-real set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom I is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(<*P*> ^ I) . (len (<*P*> ^ I)) is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . r is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . (r + 1) is set
C . (((<*P*> ^ I) . r),S) is set
[((<*P*> ^ I) . r),S] is non empty V22() set
{((<*P*> ^ I) . r),S} is non empty finite set
{((<*P*> ^ I) . r)} is non empty trivial finite 1 -element set
{{((<*P*> ^ I) . r),S},{((<*P*> ^ I) . r)}} is non empty finite V44() set
C . [((<*P*> ^ I) . r),S] is set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . (i + 1) is set
I . i is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . ((i + 1) + 1) is set
I . (i + 1) is set
(<*P*> ^ I) . r is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(<*P*> ^ I) . (r + 1) is set
C . (((<*P*> ^ I) . r),S) is set
[((<*P*> ^ I) . r),S] is non empty V22() set
{((<*P*> ^ I) . r),S} is non empty finite set
{((<*P*> ^ I) . r)} is non empty trivial finite 1 -element set
{{((<*P*> ^ I) . r),S},{((<*P*> ^ I) . r)}} is non empty finite V44() set
C . [((<*P*> ^ I) . r),S] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool T
P is Element of T
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,f)
(A,S,T,P,f,C) is ext-real Element of ExtREAL
I is non empty Relation-like NAT -defined T -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of T
len I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len I) - 1 is V79() V80() ext-real set
I . 1 is set
I . (len I) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
F₁() is non empty partial quasi_total non-empty () () () () UAStr
the carrier of F₁() is non empty set
F₃() is non empty set
bool F₃() is non empty set
[:F₃(), the carrier of F₁():] is non empty Relation-like disjoint_with_NAT set
F₅() is Element of bool F₃()
F₄() is Element of F₃()
F₆() is non empty Relation-like [:F₃(), the carrier of F₁():] -defined F₃() -valued Function-like total quasi_total (F₁(),F₃()) (F₁(),F₃()) (F₁(),F₃(),F₅())
F₂() is Element of the carrier of F₁()
[:NAT,F₃():] is non empty non trivial Relation-like non finite disjoint_with_NAT set
bool [:NAT,F₃():] is non empty non trivial non finite set
A is non empty Relation-like NAT -defined F₃() -valued Function-like total quasi_total Element of bool [:NAT,F₃():]
A . 0 is Element of F₃()
F₇((A . 0)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . T is Element of F₃()
F₇((A . T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . T is Element of F₃()
F₇((A . T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
In ((A . T),F₃()) is Element of F₃()
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (T + 1) is Element of F₃()
F₆() . ((A . T),F₂()) is Element of F₃()
[(A . T),F₂()] is non empty V22() set
{(A . T),F₂()} is non empty finite set
{(A . T)} is non empty trivial finite 1 -element set
{{(A . T),F₂()},{(A . T)}} is non empty finite V44() set
F₆() . [(A . T),F₂()] is set
F₇((A . (T + 1))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . f is Element of F₃()
F₇((A . f)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
A . f is Element of F₃()
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
P is Relation-like NAT -defined F₃() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₃()
len P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
Seg (f + 1) is non empty finite f + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
C is non empty Relation-like NAT -defined F₃() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₃()
C . 1 is set
len C is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C . (len C) is set
1 -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . (1 -' 1) is Element of F₃()
(f + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . ((f + 1) -' 1) is Element of F₃()
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . s is Element of F₃()
In ((A . s),F₃()) is Element of F₃()
F₆() . ((In ((A . s),F₃())),F₂()) is Element of F₃()
[(In ((A . s),F₃())),F₂()] is non empty V22() set
{(In ((A . s),F₃())),F₂()} is non empty finite set
{(In ((A . s),F₃()))} is non empty trivial finite 1 -element set
{{(In ((A . s),F₃())),F₂()},{(In ((A . s),F₃()))}} is non empty finite V44() set
F₆() . [(In ((A . s),F₃())),F₂()] is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C . r is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C . (r + 1) is set
F₆() . ((C . r),F₂()) is set
[(C . r),F₂()] is non empty V22() set
{(C . r),F₂()} is non empty finite set
{(C . r)} is non empty trivial finite 1 -element set
{{(C . r),F₂()},{(C . r)}} is non empty finite V44() set
F₆() . [(C . r),F₂()] is set
r -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . (r -' 1) is Element of F₃()
(r + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . ((r + 1) -' 1) is Element of F₃()
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . s is Element of F₃()
In ((A . s),F₃()) is Element of F₃()
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . r is Element of F₃()
In ((A . r),F₃()) is Element of F₃()
F₆() . ((In ((A . r),F₃())),F₂()) is Element of F₃()
[(In ((A . r),F₃())),F₂()] is non empty V22() set
{(In ((A . r),F₃())),F₂()} is non empty finite set
{(In ((A . r),F₃()))} is non empty trivial finite 1 -element set
{{(In ((A . r),F₃())),F₂()},{(In ((A . r),F₃()))}} is non empty finite V44() set
F₆() . [(In ((A . r),F₃())),F₂()] is set
F₁() is non empty partial quasi_total non-empty () () () () UAStr
the carrier of F₁() is non empty set
F₃() is non empty set
bool F₃() is non empty set
[:F₃(), the carrier of F₁():] is non empty Relation-like disjoint_with_NAT set
F₅() is Element of bool F₃()
F₄() is Element of F₃()
F₆() is non empty Relation-like [:F₃(), the carrier of F₁():] -defined F₃() -valued Function-like total quasi_total (F₁(),F₃()) (F₁(),F₃()) (F₁(),F₃(),F₅())
F₂() is Element of the carrier of F₁()
[:NAT,F₃():] is non empty non trivial Relation-like non finite disjoint_with_NAT set
bool [:NAT,F₃():] is non empty non trivial non finite set
A is non empty Relation-like NAT -defined F₃() -valued Function-like total quasi_total Element of bool [:NAT,F₃():]
A . 0 is Element of F₃()
F₇((A . 0)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . T is Element of F₃()
F₇((A . T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . T is Element of F₃()
F₇((A . T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
In ((A . T),F₃()) is Element of F₃()
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (T + 1) is Element of F₃()
F₆() . ((A . T),F₂()) is Element of F₃()
[(A . T),F₂()] is non empty V22() set
{(A . T),F₂()} is non empty finite set
{(A . T)} is non empty trivial finite 1 -element set
{{(A . T),F₂()},{(A . T)}} is non empty finite V44() set
F₆() . [(A . T),F₂()] is set
F₇((A . (T + 1))) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . f is Element of F₃()
F₇((A . f)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
f is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
A . f is Element of F₃()
f + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
P is Relation-like NAT -defined F₃() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₃()
len P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom P is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
Seg (f + 1) is non empty finite f + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
{ b₁ where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : ( 1 <= b₁ & b₁ <= f + 1 ) } is set
C is non empty Relation-like NAT -defined F₃() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₃()
C . 1 is set
len C is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C . (len C) is set
1 -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . (1 -' 1) is Element of F₃()
(f + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . ((f + 1) -' 1) is Element of F₃()
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . s is Element of F₃()
In ((A . s),F₃()) is Element of F₃()
F₆() . ((In ((A . s),F₃())),F₂()) is Element of F₃()
[(In ((A . s),F₃())),F₂()] is non empty V22() set
{(In ((A . s),F₃())),F₂()} is non empty finite set
{(In ((A . s),F₃()))} is non empty trivial finite 1 -element set
{{(In ((A . s),F₃())),F₂()},{(In ((A . s),F₃()))}} is non empty finite V44() set
F₆() . [(In ((A . s),F₃())),F₂()] is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C . r is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C . (r + 1) is set
F₆() . ((C . r),F₂()) is set
[(C . r),F₂()] is non empty V22() set
{(C . r),F₂()} is non empty finite set
{(C . r)} is non empty trivial finite 1 -element set
{{(C . r),F₂()},{(C . r)}} is non empty finite V44() set
F₆() . [(C . r),F₂()] is set
r -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . (r -' 1) is Element of F₃()
(r + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . ((r + 1) -' 1) is Element of F₃()
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
A . s is Element of F₃()
In ((A . s),F₃()) is Element of F₃()
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,T,S) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*T,S*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,S*> is set
(A,S,T) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool f
C is Element of f
I is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total (A,f) (A,f) (A,f,P)
I . (C,S) is Element of f
[C,S] is non empty V22() set
{C,S} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,S},{C}} is non empty finite V44() set
I . [C,S] is set
I . (C,(A,S,T)) is Element of f
[C,(A,S,T)] is non empty V22() set
{C,(A,S,T)} is non empty finite set
{{C,(A,S,T)},{C}} is non empty finite V44() set
I . [C,(A,S,T)] is set
s is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
dom s is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
(len s) - 0 is non empty V79() V80() ext-real positive non negative set
s . ((len s) - 0) is set
r is Element of f
I . (r,S) is Element of f
[r,S] is non empty V22() set
{r,S} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,S},{r}} is non empty finite V44() set
I . [r,S] is set
I . (r,(A,S,T)) is Element of f
[r,(A,S,T)] is non empty V22() set
{r,(A,S,T)} is non empty finite set
{{r,(A,S,T)},{r}} is non empty finite V44() set
I . [r,(A,S,T)] is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(len s) - r is V79() V80() ext-real set
s . ((len s) - r) is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len s) - (r + 1) is V79() V80() ext-real set
s . ((len s) - (r + 1)) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(i + 1) + r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (i + 1) is set
s . i is set
I . ((s . i),(A,T,S)) is set
[(s . i),(A,T,S)] is non empty V22() set
{(s . i),(A,T,S)} is non empty finite set
{(s . i)} is non empty trivial finite 1 -element set
{{(s . i),(A,T,S)},{(s . i)}} is non empty finite V44() set
I . [(s . i),(A,T,S)] is set
s is Element of f
I . (s,S) is Element of f
[s,S] is non empty V22() set
{s,S} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,S},{s}} is non empty finite V44() set
I . [s,S] is set
I . (s,(A,S,T)) is Element of f
[s,(A,S,T)] is non empty V22() set
{s,(A,S,T)} is non empty finite set
{{s,(A,S,T)},{s}} is non empty finite V44() set
I . [s,(A,S,T)] is set
I . ((I . (s,S)),T) is Element of f
[(I . (s,S)),T] is non empty V22() set
{(I . (s,S)),T} is non empty finite set
{(I . (s,S))} is non empty trivial finite 1 -element set
{{(I . (s,S)),T},{(I . (s,S))}} is non empty finite V44() set
I . [(I . (s,S)),T] is set
I . ((I . ((I . (s,S)),T)),S) is Element of f
[(I . ((I . (s,S)),T)),S] is non empty V22() set
{(I . ((I . (s,S)),T)),S} is non empty finite set
{(I . ((I . (s,S)),T))} is non empty trivial finite 1 -element set
{{(I . ((I . (s,S)),T)),S},{(I . ((I . (s,S)),T))}} is non empty finite V44() set
I . [(I . ((I . (s,S)),T)),S] is set
I . ((I . ((I . (s,S)),T)),(A,S,T)) is Element of f
[(I . ((I . (s,S)),T)),(A,S,T)] is non empty V22() set
{(I . ((I . (s,S)),T)),(A,S,T)} is non empty finite set
{{(I . ((I . (s,S)),T)),(A,S,T)},{(I . ((I . (s,S)),T))}} is non empty finite V44() set
I . [(I . ((I . (s,S)),T)),(A,S,T)] is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len s) - r is V79() V80() ext-real set
s . ((len s) - r) is set
i is Element of f
I . (i,S) is Element of f
[i,S] is non empty V22() set
{i,S} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,S},{i}} is non empty finite V44() set
I . [i,S] is set
I . (i,(A,S,T)) is Element of f
[i,(A,S,T)] is non empty V22() set
{i,(A,S,T)} is non empty finite set
{{i,(A,S,T)},{i}} is non empty finite V44() set
I . [i,(A,S,T)] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
Funcs (S,S) is non empty functional FUNCTION_DOMAIN of S,S
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
curry' f is non empty Relation-like the carrier of A -defined Funcs (S,S) -valued Function-like total quasi_total Element of bool [: the carrier of A,(Funcs (S,S)):]
[: the carrier of A,(Funcs (S,S)):] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of A,(Funcs (S,S)):] is non empty set
P is Element of the carrier of A
(curry' f) . P is Relation-like S -defined S -valued Function-like total quasi_total Element of Funcs (S,S)
C is Element of S
(((curry' f) . P),C) is set
{ ((iter (((curry' f) . P),b₁)) . C) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : C in dom (iter (((curry' f) . P),b₁)) } is set
s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (((curry' f) . P),r) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
[:S,S:] is non empty Relation-like disjoint_with_NAT set
bool [:S,S:] is non empty set
(iter (((curry' f) . P),r)) . C is Element of S
dom (iter (((curry' f) . P),r)) is non empty Element of bool S
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
iter (((curry' f) . P),r) is Relation-like Function-like set
(iter (((curry' f) . P),r)) . C is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
len i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom i is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of bool NAT
rng i is finite set
s is set
s is set
i . s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i . r is set
r -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (((curry' f) . P),(r -' 1)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
[:S,S:] is non empty Relation-like disjoint_with_NAT set
bool [:S,S:] is non empty set
(iter (((curry' f) . P),(r -' 1))) . C is Element of S
s is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
dom s is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
1 -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(r + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (((curry' f) . P),0) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
[:S,S:] is non empty Relation-like disjoint_with_NAT set
bool [:S,S:] is non empty set
id S is non empty Relation-like S -defined S -valued Function-like one-to-one total quasi_total Element of bool [:S,S:]
s . 1 is set
iter (((curry' f) . P),(1 -' 1)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
(iter (((curry' f) . P),(1 -' 1))) . C is Element of S
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(s + 1) -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
dom (curry' f) is non empty Element of bool the carrier of A
bool the carrier of A is non empty set
dom ((curry' f) . P) is Element of bool S
iter (((curry' f) . P),i) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
(iter (((curry' f) . P),i)) . C is Element of S
((curry' f) . P) . ((iter (((curry' f) . P),i)) . C) is Element of S
f . (((iter (((curry' f) . P),i)) . C),P) is Element of S
[((iter (((curry' f) . P),i)) . C),P] is non empty V22() set
{((iter (((curry' f) . P),i)) . C),P} is non empty finite set
{((iter (((curry' f) . P),i)) . C)} is non empty trivial finite 1 -element set
{{((iter (((curry' f) . P),i)) . C),P},{((iter (((curry' f) . P),i)) . C)}} is non empty finite V44() set
f . [((iter (((curry' f) . P),i)) . C),P] is set
iter (((curry' f) . P),s) is Relation-like Function-like set
((curry' f) . P) * (iter (((curry' f) . P),i)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
s . s is set
iter (((curry' f) . P),(s -' 1)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
(iter (((curry' f) . P),(s -' 1))) . C is Element of S
s . (s + 1) is set
(iter (((curry' f) . P),s)) . C is set
f . ((s . s),P) is set
[(s . s),P] is non empty V22() set
{(s . s),P} is non empty finite set
{(s . s)} is non empty trivial finite 1 -element set
{{(s . s),P},{(s . s)}} is non empty finite V44() set
f . [(s . s),P] is set
s is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
dom ((curry' f) . P) is Element of bool S
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (0 + 1) is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (r + 1) is set
(r + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . ((r + 1) + 1) is set
(r + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
dom s is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
rng s is non empty finite Element of bool S
dom f is non empty Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
[(s . (r + 1)),P] is non empty V22() set
{(s . (r + 1)),P} is non empty finite set
{(s . (r + 1))} is non empty trivial finite 1 -element set
{{(s . (r + 1)),P},{(s . (r + 1))}} is non empty finite V44() set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
iter (((curry' f) . P),i) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
[:S,S:] is non empty Relation-like disjoint_with_NAT set
bool [:S,S:] is non empty set
(iter (((curry' f) . P),i)) . C is Element of S
dom (iter (((curry' f) . P),i)) is non empty Element of bool S
s . ((r + 1) + 1) is set
f . ((s . (r + 1)),P) is set
f . [(s . (r + 1)),P] is set
((curry' f) . P) . (s . (r + 1)) is set
((curry' f) . P) * (iter (((curry' f) . P),i)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
(((curry' f) . P) * (iter (((curry' f) . P),i))) . C is Element of S
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
iter (((curry' f) . P),(i + 1)) is non empty Relation-like S -defined S -valued Function-like total quasi_total Element of bool [:S,S:]
(iter (((curry' f) . P),(i + 1))) . C is Element of S
dom (iter (((curry' f) . P),(i + 1))) is non empty Element of bool S
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (r + 1) is set
F₁() is non empty partial quasi_total non-empty () () () () UAStr
the carrier of F₁() is non empty set
F₄() is non empty set
bool F₄() is non empty set
[:F₄(), the carrier of F₁():] is non empty Relation-like disjoint_with_NAT set
F₆() is Element of bool F₄()
F₅() is Element of F₄()
F₃() is Element of the carrier of F₁()
F₂() is Element of the carrier of F₁()
(F₁(),F₃(),F₂()) is Element of the carrier of F₁()
the charact of F₁() is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of F₁() *), the carrier of F₁()) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of F₁() *), the carrier of F₁())
the carrier of F₁() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F₁()
PFuncs (( the carrier of F₁() *), the carrier of F₁()) is set
dom the charact of F₁() is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
rng the charact of F₁() is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of F₁() *), the carrier of F₁()))
bool (PFuncs (( the carrier of F₁() *), the carrier of F₁())) is non empty set
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*F₃(),F₂()*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*F₃()*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,F₃()] is non empty V22() set
{1,F₃()} is non empty finite set
{{1,F₃()},{1}} is non empty finite V44() set
{[1,F₃()]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*F₂()*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,F₂()] is non empty V22() set
{1,F₂()} is non empty finite set
{{1,F₂()},{1}} is non empty finite V44() set
{[1,F₂()]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*F₃()*> ^ <*F₂()*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*F₃(),F₂()*> is set
F₇() is non empty Relation-like [:F₄(), the carrier of F₁():] -defined F₄() -valued Function-like total quasi_total (F₁(),F₄()) (F₁(),F₄()) (F₁(),F₄(),F₆())
F₇() . (F₅(),F₂()) is Element of F₄()
[F₅(),F₂()] is non empty V22() set
{F₅(),F₂()} is non empty finite set
{F₅()} is non empty trivial finite 1 -element set
{{F₅(),F₂()},{F₅()}} is non empty finite V44() set
F₇() . [F₅(),F₂()] is set
(F₁(),F₂(),F₃()) is Element of the carrier of F₁()
In (4,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (4,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (4,(dom the charact of F₁()))) is non empty homogeneous set
<*F₂(),F₃()*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*F₂()*> ^ <*F₃()*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*F₂(),F₃()*> is set
F₇() . (F₅(),(F₁(),F₂(),F₃())) is Element of F₄()
[F₅(),(F₁(),F₂(),F₃())] is non empty V22() set
{F₅(),(F₁(),F₂(),F₃())} is non empty finite set
{{F₅(),(F₁(),F₂(),F₃())},{F₅()}} is non empty finite V44() set
F₇() . [F₅(),(F₁(),F₂(),F₃())] is set
A is non empty Relation-like NAT -defined F₄() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₄()
A . 1 is set
len A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (len A) is set
dom A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (0 + 1) is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (0 + 1) is set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (S + 1) is set
(S + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . ((S + 1) + 1) is set
(S + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . ((S + 1) + 1) is set
F₇() . ((A . (S + 1)),(F₁(),F₃(),F₂())) is set
[(A . (S + 1)),(F₁(),F₃(),F₂())] is non empty V22() set
{(A . (S + 1)),(F₁(),F₃(),F₂())} is non empty finite set
{(A . (S + 1))} is non empty trivial finite 1 -element set
{{(A . (S + 1)),(F₁(),F₃(),F₂())},{(A . (S + 1))}} is non empty finite V44() set
F₇() . [(A . (S + 1)),(F₁(),F₃(),F₂())] is set
T is Element of F₄()
F₇() . (T,F₃()) is Element of F₄()
[T,F₃()] is non empty V22() set
