REAL is non empty V12() non finite complex-membered ext-real-membered real-membered V56() set
NAT is non empty V12() epsilon-transitive epsilon-connected ordinal non finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal limit_cardinal Element of K19(REAL)
K19(REAL) is V12() non finite set
COMPLEX is non empty V12() non finite complex-membered V56() set
RAT is non empty V12() non finite complex-membered ext-real-membered real-membered rational-membered V56() set
INT is non empty V12() non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V56() set
K20(NAT,REAL) is Relation-like V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(NAT,REAL)) is V12() non finite set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
the Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{{},1} is non empty finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
NAT is non empty V12() epsilon-transitive epsilon-connected ordinal non finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal limit_cardinal set
K19(NAT) is V12() non finite set
K19(NAT) is V12() non finite set
K249(NAT) is V70() set
K20(COMPLEX,COMPLEX) is Relation-like V12() complex-valued non finite set
K19(K20(COMPLEX,COMPLEX)) is V12() non finite set
K20(K20(COMPLEX,COMPLEX),COMPLEX) is Relation-like V12() complex-valued non finite set
K19(K20(K20(COMPLEX,COMPLEX),COMPLEX)) is V12() non finite set
K20(REAL,REAL) is Relation-like V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(REAL,REAL)) is V12() non finite set
K20(K20(REAL,REAL),REAL) is Relation-like V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(K20(REAL,REAL),REAL)) is V12() non finite set
K20(RAT,RAT) is Relation-like RAT -valued V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(RAT,RAT)) is V12() non finite set
K20(K20(RAT,RAT),RAT) is Relation-like RAT -valued V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(K20(RAT,RAT),RAT)) is V12() non finite set
K20(INT,INT) is Relation-like RAT -valued INT -valued V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(INT,INT)) is V12() non finite set
K20(K20(INT,INT),INT) is Relation-like RAT -valued INT -valued V12() complex-valued ext-real-valued real-valued non finite set
K19(K20(K20(INT,INT),INT)) is V12() non finite set
K20(NAT,NAT) is Relation-like RAT -valued INT -valued V12() complex-valued ext-real-valued real-valued natural-valued non finite set
K20(K20(NAT,NAT),NAT) is Relation-like RAT -valued INT -valued V12() complex-valued ext-real-valued real-valued natural-valued non finite set
K19(K20(K20(NAT,NAT),NAT)) is V12() non finite set
2 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
3 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
rng {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V12() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued V37() decreasing non-decreasing non-increasing finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V82() V86() set
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
{{}} is functional non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
addcomplex is Relation-like K20(COMPLEX,COMPLEX) -defined COMPLEX -valued Function-like V14(K20(COMPLEX,COMPLEX)) quasi_total complex-valued having_a_unity commutative associative Element of K19(K20(K20(COMPLEX,COMPLEX),COMPLEX))
addreal is Relation-like K20(REAL,REAL) -defined REAL -valued Function-like V14(K20(REAL,REAL)) quasi_total complex-valued ext-real-valued real-valued having_a_unity commutative associative Element of K19(K20(K20(REAL,REAL),REAL))
addrat is Relation-like K20(RAT,RAT) -defined RAT -valued Function-like V14(K20(RAT,RAT)) quasi_total complex-valued ext-real-valued real-valued having_a_unity commutative associative Element of K19(K20(K20(RAT,RAT),RAT))
addint is Relation-like K20(INT,INT) -defined INT -valued Function-like V14(K20(INT,INT)) quasi_total complex-valued ext-real-valued real-valued having_a_unity commutative associative Element of K19(K20(K20(INT,INT),INT))
addnat is Relation-like K20(NAT,NAT) -defined NAT -valued INT -valued Function-like V14(K20(NAT,NAT)) quasi_total complex-valued ext-real-valued real-valued natural-valued having_a_unity commutative associative Element of K19(K20(K20(NAT,NAT),NAT))
the_unity_wrt addcomplex is complex Element of COMPLEX
the_unity_wrt addreal is ext-real complex real Element of REAL
the_unity_wrt addrat is ext-real complex real rational Element of RAT
the_unity_wrt addint is ext-real complex real integer rational Element of INT
the_unity_wrt addnat is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is finite set
p is finite set
card x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \ x is finite Element of K19(p)
K19(p) is finite V48() set
card (p \ x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(card p) - (card x) is ext-real complex real integer rational Element of INT
x is finite set
p is finite set
K20(x,p) is Relation-like finite set
K19(K20(x,p)) is finite V48() set
card x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is Relation-like x -defined p -valued Function-like quasi_total finite Element of K19(K20(x,p))
dom q is finite Element of K19(x)
K19(x) is finite V48() set
y is set
r is set
q . y is set
q . r is set
{y} is non empty V12() finite 1 -element set
x \ {y} is finite Element of K19(x)
c6 is finite set
q .: c6 is finite Element of K19(p)
K19(p) is finite V48() set
c is set
rng q is finite Element of K19(p)
b is set
q . b is set
card c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is finite Element of K19(x)
K19(x) is finite V48() set
q .: (dom q) is finite Element of K19(p)
K19(p) is finite V48() set
card (dom q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card (q .: (dom q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng q is finite Element of K19(p)
y is set
card (rng q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{y} is non empty V12() finite 1 -element set
p \ {y} is finite Element of K19(p)
card (p \ {y}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite cardinal set
p is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite cardinal set
p is set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is Relation-like Function-like set
rng p is set
dom p is set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
r is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(c6 + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(c6 + 1) - 1 is ext-real complex real integer rational Element of INT
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . c6) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
max (c,((r . c6) + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(c6 + 1) - 1 is ext-real complex real integer rational Element of INT
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(c6 + 1) - 1 is ext-real complex real integer rational Element of INT
0 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len r) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
c is set
b is set
r . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len r) - 1 is ext-real complex real integer rational Element of INT
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng p is finite set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
q is set
p . q is set