{T,F₃()} is non empty finite set
{T} is non empty trivial finite 1 -element set
{{T,F₃()},{T}} is non empty finite V44() set
F₇() . [T,F₃()] is set
f is Element of F₄()
P is Element of F₄()
F₇() . (P,F₂()) is Element of F₄()
[P,F₂()] is non empty V22() set
{P,F₂()} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,F₂()},{P}} is non empty finite V44() set
F₇() . [P,F₂()] is set
C is Element of F₄()
F₇() . (C,F₂()) is Element of F₄()
[C,F₂()] is non empty V22() set
{C,F₂()} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,F₂()},{C}} is non empty finite V44() set
F₇() . [C,F₂()] is set
C is Element of F₄()
F₇() . (C,F₂()) is Element of F₄()
[C,F₂()] is non empty V22() set
{C,F₂()} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,F₂()},{C}} is non empty finite V44() set
F₇() . [C,F₂()] is set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + S is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
S + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (S + 1) is set
T is Element of F₄()
F₇() . (T,F₂()) is Element of F₄()
[T,F₂()] is non empty V22() set
{T,F₂()} is non empty finite set
{T} is non empty trivial finite 1 -element set
{{T,F₂()},{T}} is non empty finite V44() set
F₇() . [T,F₂()] is set
F₁() is non empty partial quasi_total non-empty () () () () UAStr
the carrier of F₁() is non empty set
F₄() is non empty set
bool F₄() is non empty set
[:F₄(), the carrier of F₁():] is non empty Relation-like disjoint_with_NAT set
F₆() is Element of bool F₄()
F₇() is non empty Relation-like [:F₄(), the carrier of F₁():] -defined F₄() -valued Function-like total quasi_total (F₁(),F₄()) (F₁(),F₄()) (F₁(),F₄(),F₆())
F₅() is Element of F₄()
F₂() is Element of the carrier of F₁()
F₃() is Element of the carrier of F₁()
(F₁(),F₂(),F₃()) is Element of the carrier of F₁()
the charact of F₁() is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of F₁() *), the carrier of F₁()) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of F₁() *), the carrier of F₁())
the carrier of F₁() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F₁()
PFuncs (( the carrier of F₁() *), the carrier of F₁()) is set
dom the charact of F₁() is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (4,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
rng the charact of F₁() is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of F₁() *), the carrier of F₁()))
bool (PFuncs (( the carrier of F₁() *), the carrier of F₁())) is non empty set
the charact of F₁() . (In (4,(dom the charact of F₁()))) is non empty homogeneous set
<*F₂(),F₃()*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*F₂()*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,F₂()] is non empty V22() set
{1,F₂()} is non empty finite set
{{1,F₂()},{1}} is non empty finite V44() set
{[1,F₂()]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*F₃()*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,F₃()] is non empty V22() set
{1,F₃()} is non empty finite set
{{1,F₃()},{1}} is non empty finite V44() set
{[1,F₃()]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*F₂()*> ^ <*F₃()*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*F₂(),F₃()*> is set
F₇() . (F₅(),(F₁(),F₂(),F₃())) is Element of F₄()
[F₅(),(F₁(),F₂(),F₃())] is non empty V22() set
{F₅(),(F₁(),F₂(),F₃())} is non empty finite set
{F₅()} is non empty trivial finite 1 -element set
{{F₅(),(F₁(),F₂(),F₃())},{F₅()}} is non empty finite V44() set
F₇() . [F₅(),(F₁(),F₂(),F₃())] is set
(F₁(),F₃(),F₂()) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*F₃(),F₂()*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*F₃()*> ^ <*F₂()*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*F₃(),F₂()*> is set
F₇() . (F₅(),F₂()) is Element of F₄()
[F₅(),F₂()] is non empty V22() set
{F₅(),F₂()} is non empty finite set
{{F₅(),F₂()},{F₅()}} is non empty finite V44() set
F₇() . [F₅(),F₂()] is set
A is non empty Relation-like NAT -defined F₄() -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of F₄()
A . 1 is set
len A is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (len A) is set
dom A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (0 + 1) is set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
S + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (S + 1) is set
(S + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . ((S + 1) + 1) is set
(S + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . ((S + 1) + 1) is set
F₇() . ((A . (S + 1)),(F₁(),F₃(),F₂())) is set
[(A . (S + 1)),(F₁(),F₃(),F₂())] is non empty V22() set
{(A . (S + 1)),(F₁(),F₃(),F₂())} is non empty finite set
{(A . (S + 1))} is non empty trivial finite 1 -element set
{{(A . (S + 1)),(F₁(),F₃(),F₂())},{(A . (S + 1))}} is non empty finite V44() set
F₇() . [(A . (S + 1)),(F₁(),F₃(),F₂())] is set
T is Element of F₄()
F₇() . (T,F₃()) is Element of F₄()
[T,F₃()] is non empty V22() set
{T,F₃()} is non empty finite set
{T} is non empty trivial finite 1 -element set
{{T,F₃()},{T}} is non empty finite V44() set
F₇() . [T,F₃()] is set
f is Element of F₄()
F₇() . (f,F₂()) is Element of F₄()
[f,F₂()] is non empty V22() set
{f,F₂()} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F₂()},{f}} is non empty finite V44() set
F₇() . [f,F₂()] is set
(len A) - 0 is non empty V79() V80() ext-real positive non negative set
A . ((len A) - 0) is set
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(len A) - S is V79() V80() ext-real set
A . ((len A) - S) is set
S + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len A) - (S + 1) is V79() V80() ext-real set
A . ((len A) - (S + 1)) is set
T is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len A) - 1 is V79() V80() ext-real set
T + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
((len A) - 1) + 1 is V79() V80() ext-real set
1 + T is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . T is set
A . (T + 1) is set
F₇() . ((A . T),(F₁(),F₃(),F₂())) is set
[(A . T),(F₁(),F₃(),F₂())] is non empty V22() set
{(A . T),(F₁(),F₃(),F₂())} is non empty finite set
{(A . T)} is non empty trivial finite 1 -element set
{{(A . T),(F₁(),F₃(),F₂())},{(A . T)}} is non empty finite V44() set
F₇() . [(A . T),(F₁(),F₃(),F₂())] is set
f is Element of F₄()
F₇() . (f,F₃()) is Element of F₄()
[f,F₃()] is non empty V22() set
{f,F₃()} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F₃()},{f}} is non empty finite V44() set
F₇() . [f,F₃()] is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + I is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
A . (I + 1) is set
s is Element of F₄()
F₇() . (s,F₂()) is Element of F₄()
[s,F₂()] is non empty V22() set
{s,F₂()} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₂()},{s}} is non empty finite V44() set
F₇() . [s,F₂()] is set
P is Element of F₄()
C is Element of F₄()
F₇() . (C,F₂()) is Element of F₄()
[C,F₂()] is non empty V22() set
{C,F₂()} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,F₂()},{C}} is non empty finite V44() set
F₇() . [C,F₂()] is set
F₇() . ((F₇() . (s,F₂())),F₃()) is Element of F₄()
[(F₇() . (s,F₂())),F₃()] is non empty V22() set
{(F₇() . (s,F₂())),F₃()} is non empty finite set
{(F₇() . (s,F₂()))} is non empty trivial finite 1 -element set
{{(F₇() . (s,F₂())),F₃()},{(F₇() . (s,F₂()))}} is non empty finite V44() set
F₇() . [(F₇() . (s,F₂())),F₃()] is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
S is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + S is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(len A) - S is V79() V80() ext-real set
A is non empty partial quasi_total non-empty free () () () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
(A,T,f) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,f*> is set
(A,(A),0) is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,(A),P) is Element of bool the carrier of A
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
1 + C is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
C + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,(A),C) is Element of bool the carrier of A
dom (Den ((In (2,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . I is non empty homogeneous set
dom (Den (I,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (I,A)) . s is set
r is set
i is set
<*r,i*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
s is Element of the carrier of A
s is Element of the carrier of A
(A,s,s) is Element of the carrier of A
<*s,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*s,s*> is set
rng <*T,f*> is non empty finite Element of bool the carrier of A
{T,f} is non empty finite set
A is non empty partial quasi_total non-empty free () () () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
P is Element of the carrier of A
(A,T,f,P) is Element of the carrier of A
<*T,f,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*T*> ^ <*f*>) ^ <*P*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*T,f,P*> is set
(A,(A),0) is Element of bool the carrier of A
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,(A),C) is Element of bool the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,(A),I) is Element of bool the carrier of A
dom (Den ((In (3,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
3 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (s,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . s is non empty homogeneous set
dom (Den (s,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
r is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (s,A)) . r is set
i is set
s is set
s is set
<*i,s,s*> is non empty Relation-like NAT -defined Function-like finite 3 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*s*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
r is Element of the carrier of A
i is Element of the carrier of A
ri is Element of the carrier of A
(A,r,i,ri) is Element of the carrier of A
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,ri*> is set
rng <*T,f,P*> is non empty finite Element of bool the carrier of A
{T,f,P} is non empty finite set
A is non empty partial quasi_total non-empty free () () () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
(A,T,f) is Element of the carrier of A
<*T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> ^ <*f*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*T,f*> is set
(A,(A),0) is Element of bool the carrier of A
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(A,(A),P) is Element of bool the carrier of A
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(A,(A),C) is Element of bool the carrier of A
dom (Den ((In (4,(dom the charact of A))),A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
bool ( the carrier of A *) is non empty set
2 -tuples_on the carrier of A is non empty functional FinSequence-membered with_common_domain product-like FinSequenceSet of the carrier of A
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den (I,A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . I is non empty homogeneous set
dom (Den (I,A)) is functional FinSequence-membered Element of bool ( the carrier of A *)
s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
(Den (I,A)) . s is set
r is set
i is set
<*r,i*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
s is Element of the carrier of A
s is Element of the carrier of A
(A,s,s) is Element of the carrier of A
<*s,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*s,s*> is set
rng <*T,f*> is non empty finite Element of bool the carrier of A
{T,f} is non empty finite set
F₂() is non empty set
F₁() is non empty partial quasi_total non-empty free () () () () () () () UAStr
the carrier of F₁() is non empty set
(F₁()) is Element of bool the carrier of F₁()
bool the carrier of F₁() is non empty set
(F₁()) is Element of the carrier of F₁()
the charact of F₁() is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of F₁() *), the carrier of F₁()) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of F₁() *), the carrier of F₁())
the carrier of F₁() * is non empty functional FinSequence-membered FinSequenceSet of the carrier of F₁()
PFuncs (( the carrier of F₁() *), the carrier of F₁()) is set
dom the charact of F₁() is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (1,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
rng the charact of F₁() is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of F₁() *), the carrier of F₁()))
bool (PFuncs (( the carrier of F₁() *), the carrier of F₁())) is non empty set
the charact of F₁() . (In (1,(dom the charact of F₁()))) is non empty homogeneous set
(Den ((In (1,(dom the charact of F₁()))),F₁())) . {} is set
{(F₁())} is non empty trivial finite 1 -element set
the carrier of F₁() \ {(F₁())} is Element of bool the carrier of F₁()
In (3,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (3,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (3,(dom the charact of F₁()))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of F₁()))),F₁())) is Element of bool the carrier of F₁()
( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁()))) is Element of bool the carrier of F₁()
In (4,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (4,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (4,(dom the charact of F₁()))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of F₁()))),F₁())) is Element of bool the carrier of F₁()
(( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁())))) \ (rng (Den ((In (4,(dom the charact of F₁()))),F₁()))) is Element of bool the carrier of F₁()
{ (F₁(),b₁,b₂) where b₁, b₂ is Element of the carrier of F₁() : ( not b₁ = (F₁(),b₁,b₂) & not b₂ = (F₁(),b₁,b₂) ) } is set
((( the carrier of F₁() \ {(F₁())}) \ (rng (Den ((In (3,(dom the charact of F₁()))),F₁())))) \ (rng (Den ((In (4,(dom the charact of F₁()))),F₁())))) \ { (F₁(),b₁,b₂) where b₁, b₂ is Element of the carrier of F₁() : ( not b₁ = (F₁(),b₁,b₂) & not b₂ = (F₁(),b₁,b₂) ) } is Element of bool the carrier of F₁()
[: the carrier of F₁(),F₂():] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of F₁(),F₂():] is non empty set
F₃() is Element of F₂()
(F₁(),(F₁()),0) is Element of bool the carrier of F₁()
{ [b₁,F₄(b₁)] where b₁ is Element of the carrier of F₁() : b₁ in (F₁()) } is set
{ [(F₁(),b₁,b₂),F₅(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),a₁) & b₂ in (F₁(),(F₁()),a₁) & [b₁,b₃] in a₂ & [b₂,b₄] in a₂ ) } is set
{ [(F₁(),b₁,b₂,b₃),F₇(b₄,b₅,b₆)] where b₁, b₂, b₃ is Element of the carrier of F₁(), b₄, b₅, b₆ is Element of F₂() : ( b₁ in (F₁(),(F₁()),a₁) & b₂ in (F₁(),(F₁()),a₁) & b₃ in (F₁(),(F₁()),a₁) & [b₁,b₄] in a₂ & [b₂,b₅] in a₂ & [b₃,b₆] in a₂ ) } is set
{ [(F₁(),b₁,b₂),F₆(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),a₁) & b₂ in (F₁(),(F₁()),a₁) & [b₁,b₃] in a₂ & [b₂,b₄] in a₂ ) } is set
[(F₁()),F₃()] is non empty V22() set
{(F₁()),F₃()} is non empty finite set
{{(F₁()),F₃()},{(F₁())}} is non empty finite V44() set
{[(F₁()),F₃()]} is non empty trivial Relation-like Function-like constant finite 1 -element set
T is Relation-like Function-like set
dom T is set
T . 0 is set
Union T is set
P is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
C is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T . P is set
T . C is set
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
P + I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P + s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
P + 0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T . (P + 0) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
i + r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(i + r) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
T . ((i + r) + 1) is set
T . (i + r) is set
(T . (i + r)) \/ {[(F₁()),F₃()]} is non empty set
(F₁(),(F₁()),(i + r)) is Element of bool the carrier of F₁()
{ [(F₁(),b₁,b₂),F₅(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(i + r)) & b₂ in (F₁(),(F₁()),(i + r)) & [b₁,b₃] in T . (i + r) & [b₂,b₄] in T . (i + r) ) } is set
((T . (i + r)) \/ {[(F₁()),F₃()]}) \/ H₁(i + r,T . (i + r)) is non empty set
{ [(F₁(),b₁,b₂,b₃),F₇(b₄,b₅,b₆)] where b₁, b₂, b₃ is Element of the carrier of F₁(), b₄, b₅, b₆ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(i + r)) & b₂ in (F₁(),(F₁()),(i + r)) & b₃ in (F₁(),(F₁()),(i + r)) & [b₁,b₄] in T . (i + r) & [b₂,b₅] in T . (i + r) & [b₃,b₆] in T . (i + r) ) } is set
(((T . (i + r)) \/ {[(F₁()),F₃()]}) \/ H₁(i + r,T . (i + r))) \/ H₂(i + r,T . (i + r)) is non empty set
{ [(F₁(),b₁,b₂),F₆(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(i + r)) & b₂ in (F₁(),(F₁()),(i + r)) & [b₁,b₃] in T . (i + r) & [b₂,b₄] in T . (i + r) ) } is set
((((T . (i + r)) \/ {[(F₁()),F₃()]}) \/ H₁(i + r,T . (i + r))) \/ H₂(i + r,T . (i + r))) \/ H₃(i + r,T . (i + r)) is non empty set
P + r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
T . (P + r) is set
(T . (P + r)) \/ {[(F₁()),F₃()]} is non empty set
(F₁(),(F₁()),(P + r)) is Element of bool the carrier of F₁()
{ [(F₁(),b₁,b₂),F₅(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(P + r)) & b₂ in (F₁(),(F₁()),(P + r)) & [b₁,b₃] in T . (P + r) & [b₂,b₄] in T . (P + r) ) } is set
((T . (P + r)) \/ {[(F₁()),F₃()]}) \/ H₁(P + r,T . (P + r)) is non empty set
{ [(F₁(),b₁,b₂,b₃),F₇(b₄,b₅,b₆)] where b₁, b₂, b₃ is Element of the carrier of F₁(), b₄, b₅, b₆ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(P + r)) & b₂ in (F₁(),(F₁()),(P + r)) & b₃ in (F₁(),(F₁()),(P + r)) & [b₁,b₄] in T . (P + r) & [b₂,b₅] in T . (P + r) & [b₃,b₆] in T . (P + r) ) } is set
{ [(F₁(),b₁,b₂),F₆(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),(P + r)) & b₂ in (F₁(),(F₁()),(P + r)) & [b₁,b₃] in T . (P + r) & [b₂,b₄] in T . (P + r) ) } is set
H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r)) is set
(((T . (P + r)) \/ {[(F₁()),F₃()]}) \/ H₁(P + r,T . (P + r))) \/ (H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r))) is non empty set
H₁(P + r,T . (P + r)) \/ (H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r))) is set
((T . (P + r)) \/ {[(F₁()),F₃()]}) \/ (H₁(P + r,T . (P + r)) \/ (H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r)))) is non empty set
{[(F₁()),F₃()]} \/ (H₁(P + r,T . (P + r)) \/ (H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r)))) is non empty set
(T . (P + r)) \/ ({[(F₁()),F₃()]} \/ (H₁(P + r,T . (P + r)) \/ (H₂(P + r,T . (P + r)) \/ H₃(P + r,T . (P + r))))) is non empty set
(P + r) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
T . ((P + r) + 1) is set
P + r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
T . (P + r) is set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
P + (r + 1) is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative set
T . (P + (r + 1)) is set
{ [b₁,F₄(b₁)] where b₁ is Element of the carrier of F₁() : S₂[b₁] } is set
P is Relation-like Function-like set
dom P is set
C is set
P . C is set
[C,(P . C)] is non empty V22() set
{C,(P . C)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(P . C)},{C}} is non empty finite V44() set
I is Element of the carrier of F₁()
F₄(I) is set
[I,F₄(I)] is non empty V22() set
{I,F₄(I)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,F₄(I)},{I}} is non empty finite V44() set
C is set
I is Element of the carrier of F₁()
F₄(I) is set
[I,F₄(I)] is non empty V22() set
{I,F₄(I)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,F₄(I)},{I}} is non empty finite V44() set
rng P is set
C is set
I is set
[I,C] is non empty V22() set
{I,C} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,C},{I}} is non empty finite V44() set
s is Element of the carrier of F₁()
F₄(s) is set
[s,F₄(s)] is non empty V22() set
{s,F₄(s)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₄(s)},{s}} is non empty finite V44() set
[:(F₁(),(F₁()),0),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),0),F₂():] is non empty set
C is Relation-like (F₁(),(F₁()),0) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),0),F₂():]
C . (F₁()) is set
I is Element of the carrier of F₁()
s is Element of the carrier of F₁()
(F₁(),I,s) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*I,s*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*I,s*> is set
I is Element of the carrier of F₁()
s is Element of the carrier of F₁()
(F₁(),I,s) is Element of the carrier of F₁()
<*I,s*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*I,s*> is set
C . (F₁(),I,s) is set
C . I is set
C . s is set
F₅((C . I),(C . s)) is Element of F₂()
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
s is Element of the carrier of F₁()
(F₁(),r,i,s) is Element of the carrier of F₁()
<*r,i,s*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,s*> is set
C . (F₁(),r,i,s) is set
C . r is set
C . i is set
C . s is set
F₇((C . r),(C . i),(C . s)) is Element of F₂()
s is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),s,r) is Element of the carrier of F₁()
<*s,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*s,r*> is set
C . (F₁(),s,r) is set
C . s is set
C . r is set
F₆((C . s),(C . r)) is Element of F₂()
I is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),I) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),I),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),I),F₂():] is non empty set
T . I is set
I + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),(I + 1)) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),(I + 1)),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),(I + 1)),F₂():] is non empty set
T . (I + 1) is set
s is Relation-like (F₁(),(F₁()),I) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),I),F₂():]
s . (F₁()) is set
s \/ {[(F₁()),F₃()]} is non empty Relation-like set
{ [(F₁(),b₁,b₂),F₅(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),I) & b₂ in (F₁(),(F₁()),I) & [b₁,b₃] in s & [b₂,b₄] in s ) } is set
(s \/ {[(F₁()),F₃()]}) \/ H₁(I,s) is non empty set
{ [(F₁(),b₁,b₂,b₃),F₇(b₄,b₅,b₆)] where b₁, b₂, b₃ is Element of the carrier of F₁(), b₄, b₅, b₆ is Element of F₂() : ( b₁ in (F₁(),(F₁()),I) & b₂ in (F₁(),(F₁()),I) & b₃ in (F₁(),(F₁()),I) & [b₁,b₄] in s & [b₂,b₅] in s & [b₃,b₆] in s ) } is set
((s \/ {[(F₁()),F₃()]}) \/ H₁(I,s)) \/ H₂(I,s) is non empty set
{ [(F₁(),b₁,b₂),F₆(b₃,b₄)] where b₁, b₂ is Element of the carrier of F₁(), b₃, b₄ is Element of F₂() : ( b₁ in (F₁(),(F₁()),I) & b₂ in (F₁(),(F₁()),I) & [b₁,b₃] in s & [b₂,b₄] in s ) } is set
(((s \/ {[(F₁()),F₃()]}) \/ H₁(I,s)) \/ H₂(I,s)) \/ H₃(I,s) is non empty set
dom s is Element of bool (F₁(),(F₁()),I)
bool (F₁(),(F₁()),I) is non empty set
i is set
i is set
s is set
s is set
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
ri is Element of F₂()
r is Element of F₂()
F₅(ri,r) is Element of F₂()
[(F₁(),r,i),F₅(ri,r)] is non empty V22() set
{(F₁(),r,i),F₅(ri,r)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₅(ri,r)},{(F₁(),r,i)}} is non empty finite V44() set
[r,ri] is non empty V22() set
{r,ri} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,ri},{r}} is non empty finite V44() set
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
i is Element of the carrier of F₁()
i is Element of the carrier of F₁()
i0 is Element of the carrier of F₁()
(F₁(),i,i,i0) is Element of the carrier of F₁()
<*i,i,i0*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i0*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i0] is non empty V22() set
{1,i0} is non empty finite set
{{1,i0},{1}} is non empty finite V44() set
{[1,i0]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*i*>) ^ <*i0*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*i,i,i0*> is set
gI2 is Element of F₂()
J is Element of F₂()
a is Element of F₂()
F₇(gI2,J,a) is Element of F₂()
[(F₁(),i,i,i0),F₇(gI2,J,a)] is non empty V22() set
{(F₁(),i,i,i0),F₇(gI2,J,a)} is non empty finite set
{(F₁(),i,i,i0)} is non empty trivial finite 1 -element set
{{(F₁(),i,i,i0),F₇(gI2,J,a)},{(F₁(),i,i,i0)}} is non empty finite V44() set
[i,gI2] is non empty V22() set
{i,gI2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,gI2},{i}} is non empty finite V44() set
[i,J] is non empty V22() set
{i,J} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,J},{i}} is non empty finite V44() set
[i0,a] is non empty V22() set
{i0,a} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,a},{i0}} is non empty finite V44() set
b is Element of the carrier of F₁()
b is Element of the carrier of F₁()
(F₁(),b,b) is Element of the carrier of F₁()
<*b,b*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*b,b*> is set
c is Element of F₂()
b is Element of F₂()
F₆(c,b) is Element of F₂()
[(F₁(),b,b),F₆(c,b)] is non empty V22() set
{(F₁(),b,b),F₆(c,b)} is non empty finite set
{(F₁(),b,b)} is non empty trivial finite 1 -element set
{{(F₁(),b,b),F₆(c,b)},{(F₁(),b,b)}} is non empty finite V44() set
[b,c] is non empty V22() set
{b,c} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,c},{b}} is non empty finite V44() set
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
i is Relation-like set
s is set
s is set
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{{s,r},{s}} is non empty finite V44() set
s . s is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₅(i,i0) is Element of F₂()
[(F₁(),r,i),F₅(i,i0)] is non empty V22() set
{(F₁(),r,i),F₅(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₅(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
s . r is set
s . i is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i,i) is Element of the carrier of F₁()
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,i*> is set
i0 is Element of F₂()
gI2 is Element of F₂()
J is Element of F₂()
F₇(i0,gI2,J) is Element of F₂()
[(F₁(),r,i,i),F₇(i0,gI2,J)] is non empty V22() set
{(F₁(),r,i,i),F₇(i0,gI2,J)} is non empty finite set
{(F₁(),r,i,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i,i),F₇(i0,gI2,J)},{(F₁(),r,i,i)}} is non empty finite V44() set
[r,i0] is non empty V22() set
{r,i0} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i0},{r}} is non empty finite V44() set
[i,gI2] is non empty V22() set
{i,gI2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,gI2},{i}} is non empty finite V44() set
[i,J] is non empty V22() set
{i,J} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,J},{i}} is non empty finite V44() set
s . r is set
s . i is set
s . i is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₆(i,i0) is Element of F₂()
[(F₁(),r,i),F₆(i,i0)] is non empty V22() set
{(F₁(),r,i),F₆(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₆(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
s . r is set
s . i is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₅(i,i0) is Element of F₂()
[(F₁(),r,i),F₅(i,i0)] is non empty V22() set
{(F₁(),r,i),F₅(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₅(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i,i) is Element of the carrier of F₁()
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,i*> is set
i0 is Element of F₂()
gI2 is Element of F₂()
J is Element of F₂()
F₇(i0,gI2,J) is Element of F₂()
[(F₁(),r,i,i),F₇(i0,gI2,J)] is non empty V22() set
{(F₁(),r,i,i),F₇(i0,gI2,J)} is non empty finite set
{(F₁(),r,i,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i,i),F₇(i0,gI2,J)},{(F₁(),r,i,i)}} is non empty finite V44() set
[r,i0] is non empty V22() set
{r,i0} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i0},{r}} is non empty finite V44() set
[i,gI2] is non empty V22() set
{i,gI2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,gI2},{i}} is non empty finite V44() set
[i,J] is non empty V22() set
{i,J} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,J},{i}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₆(i,i0) is Element of F₂()
[(F₁(),r,i),F₆(i,i0)] is non empty V22() set
{(F₁(),r,i),F₆(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₆(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₅(i,i0) is Element of F₂()
[(F₁(),r,i),F₅(i,i0)] is non empty V22() set
{(F₁(),r,i),F₅(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₅(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
s . r is set
s . i is set
s . s is set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
(F₁(),gI2,J) is Element of the carrier of F₁()
<*gI2,J*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*gI2,J*> is set
a is Element of F₂()
b is Element of F₂()
F₅(a,b) is Element of F₂()
[(F₁(),gI2,J),F₅(a,b)] is non empty V22() set
{(F₁(),gI2,J),F₅(a,b)} is non empty finite set
{(F₁(),gI2,J)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J),F₅(a,b)},{(F₁(),gI2,J)}} is non empty finite V44() set
[gI2,a] is non empty V22() set
{gI2,a} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,a},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
s . gI2 is set
s . J is set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
a is Element of the carrier of F₁()
(F₁(),gI2,J,a) is Element of the carrier of F₁()
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*gI2,J,a*> is set
b is Element of F₂()
b is Element of F₂()
c is Element of F₂()
F₇(b,b,c) is Element of F₂()
[(F₁(),gI2,J,a),F₇(b,b,c)] is non empty V22() set
{(F₁(),gI2,J,a),F₇(b,b,c)} is non empty finite set
{(F₁(),gI2,J,a)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J,a),F₇(b,b,c)},{(F₁(),gI2,J,a)}} is non empty finite V44() set
[gI2,b] is non empty V22() set
{gI2,b} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,b},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
[a,c] is non empty V22() set
{a,c} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,c},{a}} is non empty finite V44() set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
(F₁(),gI2,J) is Element of the carrier of F₁()
<*gI2,J*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*gI2,J*> is set
a is Element of F₂()
b is Element of F₂()
F₆(a,b) is Element of F₂()
[(F₁(),gI2,J),F₆(a,b)] is non empty V22() set
{(F₁(),gI2,J),F₆(a,b)} is non empty finite set
{(F₁(),gI2,J)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J),F₆(a,b)},{(F₁(),gI2,J)}} is non empty finite V44() set
[gI2,a] is non empty V22() set
{gI2,a} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,a},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i,i) is Element of the carrier of F₁()
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,i*> is set
i0 is Element of F₂()
gI2 is Element of F₂()
J is Element of F₂()
F₇(i0,gI2,J) is Element of F₂()
[(F₁(),r,i,i),F₇(i0,gI2,J)] is non empty V22() set
{(F₁(),r,i,i),F₇(i0,gI2,J)} is non empty finite set
{(F₁(),r,i,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i,i),F₇(i0,gI2,J)},{(F₁(),r,i,i)}} is non empty finite V44() set
[r,i0] is non empty V22() set
{r,i0} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i0},{r}} is non empty finite V44() set
[i,gI2] is non empty V22() set
{i,gI2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,gI2},{i}} is non empty finite V44() set
[i,J] is non empty V22() set
{i,J} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,J},{i}} is non empty finite V44() set
s . i is set
s . i is set
s . r is set
s . s is set
a is Element of the carrier of F₁()
b is Element of the carrier of F₁()
(F₁(),a,b) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*a,b*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*a*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*a,b*> is set
b is Element of F₂()
c is Element of F₂()
F₅(b,c) is Element of F₂()
[(F₁(),a,b),F₅(b,c)] is non empty V22() set
{(F₁(),a,b),F₅(b,c)} is non empty finite set
{(F₁(),a,b)} is non empty trivial finite 1 -element set
{{(F₁(),a,b),F₅(b,c)},{(F₁(),a,b)}} is non empty finite V44() set
[a,b] is non empty V22() set
{a,b} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,b},{a}} is non empty finite V44() set
[b,c] is non empty V22() set
{b,c} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,c},{b}} is non empty finite V44() set
a is Element of the carrier of F₁()
b is Element of the carrier of F₁()
b is Element of the carrier of F₁()
(F₁(),a,b,b) is Element of the carrier of F₁()
<*a,b,b*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*a*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*a*> ^ <*b*>) ^ <*b*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*a,b,b*> is set
c is Element of F₂()
b is Element of F₂()
c is Element of F₂()
F₇(c,b,c) is Element of F₂()
[(F₁(),a,b,b),F₇(c,b,c)] is non empty V22() set
{(F₁(),a,b,b),F₇(c,b,c)} is non empty finite set
{(F₁(),a,b,b)} is non empty trivial finite 1 -element set
{{(F₁(),a,b,b),F₇(c,b,c)},{(F₁(),a,b,b)}} is non empty finite V44() set
[a,c] is non empty V22() set
{a,c} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,c},{a}} is non empty finite V44() set
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
[b,c] is non empty V22() set
{b,c} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,c},{b}} is non empty finite V44() set
s . a is set
s . b is set
s . b is set
a is Element of the carrier of F₁()
b is Element of the carrier of F₁()
(F₁(),a,b) is Element of the carrier of F₁()
<*a,b*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*a*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*a,b*> is set
b is Element of F₂()
c is Element of F₂()
F₆(b,c) is Element of F₂()
[(F₁(),a,b),F₆(b,c)] is non empty V22() set
{(F₁(),a,b),F₆(b,c)} is non empty finite set
{(F₁(),a,b)} is non empty trivial finite 1 -element set
{{(F₁(),a,b),F₆(b,c)},{(F₁(),a,b)}} is non empty finite V44() set
[a,b] is non empty V22() set
{a,b} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,b},{a}} is non empty finite V44() set
[b,c] is non empty V22() set
{b,c} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,c},{b}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
F₆(i,i0) is Element of F₂()
[(F₁(),r,i),F₆(i,i0)] is non empty V22() set
{(F₁(),r,i),F₆(i,i0)} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₆(i,i0)},{(F₁(),r,i)}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
s . r is set
s . i is set
s . s is set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
(F₁(),gI2,J) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*gI2,J*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*gI2,J*> is set
a is Element of F₂()
b is Element of F₂()
F₅(a,b) is Element of F₂()
[(F₁(),gI2,J),F₅(a,b)] is non empty V22() set
{(F₁(),gI2,J),F₅(a,b)} is non empty finite set
{(F₁(),gI2,J)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J),F₅(a,b)},{(F₁(),gI2,J)}} is non empty finite V44() set
[gI2,a] is non empty V22() set
{gI2,a} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,a},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
a is Element of the carrier of F₁()
(F₁(),gI2,J,a) is Element of the carrier of F₁()
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*gI2,J,a*> is set
b is Element of F₂()
b is Element of F₂()
c is Element of F₂()
F₇(b,b,c) is Element of F₂()
[(F₁(),gI2,J,a),F₇(b,b,c)] is non empty V22() set
{(F₁(),gI2,J,a),F₇(b,b,c)} is non empty finite set
{(F₁(),gI2,J,a)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J,a),F₇(b,b,c)},{(F₁(),gI2,J,a)}} is non empty finite V44() set
[gI2,b] is non empty V22() set
{gI2,b} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,b},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
[a,c] is non empty V22() set
{a,c} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,c},{a}} is non empty finite V44() set
gI2 is Element of the carrier of F₁()
J is Element of the carrier of F₁()
(F₁(),gI2,J) is Element of the carrier of F₁()
<*gI2,J*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*gI2,J*> is set
a is Element of F₂()
b is Element of F₂()
F₆(a,b) is Element of F₂()
[(F₁(),gI2,J),F₆(a,b)] is non empty V22() set
{(F₁(),gI2,J),F₆(a,b)} is non empty finite set
{(F₁(),gI2,J)} is non empty trivial finite 1 -element set
{{(F₁(),gI2,J),F₆(a,b)},{(F₁(),gI2,J)}} is non empty finite V44() set
[gI2,a] is non empty V22() set
{gI2,a} is non empty finite set
{gI2} is non empty trivial finite 1 -element set
{{gI2,a},{gI2}} is non empty finite V44() set
[J,b] is non empty V22() set
{J,b} is non empty finite set
{J} is non empty trivial finite 1 -element set
{{J,b},{J}} is non empty finite V44() set
s . gI2 is set
s . J is set
s is Relation-like Function-like set
dom s is set
s is set
s . s is set
[s,(s . s)] is non empty V22() set
{s,(s . s)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(s . s)},{s}} is non empty finite V44() set
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
(F₁(),i,ri) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*i,ri*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*i,ri*> is set
r is Element of F₂()
i is Element of F₂()
F₅(r,i) is Element of F₂()
[(F₁(),i,ri),F₅(r,i)] is non empty V22() set
{(F₁(),i,ri),F₅(r,i)} is non empty finite set
{(F₁(),i,ri)} is non empty trivial finite 1 -element set
{{(F₁(),i,ri),F₅(r,i)},{(F₁(),i,ri)}} is non empty finite V44() set
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),i,ri,r) is Element of the carrier of F₁()
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*i,ri,r*> is set
i is Element of F₂()
i is Element of F₂()
i0 is Element of F₂()
F₇(i,i,i0) is Element of F₂()
[(F₁(),i,ri,r),F₇(i,i,i0)] is non empty V22() set
{(F₁(),i,ri,r),F₇(i,i,i0)} is non empty finite set
{(F₁(),i,ri,r)} is non empty trivial finite 1 -element set
{{(F₁(),i,ri,r),F₇(i,i,i0)},{(F₁(),i,ri,r)}} is non empty finite V44() set
[i,i] is non empty V22() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
[r,i0] is non empty V22() set
{r,i0} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i0},{r}} is non empty finite V44() set
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
(F₁(),i,ri) is Element of the carrier of F₁()
<*i,ri*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*i,ri*> is set
r is Element of F₂()
i is Element of F₂()
F₆(r,i) is Element of F₂()
[(F₁(),i,ri),F₆(r,i)] is non empty V22() set
{(F₁(),i,ri),F₆(r,i)} is non empty finite set
{(F₁(),i,ri)} is non empty trivial finite 1 -element set
{{(F₁(),i,ri),F₆(r,i)},{(F₁(),i,ri)}} is non empty finite V44() set
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
s is set
s . s is set
[s,(s . s)] is non empty V22() set
{s,(s . s)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(s . s)},{s}} is non empty finite V44() set
[s,F₃()] is non empty V22() set
{s,F₃()} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₃()},{s}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
s . r is set
s . i is set
ri is Element of F₂()
[r,ri] is non empty V22() set
{r,ri} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,ri},{r}} is non empty finite V44() set
r is Element of F₂()
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
F₅(ri,r) is Element of F₂()
[s,F₅(ri,r)] is non empty V22() set
{s,F₅(ri,r)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₅(ri,r)},{s}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
(F₁(),r,i,ri) is Element of the carrier of F₁()
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,ri*> is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
(F₁(),r,i,ri) is Element of the carrier of F₁()
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,ri*> is set
s . i is set
s . ri is set
s . r is set
r is Element of F₂()
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
i is Element of F₂()
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
i is Element of F₂()
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
F₇(i,r,i) is Element of F₂()
[s,F₇(i,r,i)] is non empty V22() set
{s,F₇(i,r,i)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₇(i,r,i)},{s}} is non empty finite V44() set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
s . r is set
s . i is set
ri is Element of F₂()
[r,ri] is non empty V22() set
{r,ri} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,ri},{r}} is non empty finite V44() set
r is Element of F₂()
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
F₆(ri,r) is Element of F₂()
[s,F₆(ri,r)] is non empty V22() set
{s,F₆(ri,r)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,F₆(ri,r)},{s}} is non empty finite V44() set
rng s is set
s is set
r is set
[r,s] is non empty V22() set
{r,s} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,s},{r}} is non empty finite V44() set
ri is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),ri,r) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*ri,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*ri,r*> is set
i is Element of F₂()
i is Element of F₂()
F₅(i,i) is Element of F₂()
[(F₁(),ri,r),F₅(i,i)] is non empty V22() set
{(F₁(),ri,r),F₅(i,i)} is non empty finite set
{(F₁(),ri,r)} is non empty trivial finite 1 -element set
{{(F₁(),ri,r),F₅(i,i)},{(F₁(),ri,r)}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
ri is Element of the carrier of F₁()
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),ri,r,i) is Element of the carrier of F₁()
<*ri,r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*ri*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*ri,r,i*> is set
i is Element of F₂()
i0 is Element of F₂()
gI2 is Element of F₂()
F₇(i,i0,gI2) is Element of F₂()
[(F₁(),ri,r,i),F₇(i,i0,gI2)] is non empty V22() set
{(F₁(),ri,r,i),F₇(i,i0,gI2)} is non empty finite set
{(F₁(),ri,r,i)} is non empty trivial finite 1 -element set
{{(F₁(),ri,r,i),F₇(i,i0,gI2)},{(F₁(),ri,r,i)}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
[r,i0] is non empty V22() set
{r,i0} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i0},{r}} is non empty finite V44() set
[i,gI2] is non empty V22() set
{i,gI2} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,gI2},{i}} is non empty finite V44() set
ri is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),ri,r) is Element of the carrier of F₁()
<*ri,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*ri,r*> is set
i is Element of F₂()
i is Element of F₂()
F₆(i,i) is Element of F₂()
[(F₁(),ri,r),F₆(i,i)] is non empty V22() set
{(F₁(),ri,r),F₆(i,i)} is non empty finite set
{(F₁(),ri,r)} is non empty trivial finite 1 -element set
{{(F₁(),ri,r),F₆(i,i)},{(F₁(),ri,r)}} is non empty finite V44() set
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s is Relation-like (F₁(),(F₁()),(I + 1)) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),(I + 1)),F₂():]
H₂(I,s) \/ H₃(I,s) is set
((s \/ {[(F₁()),F₃()]}) \/ H₁(I,s)) \/ (H₂(I,s) \/ H₃(I,s)) is non empty set
H₁(I,s) \/ (H₂(I,s) \/ H₃(I,s)) is set
(s \/ {[(F₁()),F₃()]}) \/ (H₁(I,s) \/ (H₂(I,s) \/ H₃(I,s))) is non empty set
{[(F₁()),F₃()]} \/ (H₁(I,s) \/ (H₂(I,s) \/ H₃(I,s))) is non empty set
s \/ ({[(F₁()),F₃()]} \/ (H₁(I,s) \/ (H₂(I,s) \/ H₃(I,s)))) is non empty set
s . (F₁()) is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
s . r is set
s . i is set
F₅((s . r),(s . i)) is Element of F₂()
s . r is set
s . i is set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
i is Element of F₂()
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
i is Element of F₂()
[i,i] is non empty V22() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
[(F₁(),r,i),F₅((s . r),(s . i))] is non empty V22() set
{(F₁(),r,i),F₅((s . r),(s . i))} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₅((s . r),(s . i))},{(F₁(),r,i)}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
s . (F₁(),r,i) is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
ri is Element of the carrier of F₁()
(F₁(),r,i,ri) is Element of the carrier of F₁()
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*r,i,ri*> is set
s . r is set
s . i is set
s . ri is set
F₇((s . r),(s . i),(s . ri)) is Element of F₂()
s . r is set
s . i is set
s . ri is set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
J + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),J) is Element of bool the carrier of F₁()
i is Element of F₂()
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
J + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),J) is Element of bool the carrier of F₁()
i0 is Element of F₂()
[i,i0] is non empty V22() set
{i,i0} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i0},{i}} is non empty finite V44() set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
J + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),J) is Element of bool the carrier of F₁()
gI2 is Element of F₂()
[ri,gI2] is non empty V22() set
{ri,gI2} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,gI2},{ri}} is non empty finite V44() set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
J + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),J) is Element of bool the carrier of F₁()
[(F₁(),r,i,ri),F₇((s . r),(s . i),(s . ri))] is non empty V22() set
{(F₁(),r,i,ri),F₇((s . r),(s . i),(s . ri))} is non empty finite set
{(F₁(),r,i,ri)} is non empty trivial finite 1 -element set
{{(F₁(),r,i,ri),F₇((s . r),(s . i),(s . ri))},{(F₁(),r,i,ri)}} is non empty finite V44() set
J is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
J + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),J) is Element of bool the carrier of F₁()
s . (F₁(),r,i,ri) is set
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),r,i) is Element of the carrier of F₁()
<*r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*r,i*> is set
s . (F₁(),r,i) is set
s . r is set
s . i is set
F₆((s . r),(s . i)) is Element of F₂()
s . r is set
s . i is set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
i is Element of F₂()
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
i is Element of F₂()
[i,i] is non empty V22() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
[(F₁(),r,i),F₆((s . r),(s . i))] is non empty V22() set
{(F₁(),r,i),F₆((s . r),(s . i))} is non empty finite set
{(F₁(),r,i)} is non empty trivial finite 1 -element set
{{(F₁(),r,i),F₆((s . r),(s . i))},{(F₁(),r,i)}} is non empty finite V44() set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
I is set
s is set
T . s is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),r),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),r),F₂():] is non empty set
T . r is set
i is Relation-like (F₁(),(F₁()),r) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),r),F₂():]
i . (F₁()) is set
I is set
s is set
[I,s] is non empty V22() set
{I,s} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,s},{I}} is non empty finite V44() set
r is set
[I,r] is non empty V22() set
{I,r} is non empty finite set
{{I,r},{I}} is non empty finite V44() set
i is set
T . i is set
s is set
T . s is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
T . s is set
T . r is set
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),r),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),r),F₂():] is non empty set
(F₁(),(F₁()),s) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),s),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),s),F₂():] is non empty set
i is Relation-like (F₁(),(F₁()),r) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),r),F₂():]
i . (F₁()) is set
ri is Relation-like (F₁(),(F₁()),s) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),s),F₂():]
ri . (F₁()) is set
I is Relation-like Function-like set
dom I is set
s is set
I . s is set
[s,(I . s)] is non empty V22() set
{s,(I . s)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(I . s)},{s}} is non empty finite V44() set
r is set
T . r is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),i),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),i),F₂():] is non empty set
T . i is set
s is Relation-like (F₁(),(F₁()),i) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),i),F₂():]
s . (F₁()) is set
s is set
r is Element of the carrier of F₁()
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),s) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),s),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),s),F₂():] is non empty set
T . s is set
r is Relation-like (F₁(),(F₁()),s) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),s),F₂():]
r . (F₁()) is set
s is Relation-like (F₁(),(F₁()),s) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),s),F₂():]
dom s is Element of bool (F₁(),(F₁()),s)
bool (F₁(),(F₁()),s) is non empty set
s . r is set
[r,(s . r)] is non empty V22() set
{r,(s . r)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(s . r)},{r}} is non empty finite V44() set
rng I is set
s is set
r is set
I . r is set
i is Element of the carrier of F₁()
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),s) is Element of bool the carrier of F₁()
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),s) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),s),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),s),F₂():] is non empty set
T . s is set
i is Relation-like (F₁(),(F₁()),s) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),s),F₂():]
i . (F₁()) is set
r is Relation-like (F₁(),(F₁()),s) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),s),F₂():]
dom r is Element of bool (F₁(),(F₁()),s)
bool (F₁(),(F₁()),s) is non empty set
r . i is set
[i,(r . i)] is non empty V22() set
{i,(r . i)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(r . i)},{i}} is non empty finite V44() set
rng r is Element of bool F₂()
bool F₂() is non empty set
s is non empty Relation-like the carrier of F₁() -defined F₂() -valued Function-like total quasi_total Element of bool [: the carrier of F₁(),F₂():]
s . (F₁()) is Element of F₂()
r is Element of the carrier of F₁()
F₄(r) is set
[r,F₄(r)] is non empty V22() set
{r,F₄(r)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,F₄(r)},{r}} is non empty finite V44() set
s . r is Element of F₂()
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),r),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),r),F₂():] is non empty set
i is Element of the carrier of F₁()
s is Relation-like (F₁(),(F₁()),r) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),r),F₂():]
T . r is set
dom s is Element of bool (F₁(),(F₁()),r)
bool (F₁(),(F₁()),r) is non empty set
s . i is set
[i,(s . i)] is non empty V22() set
{i,(s . i)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(s . i)},{i}} is non empty finite V44() set
s . i is Element of F₂()
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),i),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),i),F₂():] is non empty set
T . i is set
s is Relation-like (F₁(),(F₁()),i) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),i),F₂():]
s . (F₁()) is set
dom s is Element of bool (F₁(),(F₁()),i)
bool (F₁(),(F₁()),i) is non empty set
s is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),s,r) is Element of the carrier of F₁()
In (2,(dom the charact of F₁())) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of F₁()
Den ((In (2,(dom the charact of F₁()))),F₁()) is Relation-like the carrier of F₁() * -defined the carrier of F₁() -valued Function-like Element of rng the charact of F₁()
the charact of F₁() . (In (2,(dom the charact of F₁()))) is non empty homogeneous set
<*s,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of F₁()))),F₁())) . <*s,r*> is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),ri) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),ri),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),ri),F₂():] is non empty set
T . ri is set
r is Relation-like (F₁(),(F₁()),ri) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),ri),F₂():]
r . (F₁()) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
dom r is Element of bool (F₁(),(F₁()),ri)
bool (F₁(),(F₁()),ri) is non empty set
r . s is set
s . s is Element of F₂()
r . r is set
s . r is Element of F₂()
r . (F₁(),s,r) is set
[(F₁(),s,r),(r . (F₁(),s,r))] is non empty V22() set
{(F₁(),s,r),(r . (F₁(),s,r))} is non empty finite set
{(F₁(),s,r)} is non empty trivial finite 1 -element set
{{(F₁(),s,r),(r . (F₁(),s,r))},{(F₁(),s,r)}} is non empty finite V44() set
F₅((s . s),(s . r)) is Element of F₂()
[(F₁(),s,r),F₅((s . s),(s . r))] is non empty V22() set
{(F₁(),s,r),F₅((s . s),(s . r))} is non empty finite set
{{(F₁(),s,r),F₅((s . s),(s . r))},{(F₁(),s,r)}} is non empty finite V44() set
s . (F₁(),s,r) is Element of F₂()
s is Element of the carrier of F₁()
r is Element of the carrier of F₁()
i is Element of the carrier of F₁()
(F₁(),s,r,i) is Element of the carrier of F₁()
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of F₁()))),F₁())) . <*s,r,i*> is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),i),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),i),F₂():] is non empty set
T . i is set
i is Relation-like (F₁(),(F₁()),i) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),i),F₂():]
i . (F₁()) is set
i0 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i0) is Element of bool the carrier of F₁()
dom i is Element of bool (F₁(),(F₁()),i)
bool (F₁(),(F₁()),i) is non empty set
i . s is set
s . s is Element of F₂()
i . r is set
s . r is Element of F₂()
i . i is set
s . i is Element of F₂()
i . (F₁(),s,r,i) is set
[(F₁(),s,r,i),(i . (F₁(),s,r,i))] is non empty V22() set
{(F₁(),s,r,i),(i . (F₁(),s,r,i))} is non empty finite set
{(F₁(),s,r,i)} is non empty trivial finite 1 -element set
{{(F₁(),s,r,i),(i . (F₁(),s,r,i))},{(F₁(),s,r,i)}} is non empty finite V44() set
F₇((s . s),(s . r),(s . i)) is Element of F₂()
[(F₁(),s,r,i),F₇((s . s),(s . r),(s . i))] is non empty V22() set
{(F₁(),s,r,i),F₇((s . s),(s . r),(s . i))} is non empty finite set
{{(F₁(),s,r,i),F₇((s . s),(s . r),(s . i))},{(F₁(),s,r,i)}} is non empty finite V44() set
s . (F₁(),s,r,i) is Element of F₂()
s is Element of the carrier of F₁()
r is Element of the carrier of F₁()
(F₁(),s,r) is Element of the carrier of F₁()
<*s,r*> is non empty Relation-like NAT -defined the carrier of F₁() -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of F₁()
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of F₁()))),F₁())) . <*s,r*> is set
s . (F₁(),s,r) is Element of F₂()
s . s is Element of F₂()
s . r is Element of F₂()
F₆((s . s),(s . r)) is Element of F₂()
ri is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
(F₁(),(F₁()),ri) is Element of bool the carrier of F₁()
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(F₁(),(F₁()),r) is Element of bool the carrier of F₁()
[:(F₁(),(F₁()),r),F₂():] is Relation-like disjoint_with_NAT set
bool [:(F₁(),(F₁()),r),F₂():] is non empty set
T . r is set
i is Relation-like (F₁(),(F₁()),r) -defined F₂() -valued Function-like total quasi_total Element of bool [:(F₁(),(F₁()),r),F₂():]
i . (F₁()) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
(F₁(),(F₁()),i) is Element of bool the carrier of F₁()
dom i is Element of bool (F₁(),(F₁()),r)
bool (F₁(),(F₁()),r) is non empty set
i . s is set
i . r is set
i . (F₁(),s,r) is set
[(F₁(),s,r),(i . (F₁(),s,r))] is non empty V22() set
{(F₁(),s,r),(i . (F₁(),s,r))} is non empty finite set
{(F₁(),s,r)} is non empty trivial finite 1 -element set
{{(F₁(),s,r),(i . (F₁(),s,r))},{(F₁(),s,r)}} is non empty finite V44() set
[(F₁(),s,r),F₆((s . s),(s . r))] is non empty V22() set
{(F₁(),s,r),F₆((s . s),(s . r))} is non empty finite set
{{(F₁(),s,r),F₆((s . s),(s . r))},{(F₁(),s,r)}} is non empty finite V44() set
A is non empty set
bool A is non empty set
S is Element of bool A
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
(T) is Element of bool the carrier of T
the carrier of T is non empty set
bool the carrier of T is non empty set
(T) is Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
[:[:A,(T):],A:] is Relation-like disjoint_with_NAT set
bool [:[:A,(T):],A:] is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
f is Relation-like [:A,(T):] -defined A -valued Function-like total quasi_total Element of bool [:[:A,(T):],A:]
P is Element of A
Funcs (A,A) is non empty functional FUNCTION_DOMAIN of A,A
A --> P is non empty Relation-like A -defined A -valued Function-like total quasi_total Element of bool [:A,A:]
[:A,A:] is non empty Relation-like disjoint_with_NAT set
bool [:A,A:] is non empty set
curry' f is Relation-like Function-like set
s is Element of the carrier of T
(curry' f) . s is set
r is non empty set
[:A,r:] is non empty Relation-like disjoint_with_NAT set
[:[:A,r:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A,r:],A:] is non empty set
s is non empty Relation-like [:A,r:] -defined A -valued Function-like total quasi_total Element of bool [:[:A,r:],A:]
curry' s is non empty Relation-like r -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [:r,(Funcs (A,A)):]
[:r,(Funcs (A,A)):] is non empty Relation-like disjoint_with_NAT set
bool [:r,(Funcs (A,A)):] is non empty set
i is Element of r
(curry' s) . i is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
id A is non empty Relation-like A -defined A -valued Function-like one-to-one total quasi_total Element of bool [:A,A:]
C is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
[: the carrier of T,(Funcs (A,A)):] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of T,(Funcs (A,A)):] is non empty set
s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
r is non empty Relation-like the carrier of T -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [: the carrier of T,(Funcs (A,A)):]
r . (T) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
Funcs ( the carrier of T,(Funcs (A,A))) is non empty functional FUNCTION_DOMAIN of the carrier of T, Funcs (A,A)
uncurry' r is Relation-like Function-like set
Funcs ([:A, the carrier of T:],A) is non empty functional FUNCTION_DOMAIN of [:A, the carrier of T:],A
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
dom r is non empty Element of bool the carrier of T
dom s is Element of bool A
i is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total Element of bool [:[:A, the carrier of T:],A:]
s is Element of A
i . (s,(T)) is Element of A
[s,(T)] is non empty V22() set
{s,(T)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(T)},{s}} is non empty finite V44() set
i . [s,(T)] is set
s . s is Element of A
s is Element of A
s is Element of the carrier of T
r is Element of the carrier of T
(T,s,r) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*s,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*s,r*> is set
i . (s,(T,s,r)) is Element of A
[s,(T,s,r)] is non empty V22() set
{s,(T,s,r)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(T,s,r)},{s}} is non empty finite V44() set
i . [s,(T,s,r)] is set
i . (s,s) is Element of A
[s,s] is non empty V22() set
{s,s} is non empty finite set
{{s,s},{s}} is non empty finite V44() set
i . [s,s] is set
i . ((i . (s,s)),r) is Element of A
[(i . (s,s)),r] is non empty V22() set
{(i . (s,s)),r} is non empty finite set
{(i . (s,s))} is non empty trivial finite 1 -element set
{{(i . (s,s)),r},{(i . (s,s))}} is non empty finite V44() set
i . [(i . (s,s)),r] is set
r . (T,s,r) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . (T,s,r)) is Element of bool A
r . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . s) is Element of bool A
r . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . r) is Element of bool A
(r . (T,s,r)) . s is Element of A
(A,A,A,(r . s),(r . r)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(r . s),(r . r)) . s is Element of A
(r . s) . s is Element of A
(r . r) . ((r . s) . s) is Element of A
(r . r) . (i . (s,s)) is Element of A
s is Element of A
s is Element of the carrier of T
i . (s,s) is Element of A
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
i . [s,s] is set
r is Element of the carrier of T
i is Element of the carrier of T
(T,s,r,i) is Element of the carrier of T
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of T))),T)) . <*s,r,i*> is set
i . (s,(T,s,r,i)) is Element of A
[s,(T,s,r,i)] is non empty V22() set
{s,(T,s,r,i)} is non empty finite set
{{s,(T,s,r,i)},{s}} is non empty finite V44() set
i . [s,(T,s,r,i)] is set
i . ((i . (s,s)),r) is Element of A
[(i . (s,s)),r] is non empty V22() set
{(i . (s,s)),r} is non empty finite set
{(i . (s,s))} is non empty trivial finite 1 -element set
{{(i . (s,s)),r},{(i . (s,s))}} is non empty finite V44() set
i . [(i . (s,s)),r] is set
i . ((i . (s,s)),i) is Element of A
[(i . (s,s)),i] is non empty V22() set
{(i . (s,s)),i} is non empty finite set
{{(i . (s,s)),i},{(i . (s,s))}} is non empty finite V44() set
i . [(i . (s,s)),i] is set
r . (T,s,r,i) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . (T,s,r,i)) is Element of bool A
r . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . s) is Element of bool A
r . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . r) is Element of bool A
r . i is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . i) is Element of bool A
(r . (T,s,r,i)) . s is Element of A
(A,A,(r . r),(r . i),S) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(r . r) | S is Relation-like S -defined A -defined A -valued Function-like set
(r . i) +* ((r . r) | S) is Relation-like Function-like set
(A,A,A,(r . s),(A,A,(r . r),(r . i),S)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₃(r . s,r . r,r . i) . s is Element of A
(r . s) . s is Element of A
(A,A,(r . r),(r . i),S) . ((r . s) . s) is Element of A
(r . r) . (i . (s,s)) is Element of A
(r . i) . (i . (s,s)) is Element of A
s is Element of A
s is Element of the carrier of T
i . (s,s) is Element of A
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
i . [s,s] is set
r is Element of the carrier of T
(T,s,r) is Element of the carrier of T
<*s,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*s,r*> is set
i . (s,(T,s,r)) is Element of A
[s,(T,s,r)] is non empty V22() set
{s,(T,s,r)} is non empty finite set
{{s,(T,s,r)},{s}} is non empty finite V44() set
i . [s,(T,s,r)] is set
i . ((i . (s,s)),r) is Element of A
[(i . (s,s)),r] is non empty V22() set
{(i . (s,s)),r} is non empty finite set
{(i . (s,s))} is non empty trivial finite 1 -element set
{{(i . (s,s)),r},{(i . (s,s))}} is non empty finite V44() set
i . [(i . (s,s)),r] is set
i . ((i . ((i . (s,s)),r)),(T,s,r)) is Element of A
[(i . ((i . (s,s)),r)),(T,s,r)] is non empty V22() set
{(i . ((i . (s,s)),r)),(T,s,r)} is non empty finite set
{(i . ((i . (s,s)),r))} is non empty trivial finite 1 -element set
{{(i . ((i . (s,s)),r)),(T,s,r)},{(i . ((i . (s,s)),r))}} is non empty finite V44() set
i . [(i . ((i . (s,s)),r)),(T,s,r)] is set
r . (T,s,r) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . (T,s,r)) is Element of bool A
r . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . s) is Element of bool A
r . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . r) is Element of bool A
(A,A,A,(r . r),(r . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (A,A,A,(r . r),(r . s)) is Element of bool A
(r . (T,s,r)) . s is Element of A
(A,(A,A,A,(r . r),(r . s)),S,C) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(r . s),(A,(A,A,A,(r . r),(r . s)),S,C)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₄(r . s,r . r) . s is Element of A
(r . s) . s is Element of A
(A,(A,A,A,(r . r),(r . s)),S,C) . ((r . s) . s) is Element of A
rng (A,A,A,(r . r),(r . s)) is Element of bool A
(A,A,A,(A,A,A,(r . r),(r . s)),C) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
i is Element of A
(A,A,A,(A,A,A,(r . r),(r . s)),C) . i is Element of A
(A,A,A,(r . r),(r . s)) . i is Element of A
C . ((A,A,A,(r . r),(r . s)) . i) is Element of A
C . i is Element of A
(A,A,A,(r . r),(r . s)) . (i . (s,s)) is Element of A
(A,(A,A,A,(r . r),(r . s)),S,C) . ((A,A,A,(r . r),(r . s)) . (i . (s,s))) is Element of A
(r . r) . (i . (s,s)) is Element of A
(r . s) . ((r . r) . (i . (s,s))) is Element of A
(A,(A,A,A,(r . r),(r . s)),S,C) . ((r . s) . ((r . r) . (i . (s,s)))) is Element of A
(A,A,A,(r . s),(A,(A,A,A,(r . r),(r . s)),S,C)) . ((r . r) . (i . (s,s))) is Element of A
(A,A,A,(r . s),(A,(A,A,A,(r . r),(r . s)),S,C)) . (i . ((i . (s,s)),r)) is Element of A
(r . (T,s,r)) . (i . ((i . (s,s)),r)) is Element of A
s is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
s | [:A,(T):] is Relation-like [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
dom s is non empty Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
bool [:A, the carrier of T:] is non empty set
dom f is Relation-like A -defined (T) -valued Element of bool [:A,(T):]
bool [:A,(T):] is non empty set
(dom s) /\ [:A,(T):] is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
s is set
r is set
i is set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
r is Element of the carrier of T
(curry' f) . r is set
Funcs ([:A,(T):],A) is non empty functional FUNCTION_DOMAIN of [:A,(T):],A
Funcs ((T),(Funcs (A,A))) is non empty functional FUNCTION_DOMAIN of (T), Funcs (A,A)
dom (curry' f) is set
i is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom i is Element of bool A
r . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . r) is Element of bool A
f . s is set
ri is Element of A
f . (ri,r) is set
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
f . [ri,r] is set
i . ri is Element of A
(r . r) . ri is Element of A
s . (ri,r) is Element of A
s . [ri,r] is set
s . s is set
s is Element of A
i is Element of the carrier of T
r is Element of the carrier of T
(T,i,r) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*i,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*i,r*> is set
s . (s,r) is Element of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
s . [s,r] is set
(T,r,i) is Element of the carrier of T
<*r,i*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*r,i*> is set
s . (s,(T,r,i)) is Element of A
[s,(T,r,i)] is non empty V22() set
{s,(T,r,i)} is non empty finite set
{{s,(T,r,i)},{s}} is non empty finite V44() set
s . [s,(T,r,i)] is set
r . (T,r,i) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . (T,r,i)) is Element of bool A
r . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (r . r) is Element of bool A
r . i is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(r . i),(r . r)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (A,A,A,(r . i),(r . r)) is Element of bool A
(r . (T,r,i)) . s is Element of A
(A,(A,A,A,(r . i),(r . r)),S,C) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(r . r),(A,(A,A,A,(r . i),(r . r)),S,C)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₄(r . r,r . i) . s is Element of A
(r . r) . s is Element of A
(A,(A,A,A,(r . i),(r . r)),S,C) . ((r . r) . s) is Element of A
rng (A,A,A,(r . i),(r . r)) is Element of bool A
rng r is non empty functional Element of bool (Funcs (A,A))
bool (Funcs (A,A)) is non empty set
curry' s is non empty Relation-like the carrier of T -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [: the carrier of T,(Funcs (A,A)):]
(curry' s) . (T,i,r) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
((A,A,A,(r . i),(r . r)),(s . (s,r))) is set
{ ((iter ((A,A,A,(r . i),(r . r)),b₁)) . (s . (s,r))) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : s . (s,r) in dom (iter ((A,A,A,(r . i),(r . r)),b₁)) } is set
C . (s . (s,r)) is Element of A
A is non empty set
bool A is non empty set
Funcs (A,A) is non empty functional FUNCTION_DOMAIN of A,A
[:(Funcs (A,A)),(Funcs (A,A)):] is non empty Relation-like disjoint_with_NAT set
bool [:(Funcs (A,A)),(Funcs (A,A)):] is non empty set
S is Element of bool A
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
(T) is Element of bool the carrier of T
the carrier of T is non empty set
bool the carrier of T is non empty set
(T) is Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
[:[:A,(T):],A:] is Relation-like disjoint_with_NAT set
bool [:[:A,(T):],A:] is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
f is Relation-like [:A,(T):] -defined A -valued Function-like total quasi_total Element of bool [:[:A,(T):],A:]
P is non empty Relation-like Funcs (A,A) -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [:(Funcs (A,A)),(Funcs (A,A)):]
curry' f is Relation-like Function-like set
I is Element of the carrier of T
(curry' f) . I is set
s is non empty set
[:A,s:] is non empty Relation-like disjoint_with_NAT set
[:[:A,s:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A,s:],A:] is non empty set
i is non empty Relation-like [:A,s:] -defined A -valued Function-like total quasi_total Element of bool [:[:A,s:],A:]
curry' i is non empty Relation-like s -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [:s,(Funcs (A,A)):]
[:s,(Funcs (A,A)):] is non empty Relation-like disjoint_with_NAT set
bool [:s,(Funcs (A,A)):] is non empty set
r is Element of s
(curry' i) . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
id A is non empty Relation-like A -defined A -valued Function-like one-to-one total quasi_total Element of bool [:A,A:]
[:A,A:] is non empty Relation-like disjoint_with_NAT set
bool [:A,A:] is non empty set
[: the carrier of T,(Funcs (A,A)):] is non empty Relation-like disjoint_with_NAT set
bool [: the carrier of T,(Funcs (A,A)):] is non empty set
I is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
s is non empty Relation-like the carrier of T -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [: the carrier of T,(Funcs (A,A)):]
s . (T) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
Funcs ( the carrier of T,(Funcs (A,A))) is non empty functional FUNCTION_DOMAIN of the carrier of T, Funcs (A,A)
uncurry' s is Relation-like Function-like set
Funcs ([:A, the carrier of T:],A) is non empty functional FUNCTION_DOMAIN of [:A, the carrier of T:],A
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
dom s is non empty Element of bool the carrier of T
dom I is Element of bool A
r is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total Element of bool [:[:A, the carrier of T:],A:]
i is Element of A
r . (i,(T)) is Element of A
[i,(T)] is non empty V22() set
{i,(T)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(T)},{i}} is non empty finite V44() set
r . [i,(T)] is set
I . i is Element of A
i is Element of A
s is Element of the carrier of T
s is Element of the carrier of T
(T,s,s) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*s,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*s,s*> is set
r . (i,(T,s,s)) is Element of A
[i,(T,s,s)] is non empty V22() set
{i,(T,s,s)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(T,s,s)},{i}} is non empty finite V44() set
r . [i,(T,s,s)] is set
r . (i,s) is Element of A
[i,s] is non empty V22() set
{i,s} is non empty finite set
{{i,s},{i}} is non empty finite V44() set
r . [i,s] is set
r . ((r . (i,s)),s) is Element of A
[(r . (i,s)),s] is non empty V22() set
{(r . (i,s)),s} is non empty finite set
{(r . (i,s))} is non empty trivial finite 1 -element set
{{(r . (i,s)),s},{(r . (i,s))}} is non empty finite V44() set
r . [(r . (i,s)),s] is set
s . (T,s,s) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . (T,s,s)) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
(s . (T,s,s)) . i is Element of A
(A,A,A,(s . s),(s . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(s . s),(s . s)) . i is Element of A
(s . s) . i is Element of A
(s . s) . ((s . s) . i) is Element of A
(s . s) . (r . (i,s)) is Element of A
i is Element of A
s is Element of the carrier of T
r . (i,s) is Element of A
[i,s] is non empty V22() set
{i,s} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s},{i}} is non empty finite V44() set
r . [i,s] is set
s is Element of the carrier of T
r is Element of the carrier of T
(T,s,s,r) is Element of the carrier of T
<*s,s,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*s*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of T))),T)) . <*s,s,r*> is set
r . (i,(T,s,s,r)) is Element of A
[i,(T,s,s,r)] is non empty V22() set
{i,(T,s,s,r)} is non empty finite set
{{i,(T,s,s,r)},{i}} is non empty finite V44() set
r . [i,(T,s,s,r)] is set
r . ((r . (i,s)),s) is Element of A
[(r . (i,s)),s] is non empty V22() set
{(r . (i,s)),s} is non empty finite set
{(r . (i,s))} is non empty trivial finite 1 -element set
{{(r . (i,s)),s},{(r . (i,s))}} is non empty finite V44() set
r . [(r . (i,s)),s] is set
r . ((r . (i,s)),r) is Element of A
[(r . (i,s)),r] is non empty V22() set
{(r . (i,s)),r} is non empty finite set
{{(r . (i,s)),r},{(r . (i,s))}} is non empty finite V44() set
r . [(r . (i,s)),r] is set
s . (T,s,s,r) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . (T,s,s,r)) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
s . r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . r) is Element of bool A
(s . (T,s,s,r)) . i is Element of A
(A,A,(s . s),(s . r),S) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(s . s) | S is Relation-like S -defined A -defined A -valued Function-like set
(s . r) +* ((s . s) | S) is Relation-like Function-like set
(A,A,A,(s . s),(A,A,(s . s),(s . r),S)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₃(s . s,s . s,s . r) . i is Element of A
(s . s) . i is Element of A
(A,A,(s . s),(s . r),S) . ((s . s) . i) is Element of A
(s . s) . (r . (i,s)) is Element of A
(s . r) . (r . (i,s)) is Element of A
i is Element of A
s is Element of the carrier of T
r . (i,s) is Element of A
[i,s] is non empty V22() set
{i,s} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s},{i}} is non empty finite V44() set
r . [i,s] is set
s is Element of the carrier of T
(T,s,s) is Element of the carrier of T
<*s,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*s,s*> is set
r . (i,(T,s,s)) is Element of A
[i,(T,s,s)] is non empty V22() set
{i,(T,s,s)} is non empty finite set
{{i,(T,s,s)},{i}} is non empty finite V44() set
r . [i,(T,s,s)] is set
r . ((r . (i,s)),s) is Element of A
[(r . (i,s)),s] is non empty V22() set
{(r . (i,s)),s} is non empty finite set
{(r . (i,s))} is non empty trivial finite 1 -element set
{{(r . (i,s)),s},{(r . (i,s))}} is non empty finite V44() set
r . [(r . (i,s)),s] is set
r . ((r . ((r . (i,s)),s)),(T,s,s)) is Element of A
[(r . ((r . (i,s)),s)),(T,s,s)] is non empty V22() set
{(r . ((r . (i,s)),s)),(T,s,s)} is non empty finite set
{(r . ((r . (i,s)),s))} is non empty trivial finite 1 -element set
{{(r . ((r . (i,s)),s)),(T,s,s)},{(r . ((r . (i,s)),s))}} is non empty finite V44() set
r . [(r . ((r . (i,s)),s)),(T,s,s)] is set
s . (T,s,s) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . (T,s,s)) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
(A,A,A,(s . s),(s . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (A,A,A,(s . s),(s . s)) is Element of bool A
(s . (T,s,s)) . i is Element of A
P . (A,A,A,(s . s),(s . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(s . s),(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s))))) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₄(s . s,s . s) . i is Element of A
(s . s) . i is Element of A
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) . ((s . s) . i) is Element of A
rng (A,A,A,(s . s),(s . s)) is Element of bool A
(A,A,A,(A,A,A,(s . s),(s . s)),(P . (A,A,A,(s . s),(s . s)))) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(s . s),(s . s)) . (r . (i,s)) is Element of A
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) . ((A,A,A,(s . s),(s . s)) . (r . (i,s))) is Element of A
(s . s) . (r . (i,s)) is Element of A
(s . s) . ((s . s) . (r . (i,s))) is Element of A
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) . ((s . s) . ((s . s) . (r . (i,s)))) is Element of A
(A,A,A,(s . s),(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s))))) . ((s . s) . (r . (i,s))) is Element of A
(A,A,A,(s . s),(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s))))) . (r . ((r . (i,s)),s)) is Element of A
(s . (T,s,s)) . (r . ((r . (i,s)),s)) is Element of A
i is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
i | [:A,(T):] is Relation-like [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
curry' i is non empty Relation-like the carrier of T -defined Funcs (A,A) -valued Function-like total quasi_total Element of bool [: the carrier of T,(Funcs (A,A)):]
dom i is non empty Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
bool [:A, the carrier of T:] is non empty set
dom f is Relation-like A -defined (T) -valued Element of bool [:A,(T):]
bool [:A,(T):] is non empty set
(dom i) /\ [:A,(T):] is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
s is set
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
ri is Element of the carrier of T
(curry' f) . ri is set
Funcs ([:A,(T):],A) is non empty functional FUNCTION_DOMAIN of [:A,(T):],A
Funcs ((T),(Funcs (A,A))) is non empty functional FUNCTION_DOMAIN of (T), Funcs (A,A)
dom (curry' f) is set
r is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom r is Element of bool A
s . ri is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . ri) is Element of bool A
f . s is set
i is Element of A
f . (i,ri) is set
[i,ri] is non empty V22() set
{i,ri} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,ri},{i}} is non empty finite V44() set
f . [i,ri] is set
r . i is Element of A
(s . ri) . i is Element of A
i . (i,ri) is Element of A
i . [i,ri] is set
i . s is set
s is Element of the carrier of T
s is Element of the carrier of T
(T,s,s) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*s,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*s,s*> is set
(T,s,s) is Element of the carrier of T
<*s,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*s,s*> is set
(curry' i) . (T,s,s) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
P . ((curry' i) . (T,s,s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
r is Element of A
i . (r,s) is Element of A
[r,s] is non empty V22() set
{r,s} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,s},{r}} is non empty finite V44() set
i . [r,s] is set
i . (r,(T,s,s)) is Element of A
[r,(T,s,s)] is non empty V22() set
{r,(T,s,s)} is non empty finite set
{{r,(T,s,s)},{r}} is non empty finite V44() set
i . [r,(T,s,s)] is set
(P . ((curry' i) . (T,s,s))) . (i . (r,s)) is Element of A
s . (T,s,s) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . (T,s,s)) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (s . s) is Element of bool A
s . s is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(s . s),(s . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
dom (A,A,A,(s . s),(s . s)) is Element of bool A
(s . (T,s,s)) . r is Element of A
P . (A,A,A,(s . s),(s . s)) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
(A,A,A,(s . s),(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s))))) is Relation-like A -defined A -valued Function-like total quasi_total Element of Funcs (A,A)
H₄(s . s,s . s) . r is Element of A
(s . s) . r is Element of A
(A,(A,A,A,(s . s),(s . s)),S,(P . (A,A,A,(s . s),(s . s)))) . ((s . s) . r) is Element of A
rng (A,A,A,(s . s),(s . s)) is Element of bool A
rng s is non empty functional Element of bool (Funcs (A,A))
bool (Funcs (A,A)) is non empty set
((A,A,A,(s . s),(s . s)),(i . (r,s))) is set
{ ((iter ((A,A,A,(s . s),(s . s)),b₁)) . (i . (r,s))) where b₁ is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT : i . (r,s) in dom (iter ((A,A,A,(s . s),(s . s)),b₁)) } is set
A is non empty set
bool A is non empty set
S is Element of bool A
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
the carrier of T is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
(T) is Element of bool the carrier of T
bool the carrier of T is non empty set
(T) is Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
f is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
f | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
P is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
P | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
I is Element of the carrier of T
s is Element of A
f . (s,I) is set
[s,I] is non empty V22() set
{s,I} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,I},{s}} is non empty finite V44() set
f . [s,I] is set
P . (s,I) is set
P . [s,I] is set
f . (s,I) is Element of A
(f | [:A,(T):]) . [s,I] is set
P . (s,I) is Element of A
I is Element of A
f . (I,(T)) is set
[I,(T)] is non empty V22() set
{I,(T)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(T)},{I}} is non empty finite V44() set
f . [I,(T)] is set
P . (I,(T)) is set
P . [I,(T)] is set
f . (I,(T)) is Element of A
P . (I,(T)) is Element of A
I is Element of the carrier of T
s is Element of the carrier of T
(T,I,s) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*I,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*I,s*> is set
r is Element of A
f . (r,(T,I,s)) is set
[r,(T,I,s)] is non empty V22() set
{r,(T,I,s)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(T,I,s)},{r}} is non empty finite V44() set
f . [r,(T,I,s)] is set
P . (r,(T,I,s)) is set
P . [r,(T,I,s)] is set
f . (r,(T,I,s)) is Element of A
f . (r,I) is Element of A
[r,I] is non empty V22() set
{r,I} is non empty finite set
{{r,I},{r}} is non empty finite V44() set
f . [r,I] is set
f . ((f . (r,I)),s) is Element of A
[(f . (r,I)),s] is non empty V22() set
{(f . (r,I)),s} is non empty finite set
{(f . (r,I))} is non empty trivial finite 1 -element set
{{(f . (r,I)),s},{(f . (r,I))}} is non empty finite V44() set
f . [(f . (r,I)),s] is set
P . (r,I) is Element of A
P . [r,I] is set
f . ((P . (r,I)),s) is Element of A
[(P . (r,I)),s] is non empty V22() set
{(P . (r,I)),s} is non empty finite set
{(P . (r,I))} is non empty trivial finite 1 -element set
{{(P . (r,I)),s},{(P . (r,I))}} is non empty finite V44() set
f . [(P . (r,I)),s] is set
P . ((P . (r,I)),s) is Element of A
P . [(P . (r,I)),s] is set
P . (r,(T,I,s)) is Element of A
I is Element of the carrier of T
s is Element of the carrier of T
r is Element of the carrier of T
(T,I,s,r) is Element of the carrier of T
<*I,s,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*I*> ^ <*s*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of T))),T)) . <*I,s,r*> is set
i is Element of A
f . (i,(T,I,s,r)) is set
[i,(T,I,s,r)] is non empty V22() set
{i,(T,I,s,r)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(T,I,s,r)},{i}} is non empty finite V44() set
f . [i,(T,I,s,r)] is set
P . (i,(T,I,s,r)) is set
P . [i,(T,I,s,r)] is set
f . (i,I) is Element of A
[i,I] is non empty V22() set
{i,I} is non empty finite set
{{i,I},{i}} is non empty finite V44() set
f . [i,I] is set
P . (i,I) is Element of A
P . [i,I] is set
f . (i,(T,I,s,r)) is Element of A
f . ((f . (i,I)),s) is Element of A
[(f . (i,I)),s] is non empty V22() set
{(f . (i,I)),s} is non empty finite set
{(f . (i,I))} is non empty trivial finite 1 -element set
{{(f . (i,I)),s},{(f . (i,I))}} is non empty finite V44() set
f . [(f . (i,I)),s] is set
P . ((f . (i,I)),s) is Element of A
P . [(f . (i,I)),s] is set
P . (i,(T,I,s,r)) is Element of A
f . (i,(T,I,s,r)) is Element of A
f . ((f . (i,I)),r) is Element of A
[(f . (i,I)),r] is non empty V22() set
{(f . (i,I)),r} is non empty finite set
{(f . (i,I))} is non empty trivial finite 1 -element set
{{(f . (i,I)),r},{(f . (i,I))}} is non empty finite V44() set
f . [(f . (i,I)),r] is set
P . ((f . (i,I)),r) is Element of A
P . [(f . (i,I)),r] is set
P . (i,(T,I,s,r)) is Element of A
I is Element of the carrier of T
s is Element of the carrier of T
(T,I,s) is Element of the carrier of T
<*I,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*I,s*> is set
r is Element of A
f . (r,(T,I,s)) is set
[r,(T,I,s)] is non empty V22() set
{r,(T,I,s)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(T,I,s)},{r}} is non empty finite V44() set
f . [r,(T,I,s)] is set
P . (r,(T,I,s)) is set
P . [r,(T,I,s)] is set
(T,s,I) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*s,I*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*s*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*s,I*> is set
f . (r,I) is Element of A
[r,I] is non empty V22() set
{r,I} is non empty finite set
{{r,I},{r}} is non empty finite V44() set
f . [r,I] is set
i is non empty Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
i . 1 is set
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (len i) is set
P . (r,I) is Element of A
P . [r,I] is set
s is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i . s is set
s + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (s + 1) is set
s is Element of A
f . (s,(T,s,I)) is Element of A
[s,(T,s,I)] is non empty V22() set
{s,(T,s,I)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(T,s,I)},{s}} is non empty finite V44() set
f . [s,(T,s,I)] is set
f . (s,s) is Element of A
[s,s] is non empty V22() set
{s,s} is non empty finite set
{{s,s},{s}} is non empty finite V44() set
f . [s,s] is set
f . ((f . (s,s)),I) is Element of A
[(f . (s,s)),I] is non empty V22() set
{(f . (s,s)),I} is non empty finite set
{(f . (s,s))} is non empty trivial finite 1 -element set
{{(f . (s,s)),I},{(f . (s,s))}} is non empty finite V44() set
f . [(f . (s,s)),I] is set
P . (s,s) is Element of A
P . [s,s] is set
f . ((P . (s,s)),I) is Element of A
[(P . (s,s)),I] is non empty V22() set
{(P . (s,s)),I} is non empty finite set
{(P . (s,s))} is non empty trivial finite 1 -element set
{{(P . (s,s)),I},{(P . (s,s))}} is non empty finite V44() set
f . [(P . (s,s)),I] is set
P . ((P . (s,s)),I) is Element of A
P . [(P . (s,s)),I] is set
P . ((i . s),(T,s,I)) is set
[(i . s),(T,s,I)] is non empty V22() set
{(i . s),(T,s,I)} is non empty finite set
{(i . s)} is non empty trivial finite 1 -element set
{{(i . s),(T,s,I)},{(i . s)}} is non empty finite V44() set
P . [(i . s),(T,s,I)] is set
f . (r,(T,I,s)) is Element of A
P . (r,(T,I,s)) is Element of A
I is Element of [:A, the carrier of T:]
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
i is Element of the carrier of T
s is Element of A
f . (s,i) is Element of A
[s,i] is non empty V22() set
{s,i} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,i},{s}} is non empty finite V44() set
f . [s,i] is set
P . (s,i) is Element of A
P . [s,i] is set
f . I is Element of A
P . I is Element of A
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
[:S,(A):] is Relation-like disjoint_with_NAT set
[:S,{(A)}:] is non empty Relation-like disjoint_with_NAT set
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
bool [:S, the carrier of A:] is non empty set
{ b₁ where b₁ is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:] : S₁[b₁] } is set
meet { b₁ where b₁ is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:] : S₁[b₁] } is set
[#] [:S, the carrier of A:] is non empty non proper Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
s is Element of S
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
f . (s,r) is Element of S
f . [s,r] is set
i is Element of the carrier of A
[(f . (s,r)),i] is non empty V22() set
{(f . (s,r)),i} is non empty finite set
{(f . (s,r))} is non empty trivial finite 1 -element set
{{(f . (s,r)),i},{(f . (s,r))}} is non empty finite V44() set
(A,r,i) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
s is Element of S
s is Element of the carrier of A
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
f . (s,s) is Element of S
f . [s,s] is set
r is Element of the carrier of A
[(f . (s,s)),r] is non empty V22() set
{(f . (s,s)),r} is non empty finite set
{(f . (s,s))} is non empty trivial finite 1 -element set
{{(f . (s,s)),r},{(f . (s,s))}} is non empty finite V44() set
i is Element of the carrier of A
(A,s,r,i) is Element of the carrier of A
<*s,r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*s*> ^ <*r*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*s,r,i*> is set
[s,(A,s,r,i)] is non empty V22() set
{s,(A,s,r,i)} is non empty finite set
{{s,(A,s,r,i)},{s}} is non empty finite V44() set
ri is Element of S
r is Element of the carrier of A
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
f . (ri,r) is Element of S
f . [ri,r] is set
i is Element of the carrier of A
[(f . (ri,r)),i] is non empty V22() set
{(f . (ri,r)),i} is non empty finite set
{(f . (ri,r))} is non empty trivial finite 1 -element set
{{(f . (ri,r)),i},{(f . (ri,r))}} is non empty finite V44() set
i is Element of the carrier of A
(A,r,i,i) is Element of the carrier of A
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,i*> is set
[ri,(A,r,i,i)] is non empty V22() set
{ri,(A,r,i,i)} is non empty finite set
{{ri,(A,r,i,i)},{ri}} is non empty finite V44() set
i0 is Element of S
gI2 is Element of the carrier of A
[i0,gI2] is non empty V22() set
{i0,gI2} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,gI2},{i0}} is non empty finite V44() set
a is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
a . 1 is set
f . (i0,gI2) is Element of S
f . [i0,gI2] is set
len a is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
a . (len a) is set
J is Element of the carrier of A
(A,J,gI2) is Element of the carrier of A
<*J,gI2*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> ^ <*gI2*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*J,gI2*> is set
(A,gI2,J) is Element of the carrier of A
<*gI2,J*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*gI2,J*> is set
[i0,(A,gI2,J)] is non empty V22() set
{i0,(A,gI2,J)} is non empty finite set
{{i0,(A,gI2,J)},{i0}} is non empty finite V44() set
I is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
s is set
r is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
s is set
r is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
s is Element of S
i is Element of the carrier of A
[s,i] is non empty V22() set
{s,i} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,i},{s}} is non empty finite V44() set
f . (s,i) is Element of S
f . [s,i] is set
s is Element of the carrier of A
[(f . (s,i)),s] is non empty V22() set
{(f . (s,i)),s} is non empty finite set
{(f . (s,i))} is non empty trivial finite 1 -element set
{{(f . (s,i)),s},{(f . (s,i))}} is non empty finite V44() set
s is set
(A,i,s) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*i,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,s*> is set
[s,(A,i,s)] is non empty V22() set
{s,(A,i,s)} is non empty finite set
{{s,(A,i,s)},{s}} is non empty finite V44() set
r is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{{s,r},{s}} is non empty finite V44() set
f . (s,r) is Element of S
f . [s,r] is set
[(f . (s,r)),i] is non empty V22() set
{(f . (s,r)),i} is non empty finite set
{(f . (s,r))} is non empty trivial finite 1 -element set
{{(f . (s,r)),i},{(f . (s,r))}} is non empty finite V44() set
s is set
(A,r,i,s) is Element of the carrier of A
<*r,i,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(<*r*> ^ <*i*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,s*> is set
[s,(A,r,i,s)] is non empty V22() set
{s,(A,r,i,s)} is non empty finite set
{{s,(A,r,i,s)},{s}} is non empty finite V44() set
r is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
[(f . (s,r)),s] is non empty V22() set
{(f . (s,r)),s} is non empty finite set
{{(f . (s,r)),s},{(f . (s,r))}} is non empty finite V44() set
s is set
r is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
(A,i,r) is Element of the carrier of A
<*i,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,r*> is set
s is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
r is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s . i is set
[(s . i),(A,i,r)] is non empty V22() set
{(s . i),(A,i,r)} is non empty finite set
{(s . i)} is non empty trivial finite 1 -element set
{{(s . i),(A,i,r)},{(s . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (i + 1) is set
f . ((s . i),(A,i,r)) is set
f . [(s . i),(A,i,r)] is set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den ((In (4,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
i is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
s is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
P is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
C is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool T
P is Element of T
[P,S] is non empty V22() set
{P,S} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,S},{P}} is non empty finite V44() set
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,f)
(A,T,f,C) is Relation-like T -defined the carrier of A -valued Element of bool [:T, the carrier of A:]
bool [:T, the carrier of A:] is non empty set
[:T,(A):] is Relation-like disjoint_with_NAT set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
S is Element of the carrier of A
T is non empty set
bool T is non empty set
[:T, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool T
f is Element of T
[f,S] is non empty V22() set
{f,S} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,S},{f}} is non empty finite V44() set
C is non empty Relation-like [:T, the carrier of A:] -defined T -valued Function-like total quasi_total (A,T) (A,T) (A,T,P)
(A,T,P,C) is Relation-like T -defined the carrier of A -valued Element of bool [:T, the carrier of A:]
bool [:T, the carrier of A:] is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is Element of S
[f,(A)] is non empty V22() set
{f,(A)} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,(A)},{f}} is non empty finite V44() set
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
(A,S,T,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
{(A)} is non empty trivial finite 1 -element set
[:S,{(A)}:] is non empty Relation-like disjoint_with_NAT set
A is non empty partial quasi_total non-empty () () () () UAStr
(A) is Element of the carrier of A
the carrier of A is non empty set
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
f is Element of bool S
T is Element of S
[T,(A)] is non empty V22() set
{T,(A)} is non empty finite set
{T} is non empty trivial finite 1 -element set
{{T,(A)},{T}} is non empty finite V44() set
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,f)
(A,S,f,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
(A) is (A) Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool f
C is Element of f
[C,(A,S,T)] is non empty V22() set
{C,(A,S,T)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(A,S,T)},{C}} is non empty finite V44() set
[C,S] is non empty V22() set
{C,S} is non empty finite set
{{C,S},{C}} is non empty finite V44() set
I is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total (A,f) (A,f) (A,f,P)
(A,f,P,I) is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
bool [:f, the carrier of A:] is non empty set
I . (C,S) is Element of f
I . [C,S] is set
[(I . (C,S)),T] is non empty V22() set
{(I . (C,S)),T} is non empty finite set
{(I . (C,S))} is non empty trivial finite 1 -element set
{{(I . (C,S)),T},{(I . (C,S))}} is non empty finite V44() set
{[C,(A,S,T)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(A,f,P,I) \ {[C,(A,S,T)]} is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is (A) Element of the carrier of A
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
[:f,(A):] is Relation-like disjoint_with_NAT set
r is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
i is set
s is set
[i,s] is non empty V22() set
{i,s} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s},{i}} is non empty finite V44() set
[:f,{(A)}:] is non empty Relation-like disjoint_with_NAT set
i is set
s is set
[i,s] is non empty V22() set
{i,s} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,s},{i}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
ri is Element of the carrier of A
(A,r,i,ri) is Element of the carrier of A
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,ri*> is set
[s,(A,r,i,ri)] is non empty V22() set
{s,(A,r,i,ri)} is non empty finite set
{{s,(A,r,i,ri)},{s}} is non empty finite V44() set
[(I . (s,r)),ri] is non empty V22() set
{(I . (s,r)),ri} is non empty finite set
{{(I . (s,r)),ri},{(I . (s,r))}} is non empty finite V44() set
(A,i,r) is Element of the carrier of A
<*i,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,r*> is set
r is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
[(r . i),(A,i,r)] is non empty V22() set
{(r . i),(A,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(A,i,r)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
I . ((r . i),(A,i,r)) is set
I . [(r . i),(A,i,r)] is set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
(Den ((In (4,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
s is Element of f
i is Element of the carrier of A
[s,i] is non empty V22() set
{s,i} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,i},{s}} is non empty finite V44() set
I . (s,i) is Element of f
I . [s,i] is set
ri is Element of the carrier of A
[(I . (s,i)),ri] is non empty V22() set
{(I . (s,i)),ri} is non empty finite set
{(I . (s,i))} is non empty trivial finite 1 -element set
{{(I . (s,i)),ri},{(I . (s,i))}} is non empty finite V44() set
(A,i,ri) is Element of the carrier of A
<*i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,ri*> is set
[s,(A,i,ri)] is non empty V22() set
{s,(A,i,ri)} is non empty finite set
{{s,(A,i,ri)},{s}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
ri is Element of f
r is Element of the carrier of A
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
I . (ri,r) is Element of f
I . [ri,r] is set
i is Element of the carrier of A
[(I . (ri,r)),i] is non empty V22() set
{(I . (ri,r)),i} is non empty finite set
{(I . (ri,r))} is non empty trivial finite 1 -element set
{{(I . (ri,r)),i},{(I . (ri,r))}} is non empty finite V44() set
i is Element of the carrier of A
(A,r,i,i) is Element of the carrier of A
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,i*> is set
[ri,(A,r,i,i)] is non empty V22() set
{ri,(A,r,i,i)} is non empty finite set
{{ri,(A,r,i,i)},{ri}} is non empty finite V44() set
i0 is Element of f
gI2 is Element of the carrier of A
[i0,gI2] is non empty V22() set
{i0,gI2} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,gI2},{i0}} is non empty finite V44() set
I . (i0,gI2) is Element of f
I . [i0,gI2] is set
a is Element of the carrier of A
[(I . (i0,gI2)),a] is non empty V22() set
{(I . (i0,gI2)),a} is non empty finite set
{(I . (i0,gI2))} is non empty trivial finite 1 -element set
{{(I . (i0,gI2)),a},{(I . (i0,gI2))}} is non empty finite V44() set
J is Element of the carrier of A
(A,gI2,J,a) is Element of the carrier of A
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*gI2,J,a*> is set
[i0,(A,gI2,J,a)] is non empty V22() set
{i0,(A,gI2,J,a)} is non empty finite set
{{i0,(A,gI2,J,a)},{i0}} is non empty finite V44() set
b is Element of f
b is Element of the carrier of A
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
b is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
b . 1 is set
I . (b,b) is Element of f
I . [b,b] is set
len b is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
b . (len b) is set
c is Element of the carrier of A
(A,c,b) is Element of the carrier of A
<*c,b*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*c*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,c] is non empty V22() set
{1,c} is non empty finite set
{{1,c},{1}} is non empty finite V44() set
{[1,c]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*c*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*c,b*> is set
(A,b,c) is Element of the carrier of A
<*b,c*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*b*> ^ <*c*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*b,c*> is set
[b,(A,b,c)] is non empty V22() set
{b,(A,b,c)} is non empty finite set
{{b,(A,b,c)},{b}} is non empty finite V44() set
s is Element of f
i is Element of the carrier of A
[s,i] is non empty V22() set
{s,i} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,i},{s}} is non empty finite V44() set
I . (s,i) is Element of f
I . [s,i] is set
ri is Element of the carrier of A
[(I . (s,i)),ri] is non empty V22() set
{(I . (s,i)),ri} is non empty finite set
{(I . (s,i))} is non empty trivial finite 1 -element set
{{(I . (s,i)),ri},{(I . (s,i))}} is non empty finite V44() set
(A,i,ri) is Element of the carrier of A
<*i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,ri*> is set
[s,(A,i,ri)] is non empty V22() set
{s,(A,i,ri)} is non empty finite set
{{s,(A,i,ri)},{s}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
ri is Element of f
r is Element of the carrier of A
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
I . (ri,r) is Element of f
I . [ri,r] is set
i is Element of the carrier of A
[(I . (ri,r)),i] is non empty V22() set
{(I . (ri,r)),i} is non empty finite set
{(I . (ri,r))} is non empty trivial finite 1 -element set
{{(I . (ri,r)),i},{(I . (ri,r))}} is non empty finite V44() set
i is Element of the carrier of A
(A,r,i,i) is Element of the carrier of A
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,i*> is set
[ri,(A,r,i,i)] is non empty V22() set
{ri,(A,r,i,i)} is non empty finite set
{{ri,(A,r,i,i)},{ri}} is non empty finite V44() set
i0 is Element of f
gI2 is Element of the carrier of A
[i0,gI2] is non empty V22() set
{i0,gI2} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,gI2},{i0}} is non empty finite V44() set
I . (i0,gI2) is Element of f
I . [i0,gI2] is set
a is Element of the carrier of A
[(I . (i0,gI2)),a] is non empty V22() set
{(I . (i0,gI2)),a} is non empty finite set
{(I . (i0,gI2))} is non empty trivial finite 1 -element set
{{(I . (i0,gI2)),a},{(I . (i0,gI2))}} is non empty finite V44() set
J is Element of the carrier of A
(A,gI2,J,a) is Element of the carrier of A
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*gI2,J,a*> is set
[i0,(A,gI2,J,a)] is non empty V22() set
{i0,(A,gI2,J,a)} is non empty finite set
{{i0,(A,gI2,J,a)},{i0}} is non empty finite V44() set
b is Element of f
b is Element of the carrier of A
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
b is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
b . 1 is set
I . (b,b) is Element of f
I . [b,b] is set
len b is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
b . (len b) is set
c is Element of the carrier of A
(A,c,b) is Element of the carrier of A
<*c,b*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*c*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,c] is non empty V22() set
{1,c} is non empty finite set
{{1,c},{1}} is non empty finite V44() set
{[1,c]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*c*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*c,b*> is set
(A,b,c) is Element of the carrier of A
<*b,c*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*b*> ^ <*c*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*b,c*> is set
[b,(A,b,c)] is non empty V22() set
{b,(A,b,c)} is non empty finite set
{{b,(A,b,c)},{b}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is (A) Element of the carrier of A
T is (A) Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*S,T*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of f
[P,(A,S,T)] is non empty V22() set
{P,(A,S,T)} is non empty finite set
{P} is non empty trivial finite 1 -element set
{{P,(A,S,T)},{P}} is non empty finite V44() set
C is Element of bool f
I is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total (A,f) (A,f) (A,f,C)
(A,f,C,I) is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
bool [:f, the carrier of A:] is non empty set
[P,S] is non empty V22() set
{P,S} is non empty finite set
{{P,S},{P}} is non empty finite V44() set
I . (P,S) is Element of f
I . [P,S] is set
[(I . (P,S)),T] is non empty V22() set
{(I . (P,S)),T} is non empty finite set
{(I . (P,S))} is non empty trivial finite 1 -element set
{{(I . (P,S)),T},{(I . (P,S))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
f is Element of the carrier of A
(A,S,T,f) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
P is non empty set
bool P is non empty set
[:P, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
C is Element of bool P
I is Element of P
[I,(A,S,T,f)] is non empty V22() set
{I,(A,S,T,f)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(A,S,T,f)},{I}} is non empty finite V44() set
[I,S] is non empty V22() set
{I,S} is non empty finite set
{{I,S},{I}} is non empty finite V44() set
s is non empty Relation-like [:P, the carrier of A:] -defined P -valued Function-like total quasi_total (A,P) (A,P) (A,P,C)
(A,P,C,s) is Relation-like P -defined the carrier of A -valued Element of bool [:P, the carrier of A:]
bool [:P, the carrier of A:] is non empty set
s . (I,S) is Element of P
s . [I,S] is set
[(s . (I,S)),T] is non empty V22() set
{(s . (I,S)),T} is non empty finite set
{(s . (I,S))} is non empty trivial finite 1 -element set
{{(s . (I,S)),T},{(s . (I,S))}} is non empty finite V44() set
[(s . (I,S)),f] is non empty V22() set
{(s . (I,S)),f} is non empty finite set
{{(s . (I,S)),f},{(s . (I,S))}} is non empty finite V44() set
{[I,(A,S,T,f)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(A,P,C,s) \ {[I,(A,S,T,f)]} is Relation-like P -defined the carrier of A -valued Element of bool [:P, the carrier of A:]
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is (A) Element of the carrier of A
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
[:P,(A):] is Relation-like disjoint_with_NAT set
s is Relation-like P -defined the carrier of A -valued Element of bool [:P, the carrier of A:]
r is set
i is set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
[:P,{(A)}:] is non empty Relation-like disjoint_with_NAT set
r is set
i is set
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
r is Element of P
ri is Element of the carrier of A
[r,ri] is non empty V22() set
{r,ri} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,ri},{r}} is non empty finite V44() set
s . (r,ri) is Element of P
s . [r,ri] is set
r is Element of the carrier of A
[(s . (r,ri)),r] is non empty V22() set
{(s . (r,ri)),r} is non empty finite set
{(s . (r,ri))} is non empty trivial finite 1 -element set
{{(s . (r,ri)),r},{(s . (r,ri))}} is non empty finite V44() set
(A,ri,r) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*ri,r*> is set
[r,(A,ri,r)] is non empty V22() set
{r,(A,ri,r)} is non empty finite set
{{r,(A,ri,r)},{r}} is non empty finite V44() set
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
(A,ri,i) is Element of the carrier of A
<*ri,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*ri,i*> is set
i is non empty Relation-like NAT -defined P -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of P
i . 1 is set
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (len i) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i . i is set
[(i . i),(A,ri,i)] is non empty V22() set
{(i . i),(A,ri,i)} is non empty finite set
{(i . i)} is non empty trivial finite 1 -element set
{{(i . i),(A,ri,i)},{(i . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (i + 1) is set
s . ((i . i),(A,ri,i)) is set
s . [(i . i),(A,ri,i)] is set
(A,i,ri) is Element of the carrier of A
<*i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*i,ri*> is set
[r,(A,i,ri)] is non empty V22() set
{r,(A,i,ri)} is non empty finite set
{{r,(A,i,ri)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
ri is Element of the carrier of A
[(s . (r,i)),ri] is non empty V22() set
{(s . (r,i)),ri} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),ri},{(s . (r,i))}} is non empty finite V44() set
r is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
r is Element of the carrier of A
[(s . (r,i)),r] is non empty V22() set
{(s . (r,i)),r} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),r},{(s . (r,i))}} is non empty finite V44() set
ri is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
ri is Element of the carrier of A
[(s . (r,i)),ri] is non empty V22() set
{(s . (r,i)),ri} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),ri},{(s . (r,i))}} is non empty finite V44() set
r is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
r is Element of the carrier of A
[(s . (r,i)),r] is non empty V22() set
{(s . (r,i)),r} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),r},{(s . (r,i))}} is non empty finite V44() set
ri is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
ri is Element of the carrier of A
[(s . (r,i)),ri] is non empty V22() set
{(s . (r,i)),ri} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),ri},{(s . (r,i))}} is non empty finite V44() set
r is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
r is Element of P
i is Element of the carrier of A
[r,i] is non empty V22() set
{r,i} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,i},{r}} is non empty finite V44() set
s . (r,i) is Element of P
s . [r,i] is set
r is Element of the carrier of A
[(s . (r,i)),r] is non empty V22() set
{(s . (r,i)),r} is non empty finite set
{(s . (r,i))} is non empty trivial finite 1 -element set
{{(s . (r,i)),r},{(s . (r,i))}} is non empty finite V44() set
ri is Element of the carrier of A
(A,i,ri,r) is Element of the carrier of A
<*i,ri,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*i*> ^ <*ri*>) ^ <*r*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*i,ri,r*> is set
[r,(A,i,ri,r)] is non empty V22() set
{r,(A,i,ri,r)} is non empty finite set
{{r,(A,i,ri,r)},{r}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is (A) Element of the carrier of A
T is (A) Element of the carrier of A
f is (A) Element of the carrier of A
(A,S,T,f) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,f*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*f*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,f] is non empty V22() set
{1,f} is non empty finite set
{{1,f},{1}} is non empty finite V44() set
{[1,f]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*f*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,f*> is set
P is non empty set
bool P is non empty set
[:P, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
C is Element of P
[C,(A,S,T,f)] is non empty V22() set
{C,(A,S,T,f)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(A,S,T,f)},{C}} is non empty finite V44() set
I is Element of bool P
s is non empty Relation-like [:P, the carrier of A:] -defined P -valued Function-like total quasi_total (A,P) (A,P) (A,P,I)
(A,P,I,s) is Relation-like P -defined the carrier of A -valued Element of bool [:P, the carrier of A:]
bool [:P, the carrier of A:] is non empty set
[C,S] is non empty V22() set
{C,S} is non empty finite set
{{C,S},{C}} is non empty finite V44() set
s . (C,S) is Element of P
s . [C,S] is set
[(s . (C,S)),T] is non empty V22() set
{(s . (C,S)),T} is non empty finite set
{(s . (C,S))} is non empty trivial finite 1 -element set
{{(s . (C,S)),T},{(s . (C,S))}} is non empty finite V44() set
[(s . (C,S)),f] is non empty V22() set
{(s . (C,S)),f} is non empty finite set
{{(s . (C,S)),f},{(s . (C,S))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is (A) Element of the carrier of A
T is (A) Element of the carrier of A
(A,S,T) is Element of the carrier of A
(A) is (A) Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
(A,S,T,(A)) is (A) Element of the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*S,T,(A)*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*(A)*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,(A)] is non empty V22() set
{1,(A)} is non empty finite set
{{1,(A)},{1}} is non empty finite V44() set
{[1,(A)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*S*> ^ <*T*>) ^ <*(A)*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*S,T,(A)*> is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
(A,T,S) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*T,S*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,S*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool f
C is Element of f
[C,(A,S,T)] is non empty V22() set
{C,(A,S,T)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(A,S,T)},{C}} is non empty finite V44() set
[C,S] is non empty V22() set
{C,S} is non empty finite set
{{C,S},{C}} is non empty finite V44() set
I is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total (A,f) (A,f) (A,f,P)
(A,f,P,I) is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
bool [:f, the carrier of A:] is non empty set
I . (C,S) is Element of f
I . [C,S] is set
{[C,(A,S,T)]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(A,f,P,I) \ {[C,(A,S,T)]} is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
(A) is Element of bool the carrier of A
bool the carrier of A is non empty set
(A) is (A) Element of the carrier of A
In (1,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (1,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (1,(dom the charact of A))) is non empty homogeneous set
(Den ((In (1,(dom the charact of A))),A)) . {} is set
{(A)} is non empty trivial finite 1 -element set
the carrier of A \ {(A)} is Element of bool the carrier of A
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of A))),A)) is Element of bool the carrier of A
( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A))) is Element of bool the carrier of A
rng (Den ((In (4,(dom the charact of A))),A)) is Element of bool the carrier of A
(( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A))) is Element of bool the carrier of A
{ (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is set
((( the carrier of A \ {(A)}) \ (rng (Den ((In (3,(dom the charact of A))),A)))) \ (rng (Den ((In (4,(dom the charact of A))),A)))) \ { (A,b₁,b₂) where b₁, b₂ is Element of the carrier of A : ( not b₁ = (A,b₁,b₂) & not b₂ = (A,b₁,b₂) ) } is Element of bool the carrier of A
[:f,(A):] is Relation-like disjoint_with_NAT set
s is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
[:f,{(A)}:] is non empty Relation-like disjoint_with_NAT set
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
s is Element of f
i is Element of the carrier of A
[s,i] is non empty V22() set
{s,i} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,i},{s}} is non empty finite V44() set
I . (s,i) is Element of f
I . [s,i] is set
ri is Element of the carrier of A
[(I . (s,i)),ri] is non empty V22() set
{(I . (s,i)),ri} is non empty finite set
{(I . (s,i))} is non empty trivial finite 1 -element set
{{(I . (s,i)),ri},{(I . (s,i))}} is non empty finite V44() set
(A,i,ri) is Element of the carrier of A
<*i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*ri*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,ri] is non empty V22() set
{1,ri} is non empty finite set
{{1,ri},{1}} is non empty finite V44() set
{[1,ri]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,ri*> is set
[s,(A,i,ri)] is non empty V22() set
{s,(A,i,ri)} is non empty finite set
{{s,(A,i,ri)},{s}} is non empty finite V44() set
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
(A,r,i,ri) is Element of the carrier of A
<*r,i,ri*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(<*r*> ^ <*i*>) ^ <*ri*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,ri*> is set
[s,(A,r,i,ri)] is non empty V22() set
{s,(A,r,i,ri)} is non empty finite set
{{s,(A,r,i,ri)},{s}} is non empty finite V44() set
[(I . (s,r)),ri] is non empty V22() set
{(I . (s,r)),ri} is non empty finite set
{{(I . (s,r)),ri},{(I . (s,r))}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
(A,i,r) is Element of the carrier of A
<*i,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,r*> is set
r is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
[(r . i),(A,i,r)] is non empty V22() set
{(r . i),(A,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(A,i,r)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
I . ((r . i),(A,i,r)) is set
I . [(r . i),(A,i,r)] is set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
ri is Element of f
r is Element of the carrier of A
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
I . (ri,r) is Element of f
I . [ri,r] is set
i is Element of the carrier of A
[(I . (ri,r)),i] is non empty V22() set
{(I . (ri,r)),i} is non empty finite set
{(I . (ri,r))} is non empty trivial finite 1 -element set
{{(I . (ri,r)),i},{(I . (ri,r))}} is non empty finite V44() set
i is Element of the carrier of A
(A,r,i,i) is Element of the carrier of A
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,i*> is set
[ri,(A,r,i,i)] is non empty V22() set
{ri,(A,r,i,i)} is non empty finite set
{{ri,(A,r,i,i)},{ri}} is non empty finite V44() set
i0 is Element of f
gI2 is Element of the carrier of A
[i0,gI2] is non empty V22() set
{i0,gI2} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,gI2},{i0}} is non empty finite V44() set
I . (i0,gI2) is Element of f
I . [i0,gI2] is set
a is Element of the carrier of A
[(I . (i0,gI2)),a] is non empty V22() set
{(I . (i0,gI2)),a} is non empty finite set
{(I . (i0,gI2))} is non empty trivial finite 1 -element set
{{(I . (i0,gI2)),a},{(I . (i0,gI2))}} is non empty finite V44() set
J is Element of the carrier of A
(A,gI2,J,a) is Element of the carrier of A
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*gI2,J,a*> is set
[i0,(A,gI2,J,a)] is non empty V22() set
{i0,(A,gI2,J,a)} is non empty finite set
{{i0,(A,gI2,J,a)},{i0}} is non empty finite V44() set
b is Element of f
b is Element of the carrier of A
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
b is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
b . 1 is set
I . (b,b) is Element of f
I . [b,b] is set
len b is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
b . (len b) is set
c is Element of the carrier of A
(A,c,b) is Element of the carrier of A
<*c,b*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*c*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,c] is non empty V22() set
{1,c} is non empty finite set
{{1,c},{1}} is non empty finite V44() set
{[1,c]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*c*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*c,b*> is set
(A,b,c) is Element of the carrier of A
<*b,c*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*b*> ^ <*c*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*b,c*> is set
[b,(A,b,c)] is non empty V22() set
{b,(A,b,c)} is non empty finite set
{{b,(A,b,c)},{b}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
(A,i,r) is Element of the carrier of A
<*i,r*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*i,r*> is set
r is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
[(r . i),(A,i,r)] is non empty V22() set
{(r . i),(A,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(A,i,r)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
I . ((r . i),(A,i,r)) is set
I . [(r . i),(A,i,r)] is set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
s is Element of f
r is Element of the carrier of A
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
I . (s,r) is Element of f
I . [s,r] is set
i is Element of the carrier of A
[(I . (s,r)),i] is non empty V22() set
{(I . (s,r)),i} is non empty finite set
{(I . (s,r))} is non empty trivial finite 1 -element set
{{(I . (s,r)),i},{(I . (s,r))}} is non empty finite V44() set
(A,r,i) is Element of the carrier of A
<*r,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*r,i*> is set
[s,(A,r,i)] is non empty V22() set
{s,(A,r,i)} is non empty finite set
{{s,(A,r,i)},{s}} is non empty finite V44() set
ri is Element of f
r is Element of the carrier of A
[ri,r] is non empty V22() set
{ri,r} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,r},{ri}} is non empty finite V44() set
I . (ri,r) is Element of f
I . [ri,r] is set
i is Element of the carrier of A
[(I . (ri,r)),i] is non empty V22() set
{(I . (ri,r)),i} is non empty finite set
{(I . (ri,r))} is non empty trivial finite 1 -element set
{{(I . (ri,r)),i},{(I . (ri,r))}} is non empty finite V44() set
i is Element of the carrier of A
(A,r,i,i) is Element of the carrier of A
<*r,i,i*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*i*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*r,i,i*> is set
[ri,(A,r,i,i)] is non empty V22() set
{ri,(A,r,i,i)} is non empty finite set
{{ri,(A,r,i,i)},{ri}} is non empty finite V44() set
i0 is Element of f
gI2 is Element of the carrier of A
[i0,gI2] is non empty V22() set
{i0,gI2} is non empty finite set
{i0} is non empty trivial finite 1 -element set
{{i0,gI2},{i0}} is non empty finite V44() set
I . (i0,gI2) is Element of f
I . [i0,gI2] is set
a is Element of the carrier of A
[(I . (i0,gI2)),a] is non empty V22() set
{(I . (i0,gI2)),a} is non empty finite set
{(I . (i0,gI2))} is non empty trivial finite 1 -element set
{{(I . (i0,gI2)),a},{(I . (i0,gI2))}} is non empty finite V44() set
J is Element of the carrier of A
(A,gI2,J,a) is Element of the carrier of A
<*gI2,J,a*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*gI2*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,gI2] is non empty V22() set
{1,gI2} is non empty finite set
{{1,gI2},{1}} is non empty finite V44() set
{[1,gI2]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*J*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,J] is non empty V22() set
{1,J} is non empty finite set
{{1,J},{1}} is non empty finite V44() set
{[1,J]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*gI2*> ^ <*J*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*a*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,a] is non empty V22() set
{1,a} is non empty finite set
{{1,a},{1}} is non empty finite V44() set
{[1,a]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*gI2*> ^ <*J*>) ^ <*a*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*gI2,J,a*> is set
[i0,(A,gI2,J,a)] is non empty V22() set
{i0,(A,gI2,J,a)} is non empty finite set
{{i0,(A,gI2,J,a)},{i0}} is non empty finite V44() set
b is Element of f
b is Element of the carrier of A
[b,b] is non empty V22() set
{b,b} is non empty finite set
{b} is non empty trivial finite 1 -element set
{{b,b},{b}} is non empty finite V44() set
b is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
b . 1 is set
I . (b,b) is Element of f
I . [b,b] is set
len b is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
b . (len b) is set
c is Element of the carrier of A
(A,c,b) is Element of the carrier of A
<*c,b*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*c*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,c] is non empty V22() set
{1,c} is non empty finite set
{{1,c},{1}} is non empty finite V44() set
{[1,c]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*b*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,b] is non empty V22() set
{1,b} is non empty finite set
{{1,b},{1}} is non empty finite V44() set
{[1,b]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*c*> ^ <*b*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*c,b*> is set
(A,b,c) is Element of the carrier of A
<*b,c*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*b*> ^ <*c*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*b,c*> is set
[b,(A,b,c)] is non empty V22() set
{b,(A,b,c)} is non empty finite set
{{b,(A,b,c)},{b}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is Element of the carrier of A
T is Element of the carrier of A
(A,S,T) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*S,T*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*S*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,S] is non empty V22() set
{1,S} is non empty finite set
{{1,S},{1}} is non empty finite V44() set
{[1,S]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*T*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,T] is non empty V22() set
{1,T} is non empty finite set
{{1,T},{1}} is non empty finite V44() set
{[1,T]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*S*> ^ <*T*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*S,T*> is set
f is non empty set
bool f is non empty set
[:f, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
P is Element of bool f
C is Element of f
[C,(A,S,T)] is non empty V22() set
{C,(A,S,T)} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,(A,S,T)},{C}} is non empty finite V44() set
I is non empty Relation-like [:f, the carrier of A:] -defined f -valued Function-like total quasi_total (A,f) (A,f) (A,f,P)
(A,f,P,I) is Relation-like f -defined the carrier of A -valued Element of bool [:f, the carrier of A:]
bool [:f, the carrier of A:] is non empty set
I . (C,S) is Element of f
[C,S] is non empty V22() set
{C,S} is non empty finite set
{{C,S},{C}} is non empty finite V44() set
I . [C,S] is set
[(I . (C,S)),T] is non empty V22() set
{(I . (C,S)),T} is non empty finite set
{(I . (C,S))} is non empty trivial finite 1 -element set
{{(I . (C,S)),T},{(I . (C,S))}} is non empty finite V44() set
(A,T,S) is Element of the carrier of A
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*T,S*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*T*> ^ <*S*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*T,S*> is set
i is non empty Relation-like NAT -defined f -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of f
i . 1 is set
len i is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
i . (len i) is set
[(i . 1),(A,T,S)] is non empty V22() set
{(i . 1),(A,T,S)} is non empty finite set
{(i . 1)} is non empty trivial finite 1 -element set
{{(i . 1),(A,T,S)},{(i . 1)}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is Element of S
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
(A,S,T,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
I is (A) Element of the carrier of A
C is (A) Element of the carrier of A
(A,I,C) is (A) Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*I,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*I,C*> is set
P . (f,C) is Element of S
[f,C] is non empty V22() set
{f,C} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,C},{f}} is non empty finite V44() set
P . [f,C] is set
(A,C,I) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*C,I*> is set
[f,(A,C,I)] is non empty V22() set
{f,(A,C,I)} is non empty finite set
{{f,(A,C,I)},{f}} is non empty finite V44() set
s is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s . r is set
[(s . r),(A,I,C)] is non empty V22() set
{(s . r),(A,I,C)} is non empty finite set
{(s . r)} is non empty trivial finite 1 -element set
{{(s . r),(A,I,C)},{(s . r)}} is non empty finite V44() set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (r + 1) is set
P . ((s . r),(A,I,C)) is set
P . [(s . r),(A,I,C)] is set
A is non empty set
bool A is non empty set
S is Element of bool A
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
the carrier of T is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
(T) is Element of bool the carrier of T
bool the carrier of T is non empty set
(T) is (T) Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
f is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
f | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
P is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
P | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
(T,A,S,f) is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
bool [:A, the carrier of T:] is non empty set
(T,A,S,P) is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
r is Element of the carrier of T
i is Element of A
[i,r] is non empty V22() set
{i,r} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,r},{i}} is non empty finite V44() set
f . (i,r) is Element of A
f . [i,r] is set
P . (i,r) is Element of A
P . [i,r] is set
(f | [:A,(T):]) . [i,r] is set
r is Element of A
[r,(T)] is non empty V22() set
{r,(T)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(T)},{r}} is non empty finite V44() set
f . (r,(T)) is Element of A
f . [r,(T)] is set
P . (r,(T)) is Element of A
P . [r,(T)] is set
r is Element of the carrier of T
i is Element of the carrier of T
(T,r,i) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*r,i*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*r,i*> is set
s is Element of A
[s,(T,r,i)] is non empty V22() set
{s,(T,r,i)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(T,r,i)},{s}} is non empty finite V44() set
f . (s,(T,r,i)) is Element of A
f . [s,(T,r,i)] is set
P . (s,(T,r,i)) is Element of A
P . [s,(T,r,i)] is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{{s,r},{s}} is non empty finite V44() set
f . (s,r) is Element of A
f . [s,r] is set
[(f . (s,r)),i] is non empty V22() set
{(f . (s,r)),i} is non empty finite set
{(f . (s,r))} is non empty trivial finite 1 -element set
{{(f . (s,r)),i},{(f . (s,r))}} is non empty finite V44() set
P . (s,r) is Element of A
P . [s,r] is set
f . ((f . (s,r)),i) is Element of A
f . [(f . (s,r)),i] is set
P . ((f . (s,r)),i) is Element of A
P . [(f . (s,r)),i] is set
r is Element of the carrier of T
i is Element of the carrier of T
s is Element of the carrier of T
(T,r,i,s) is Element of the carrier of T
<*r,i,s*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*r*> ^ <*i*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of T))),T)) . <*r,i,s*> is set
r is Element of A
[r,(T,r,i,s)] is non empty V22() set
{r,(T,r,i,s)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(T,r,i,s)},{r}} is non empty finite V44() set
f . (r,(T,r,i,s)) is Element of A
f . [r,(T,r,i,s)] is set
P . (r,(T,r,i,s)) is Element of A
P . [r,(T,r,i,s)] is set
[r,r] is non empty V22() set
{r,r} is non empty finite set
{{r,r},{r}} is non empty finite V44() set
f . (r,r) is Element of A
f . [r,r] is set
[(f . (r,r)),i] is non empty V22() set
{(f . (r,r)),i} is non empty finite set
{(f . (r,r))} is non empty trivial finite 1 -element set
{{(f . (r,r)),i},{(f . (r,r))}} is non empty finite V44() set
[(f . (r,r)),s] is non empty V22() set
{(f . (r,r)),s} is non empty finite set
{{(f . (r,r)),s},{(f . (r,r))}} is non empty finite V44() set
P . (r,r) is Element of A
P . [r,r] is set
f . ((f . (r,r)),i) is Element of A
f . [(f . (r,r)),i] is set
P . ((f . (r,r)),i) is Element of A
P . [(f . (r,r)),i] is set
f . ((f . (r,r)),s) is Element of A
f . [(f . (r,r)),s] is set
P . ((f . (r,r)),s) is Element of A
P . [(f . (r,r)),s] is set
r is Element of the carrier of T
i is Element of the carrier of T
(T,r,i) is Element of the carrier of T
<*r,i*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*r*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,r] is non empty V22() set
{1,r} is non empty finite set
{{1,r},{1}} is non empty finite V44() set
{[1,r]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,i] is non empty V22() set
{1,i} is non empty finite set
{{1,i},{1}} is non empty finite V44() set
{[1,i]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*r*> ^ <*i*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of T))),T)) . <*r,i*> is set
s is Element of A
[s,(T,r,i)] is non empty V22() set
{s,(T,r,i)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(T,r,i)},{s}} is non empty finite V44() set
f . (s,(T,r,i)) is Element of A
f . [s,(T,r,i)] is set
P . (s,(T,r,i)) is Element of A
P . [s,(T,r,i)] is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{{s,r},{s}} is non empty finite V44() set
f . (s,r) is Element of A
f . [s,r] is set
P . (s,r) is Element of A
P . [s,r] is set
(T,i,r) is Element of the carrier of T
In (2,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (2,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (2,(dom the charact of T))) is non empty homogeneous set
<*i,r*> is non empty Relation-like NAT -defined the carrier of T -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of T
<*i*> ^ <*r*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of T))),T)) . <*i,r*> is set
r is non empty Relation-like NAT -defined A -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of A
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
f . ((r . i),(T,i,r)) is set
[(r . i),(T,i,r)] is non empty V22() set
{(r . i),(T,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(T,i,r)},{(r . i)}} is non empty finite V44() set
f . [(r . i),(T,i,r)] is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
[(r . i),(T,i,r)] is non empty V22() set
{(r . i),(T,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(T,i,r)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
P . ((r . i),(T,i,r)) is set
P . [(r . i),(T,i,r)] is set
ri is Element of A
[ri,i] is non empty V22() set
{ri,i} is non empty finite set
{ri} is non empty trivial finite 1 -element set
{{ri,i},{ri}} is non empty finite V44() set
f . (ri,i) is Element of A
f . [ri,i] is set
[(f . (ri,i)),r] is non empty V22() set
{(f . (ri,i)),r} is non empty finite set
{(f . (ri,i))} is non empty trivial finite 1 -element set
{{(f . (ri,i)),r},{(f . (ri,i))}} is non empty finite V44() set
f . ((r . i),(T,i,r)) is set
f . [(r . i),(T,i,r)] is set
f . ((f . (ri,i)),r) is Element of A
f . [(f . (ri,i)),r] is set
P . ((f . (ri,i)),r) is Element of A
P . [(f . (ri,i)),r] is set
P . (ri,i) is Element of A
P . [ri,i] is set
P . ((P . (ri,i)),r) is Element of A
[(P . (ri,i)),r] is non empty V22() set
{(P . (ri,i)),r} is non empty finite set
{(P . (ri,i))} is non empty trivial finite 1 -element set
{{(P . (ri,i)),r},{(P . (ri,i))}} is non empty finite V44() set
P . [(P . (ri,i)),r] is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
P . ((r . i),(T,i,r)) is set
[(r . i),(T,i,r)] is non empty V22() set
{(r . i),(T,i,r)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(T,i,r)},{(r . i)}} is non empty finite V44() set
P . [(r . i),(T,i,r)] is set
r is Element of the carrier of T
A is non empty set
bool A is non empty set
S is Element of bool A
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
the carrier of T is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
(T) is Element of bool the carrier of T
bool the carrier of T is non empty set
(T) is (T) Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
f is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
f | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
P is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
P | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
(T,A,S,f) is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
bool [:A, the carrier of T:] is non empty set
(T,A,S,P) is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
s is set
r is set
[s,r] is non empty V22() set
{s,r} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,r},{s}} is non empty finite V44() set
A is non empty set
bool A is non empty set
T is non empty partial quasi_total non-empty free () () () () () () () UAStr
the carrier of T is non empty set
[:A, the carrier of T:] is non empty Relation-like disjoint_with_NAT set
S is Element of bool A
f is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
(T) is Element of bool the carrier of T
bool the carrier of T is non empty set
(T) is (T) Element of the carrier of T
the charact of T is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of T *), the carrier of T) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of T *), the carrier of T)
the carrier of T * is non empty functional FinSequence-membered FinSequenceSet of the carrier of T
PFuncs (( the carrier of T *), the carrier of T) is set
dom the charact of T is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (1,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (1,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
rng the charact of T is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of T *), the carrier of T))
bool (PFuncs (( the carrier of T *), the carrier of T)) is non empty set
the charact of T . (In (1,(dom the charact of T))) is non empty homogeneous set
(Den ((In (1,(dom the charact of T))),T)) . {} is set
{(T)} is non empty trivial finite 1 -element set
the carrier of T \ {(T)} is Element of bool the carrier of T
In (3,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (3,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (3,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (3,(dom the charact of T))),T)) is Element of bool the carrier of T
( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T))) is Element of bool the carrier of T
In (4,(dom the charact of T)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of T
Den ((In (4,(dom the charact of T))),T) is Relation-like the carrier of T * -defined the carrier of T -valued Function-like Element of rng the charact of T
the charact of T . (In (4,(dom the charact of T))) is non empty homogeneous set
rng (Den ((In (4,(dom the charact of T))),T)) is Element of bool the carrier of T
(( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T))) is Element of bool the carrier of T
{ (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is set
((( the carrier of T \ {(T)}) \ (rng (Den ((In (3,(dom the charact of T))),T)))) \ (rng (Den ((In (4,(dom the charact of T))),T)))) \ { (T,b₁,b₂) where b₁, b₂ is Element of the carrier of T : ( not b₁ = (T,b₁,b₂) & not b₂ = (T,b₁,b₂) ) } is Element of bool the carrier of T
[:A,(T):] is Relation-like disjoint_with_NAT set
f | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
[:[:A, the carrier of T:],A:] is non empty Relation-like disjoint_with_NAT set
bool [:[:A, the carrier of T:],A:] is non empty set
P is non empty Relation-like [:A, the carrier of T:] -defined A -valued Function-like total quasi_total (T,A) (T,A) (T,A,S)
P | [:A,(T):] is Relation-like [:A, the carrier of T:] -defined [:A,(T):] -defined [:A, the carrier of T:] -defined A -valued Function-like Element of bool [:[:A, the carrier of T:],A:]
C is Element of A
I is Element of the carrier of T
[C,I] is non empty V22() set
{C,I} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,I},{C}} is non empty finite V44() set
(T,A,S,f) is Relation-like A -defined the carrier of T -valued Element of bool [:A, the carrier of T:]
bool [:A, the carrier of T:] is non empty set
f . (C,I) is Element of A
f . [C,I] is set
P . (C,I) is Element of A
P . [C,I] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is (A) Element of the carrier of A
C is Element of S
[C,P] is non empty V22() set
{C,P} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,P},{C}} is non empty finite V44() set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is Element of S
[C,P] is non empty V22() set
{C,P} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,P},{C}} is non empty finite V44() set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is Element of S
[C,P] is non empty V22() set
{C,P} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,P},{C}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is set
I is Element of S
[I,P] is non empty V22() set
{I,P} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,P},{I}} is non empty finite V44() set
S is non empty set
bool S is non empty set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
P is (A) Element of the carrier of A
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
C is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
C is Element of S
f . (C,P) is Element of S
[C,P] is non empty V22() set
{C,P} is non empty finite set
{C} is non empty trivial finite 1 -element set
{{C,P},{C}} is non empty finite V44() set
f . [C,P] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is set
C is Element of the carrier of A
I is Element of the carrier of A
(A,C,I) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*C,I*> is set
s is Element of S
f . (s,(A,C,I)) is Element of S
[s,(A,C,I)] is non empty V22() set
{s,(A,C,I)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(A,C,I)},{s}} is non empty finite V44() set
f . [s,(A,C,I)] is set
f . (s,C) is Element of S
[s,C] is non empty V22() set
{s,C} is non empty finite set
{{s,C},{s}} is non empty finite V44() set
f . [s,C] is set
f . ((f . (s,C)),I) is Element of S
[(f . (s,C)),I] is non empty V22() set
{(f . (s,C)),I} is non empty finite set
{(f . (s,C))} is non empty trivial finite 1 -element set
{{(f . (s,C)),I},{(f . (s,C))}} is non empty finite V44() set
f . [(f . (s,C)),I] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
C is Element of the carrier of A
(A,P,C) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*P,C*> is set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
I is Element of S
[I,(A,P,C)] is non empty V22() set
{I,(A,P,C)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(A,P,C)},{I}} is non empty finite V44() set
[I,P] is non empty V22() set
{I,P} is non empty finite set
{{I,P},{I}} is non empty finite V44() set
f . (I,P) is Element of S
f . [I,P] is set
[(f . (I,P)),C] is non empty V22() set
{(f . (I,P)),C} is non empty finite set
{(f . (I,P))} is non empty trivial finite 1 -element set
{{(f . (I,P)),C},{(f . (I,P))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is set
C is Element of the carrier of A
I is Element of the carrier of A
(A,C,I) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*C,I*> is set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
s is Element of S
[s,(A,C,I)] is non empty V22() set
{s,(A,C,I)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(A,C,I)},{s}} is non empty finite V44() set
[s,C] is non empty V22() set
{s,C} is non empty finite set
{{s,C},{s}} is non empty finite V44() set
f . (s,C) is Element of S
f . [s,C] is set
[(f . (s,C)),I] is non empty V22() set
{(f . (s,C)),I} is non empty finite set
{(f . (s,C))} is non empty trivial finite 1 -element set
{{(f . (s,C)),I},{(f . (s,C))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
C is Element of the carrier of A
I is Element of the carrier of A
(A,P,C,I) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*P,C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*P*> ^ <*C*>) ^ <*I*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*P,C,I*> is set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
s is Element of S
[s,(A,P,C,I)] is non empty V22() set
{s,(A,P,C,I)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(A,P,C,I)},{s}} is non empty finite V44() set
f . (s,P) is Element of S
[s,P] is non empty V22() set
{s,P} is non empty finite set
{{s,P},{s}} is non empty finite V44() set
f . [s,P] is set
[(f . (s,P)),C] is non empty V22() set
{(f . (s,P)),C} is non empty finite set
{(f . (s,P))} is non empty trivial finite 1 -element set
{{(f . (s,P)),C},{(f . (s,P))}} is non empty finite V44() set
[(f . (s,P)),I] is non empty V22() set
{(f . (s,P)),I} is non empty finite set
{{(f . (s,P)),I},{(f . (s,P))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is set
C is Element of the carrier of A
I is Element of the carrier of A
s is Element of the carrier of A
(A,C,I,s) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (3,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (3,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (3,(dom the charact of A))) is non empty homogeneous set
<*C,I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 3 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
(<*C*> ^ <*I*>) ^ <*s*> is non empty Relation-like NAT -defined Function-like finite (1 + 1) + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (3,(dom the charact of A))),A)) . <*C,I,s*> is set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
r is Element of S
[r,(A,C,I,s)] is non empty V22() set
{r,(A,C,I,s)} is non empty finite set
{r} is non empty trivial finite 1 -element set
{{r,(A,C,I,s)},{r}} is non empty finite V44() set
f . (r,C) is Element of S
[r,C] is non empty V22() set
{r,C} is non empty finite set
{{r,C},{r}} is non empty finite V44() set
f . [r,C] is set
[(f . (r,C)),I] is non empty V22() set
{(f . (r,C)),I} is non empty finite set
{(f . (r,C))} is non empty trivial finite 1 -element set
{{(f . (r,C)),I},{(f . (r,C))}} is non empty finite V44() set
[(f . (r,C)),s] is non empty V22() set
{(f . (r,C)),s} is non empty finite set
{{(f . (r,C)),s},{(f . (r,C))}} is non empty finite V44() set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is Element of S
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
(A,S,T,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is Element of the carrier of A
I is Element of the carrier of A
(A,I,C) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*I,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*I,C*> is set
P . (f,C) is Element of S
[f,C] is non empty V22() set
{f,C} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,C},{f}} is non empty finite V44() set
P . [f,C] is set
(A,C,I) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*C,I*> is set
[f,(A,C,I)] is non empty V22() set
{f,(A,C,I)} is non empty finite set
{{f,(A,C,I)},{f}} is non empty finite V44() set
s is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
s . 1 is set
len s is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (len s) is set
r is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
s . r is set
i is Element of S
[i,I] is non empty V22() set
{i,I} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,I},{i}} is non empty finite V44() set
P . (i,I) is Element of S
P . [i,I] is set
[(P . (i,I)),C] is non empty V22() set
{(P . (i,I)),C} is non empty finite set
{(P . (i,I))} is non empty trivial finite 1 -element set
{{(P . (i,I)),C},{(P . (i,I))}} is non empty finite V44() set
[(s . r),(A,I,C)] is non empty V22() set
{(s . r),(A,I,C)} is non empty finite set
{(s . r)} is non empty trivial finite 1 -element set
{{(s . r),(A,I,C)},{(s . r)}} is non empty finite V44() set
r + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s . (r + 1) is set
P . ((s . r),(A,I,C)) is set
P . [(s . r),(A,I,C)] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is Element of S
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
(A,S,T,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is set
I is Element of the carrier of A
s is Element of the carrier of A
(A,s,I) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*s,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*s,I*> is set
P . (f,I) is Element of S
[f,I] is non empty V22() set
{f,I} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,I},{f}} is non empty finite V44() set
P . [f,I] is set
(A,I,s) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*I,s*> is set
[f,(A,I,s)] is non empty V22() set
{f,(A,I,s)} is non empty finite set
{{f,(A,I,s)},{f}} is non empty finite V44() set
r is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (0 + 1) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((i + 1) + 1) is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((i + 1) + 1) is set
P . ((r . (i + 1)),(A,s,I)) is set
[(r . (i + 1)),(A,s,I)] is non empty V22() set
{(r . (i + 1)),(A,s,I)} is non empty finite set
{(r . (i + 1))} is non empty trivial finite 1 -element set
{{(r . (i + 1)),(A,s,I)},{(r . (i + 1))}} is non empty finite V44() set
P . [(r . (i + 1)),(A,s,I)] is set
s is Element of S
P . (s,s) is Element of S
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
P . [s,s] is set
P . ((P . (s,s)),I) is Element of S
[(P . (s,s)),I] is non empty V22() set
{(P . (s,s)),I} is non empty finite set
{(P . (s,s))} is non empty trivial finite 1 -element set
{{(P . (s,s)),I},{(P . (s,s))}} is non empty finite V44() set
P . [(P . (s,s)),I] is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
i -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(i -' 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is Element of S
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
P . (s,s) is Element of S
P . [s,s] is set
[(P . (s,s)),I] is non empty V22() set
{(P . (s,s)),I} is non empty finite set
{(P . (s,s))} is non empty trivial finite 1 -element set
{{(P . (s,s)),I},{(P . (s,s))}} is non empty finite V44() set
[(r . i),(A,s,I)] is non empty V22() set
{(r . i),(A,s,I)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(A,s,I)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
P . ((r . i),(A,s,I)) is set
P . [(r . i),(A,s,I)] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is Element of S
P is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
(A,S,T,P) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
C is set
I is Element of the carrier of A
s is Element of the carrier of A
(A,s,I) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*s,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*s*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,s] is non empty V22() set
{1,s} is non empty finite set
{{1,s},{1}} is non empty finite V44() set
{[1,s]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*s*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*s,I*> is set
P . (f,I) is Element of S
[f,I] is non empty V22() set
{f,I} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,I},{f}} is non empty finite V44() set
P . [f,I] is set
(A,I,s) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*I,s*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> ^ <*s*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*I,s*> is set
[f,(A,I,s)] is non empty V22() set
{f,(A,I,s)} is non empty finite set
{{f,(A,I,s)},{f}} is non empty finite V44() set
r is non empty Relation-like NAT -defined S -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of S
r . 1 is set
len r is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (len r) is set
0 + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (0 + 1) is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((i + 1) + 1) is set
(i + 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . ((i + 1) + 1) is set
P . ((r . (i + 1)),(A,s,I)) is set
[(r . (i + 1)),(A,s,I)] is non empty V22() set
{(r . (i + 1)),(A,s,I)} is non empty finite set
{(r . (i + 1))} is non empty trivial finite 1 -element set
{{(r . (i + 1)),(A,s,I)},{(r . (i + 1))}} is non empty finite V44() set
P . [(r . (i + 1)),(A,s,I)] is set
s is Element of S
P . (s,s) is Element of S
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
P . [s,s] is set
P . ((P . (s,s)),I) is Element of S
[(P . (s,s)),I] is non empty V22() set
{(P . (s,s)),I} is non empty finite set
{(P . (s,s))} is non empty trivial finite 1 -element set
{{(P . (s,s)),I},{(P . (s,s))}} is non empty finite V44() set
P . [(P . (s,s)),I] is set
i is ordinal natural V22() finite cardinal V79() V80() ext-real non negative set
r . i is set
i -' 1 is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V146() V147() V148() Element of NAT
(i -' 1) + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
s is Element of S
[s,s] is non empty V22() set
{s,s} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,s},{s}} is non empty finite V44() set
P . (s,s) is Element of S
P . [s,s] is set
[(P . (s,s)),I] is non empty V22() set
{(P . (s,s)),I} is non empty finite set
{(P . (s,s))} is non empty trivial finite 1 -element set
{{(P . (s,s)),I},{(P . (s,s))}} is non empty finite V44() set
[(r . i),(A,s,I)] is non empty V22() set
{(r . i),(A,s,I)} is non empty finite set
{(r . i)} is non empty trivial finite 1 -element set
{{(r . i),(A,s,I)},{(r . i)}} is non empty finite V44() set
i + 1 is non empty ordinal natural V22() finite cardinal V79() V80() ext-real positive non negative V89() V90() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of NAT
r . (i + 1) is set
P . ((r . i),(A,s,I)) is set
P . [(r . i),(A,s,I)] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is Element of the carrier of A
C is Element of the carrier of A
(A,C,P) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*C,P*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*P*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,P] is non empty V22() set
{1,P} is non empty finite set
{{1,P},{1}} is non empty finite V44() set
{[1,P]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> ^ <*P*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*C,P*> is set
(A,P,C) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*P,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*P*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*P,C*> is set
I is Element of S
[I,(A,P,C)] is non empty V22() set
{I,(A,P,C)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(A,P,C)},{I}} is non empty finite V44() set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
f . (I,P) is Element of S
[I,P] is non empty V22() set
{I,P} is non empty finite set
{{I,P},{I}} is non empty finite V44() set
f . [I,P] is set
A is non empty partial quasi_total non-empty () () () () UAStr
the carrier of A is non empty set
S is non empty set
bool S is non empty set
[:S, the carrier of A:] is non empty Relation-like disjoint_with_NAT set
T is Element of bool S
f is non empty Relation-like [:S, the carrier of A:] -defined S -valued Function-like total quasi_total (A,S) (A,S) (A,S,T)
P is set
C is Element of the carrier of A
I is Element of the carrier of A
(A,I,C) is Element of the carrier of A
the charact of A is non empty Relation-like non-empty NAT -defined PFuncs (( the carrier of A *), the carrier of A) -valued Function-like finite Function-yielding FinSequence-like FinSubsequence-like homogeneous quasi_total FinSequence of PFuncs (( the carrier of A *), the carrier of A)
the carrier of A * is non empty functional FinSequence-membered FinSequenceSet of the carrier of A
PFuncs (( the carrier of A *), the carrier of A) is set
dom the charact of A is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V144() V145() V146() V147() V148() Element of bool NAT
In (2,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (2,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
rng the charact of A is non empty with_non-empty_elements finite Element of bool (PFuncs (( the carrier of A *), the carrier of A))
bool (PFuncs (( the carrier of A *), the carrier of A)) is non empty set
the charact of A . (In (2,(dom the charact of A))) is non empty homogeneous set
<*I,C*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*I*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,I] is non empty V22() set
{1,I} is non empty finite set
{{1,I},{1}} is non empty finite V44() set
{[1,I]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*C*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like set
[1,C] is non empty V22() set
{1,C} is non empty finite set
{{1,C},{1}} is non empty finite V44() set
{[1,C]} is non empty trivial Relation-like Function-like constant finite 1 -element set
<*I*> ^ <*C*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (2,(dom the charact of A))),A)) . <*I,C*> is set
(A,C,I) is Element of the carrier of A
In (4,(dom the charact of A)) is ordinal natural V22() finite cardinal V79() V80() ext-real non negative V89() V90() Element of dom the charact of A
Den ((In (4,(dom the charact of A))),A) is Relation-like the carrier of A * -defined the carrier of A -valued Function-like Element of rng the charact of A
the charact of A . (In (4,(dom the charact of A))) is non empty homogeneous set
<*C,I*> is non empty Relation-like NAT -defined the carrier of A -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of A
<*C*> ^ <*I*> is non empty Relation-like NAT -defined Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like set
(Den ((In (4,(dom the charact of A))),A)) . <*C,I*> is set
s is Element of S
[s,(A,C,I)] is non empty V22() set
{s,(A,C,I)} is non empty finite set
{s} is non empty trivial finite 1 -element set
{{s,(A,C,I)},{s}} is non empty finite V44() set
(A,S,T,f) is Relation-like S -defined the carrier of A -valued Element of bool [:S, the carrier of A:]
bool [:S, the carrier of A:] is non empty set
f . (s,C) is Element of S
[s,C] is non empty V22() set
{s,C} is non empty finite set
{{s,C},{s}} is non empty finite V44() set
f . [s,C] is set