x is set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng p is finite set
q is set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . y is set
F2() is non empty set
F1() is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x . p is set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . q is set
F3(q) is Element of F2()
p is Relation-like NAT -defined F2() -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p . q is set
F3(q) is Element of F2()
the Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
x is Relation-like Function-like T-Sequence-like complex-valued set
- x is Relation-like Function-like complex-valued set
- 1 is ext-real non positive complex real set
(- 1) (#) x is Relation-like Function-like complex-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom (- x) is set
x " is Relation-like Function-like complex-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom (x ") is set
x ^2 is Relation-like Function-like complex-valued set
x (#) x is Relation-like Function-like complex-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom (x ^2) is set
|.x.| is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom |.x.| is set
p is Relation-like Function-like T-Sequence-like complex-valued set
x + p is Relation-like Function-like complex-valued set
dom (x + p) is set
dom x is epsilon-transitive epsilon-connected ordinal set
dom p is epsilon-transitive epsilon-connected ordinal set
(dom x) /\ (dom p) is epsilon-transitive epsilon-connected ordinal set
x - p is Relation-like Function-like complex-valued set
- p is Relation-like Function-like T-Sequence-like complex-valued set
(- 1) (#) p is Relation-like Function-like complex-valued set
x + (- p) is Relation-like Function-like T-Sequence-like complex-valued set
x (#) p is Relation-like Function-like complex-valued set
dom (x (#) p) is set
dom x is epsilon-transitive epsilon-connected ordinal set
dom p is epsilon-transitive epsilon-connected ordinal set
(dom x) /\ (dom p) is epsilon-transitive epsilon-connected ordinal set
x /" p is Relation-like Function-like complex-valued set
p " is Relation-like Function-like T-Sequence-like complex-valued set
x (#) (p ") is Relation-like Function-like T-Sequence-like complex-valued set
x is Relation-like Function-like complex-valued finite set
- x is Relation-like Function-like complex-valued set
(- 1) (#) x is Relation-like Function-like complex-valued set
dom x is finite set
dom (- x) is set
x " is Relation-like Function-like complex-valued set
dom x is finite set
dom (x ") is set
x ^2 is Relation-like Function-like complex-valued set
x (#) x is Relation-like Function-like complex-valued set
dom x is finite set
dom (x ^2) is set
|.x.| is Relation-like Function-like complex-valued ext-real-valued real-valued set
dom x is finite set
dom |.x.| is set
p is Relation-like Function-like complex-valued set
x + p is Relation-like Function-like complex-valued set
dom (x + p) is set
dom x is finite set
dom p is set
(dom x) /\ (dom p) is finite set
x - p is Relation-like Function-like complex-valued set
- p is Relation-like Function-like complex-valued set
(- 1) (#) p is Relation-like Function-like complex-valued set
x + (- p) is Relation-like Function-like complex-valued finite set
x (#) p is Relation-like Function-like complex-valued set
dom (x (#) p) is set
dom x is finite set
dom p is set
(dom x) /\ (dom p) is finite set
x /" p is Relation-like Function-like complex-valued set
p " is Relation-like Function-like complex-valued set
x (#) (p ") is Relation-like Function-like complex-valued finite set
p /" x is Relation-like Function-like complex-valued set
p (#) (x ") is Relation-like Function-like complex-valued finite set
x is Relation-like Function-like T-Sequence-like complex-valued set
p is complex set
p + x is Relation-like Function-like complex-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom (p + x) is set
x - p is Relation-like Function-like complex-valued set
- p is complex set
(- p) + x is Relation-like Function-like T-Sequence-like complex-valued set
p (#) x is Relation-like Function-like complex-valued set
dom x is epsilon-transitive epsilon-connected ordinal set
dom (p (#) x) is set
x is Relation-like Function-like complex-valued finite set
p is complex set
p + x is Relation-like Function-like complex-valued set
dom x is finite set
dom (p + x) is set
x - p is Relation-like Function-like complex-valued set
- p is complex set
(- p) + x is Relation-like Function-like complex-valued finite set
p (#) x is Relation-like Function-like complex-valued set
dom x is finite set
dom (p (#) x) is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
p . y is set
y + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) - (y + 1) is ext-real complex real integer rational Element of INT
x . ((len x) - (y + 1)) is set
q . y is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng x is finite set
rng (x) is finite set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is set
q is set
x . q is set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) -' (y + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) - (y + 1) is ext-real complex real integer rational Element of INT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . r is set
r + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) - (r + 1) is ext-real complex real integer rational Element of INT
x . ((len x) - (r + 1)) is set
p is set
q is set
(x) . q is set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) -' (y + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) - (y + 1) is ext-real complex real integer rational Element of INT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . y is set
x . ((len x) - (y + 1)) is set
x is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng p is finite Element of K19(x)
K19(x) is set
rng (p) is finite set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) -' p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . r is set
r + p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . (r + p) is set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q . r is set
r + p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . (r + p) is set
y . r is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(len p) -' x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) - x is ext-real complex real Element of REAL
(len p) -' x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(q,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(q,p) . x is set
q . (x + p) is set
(len q) - p is ext-real complex real Element of REAL
len (q,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (q,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x is Relation-like NAT -defined Function-like one-to-one T-Sequence-like finite V86() set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is set
dom (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is set
(x,p) . q is set
(x,p) . y is set
dom (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) -' p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) - p is ext-real complex real Element of REAL
r + p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c6 + p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x,p) . c6 is set
x . (c6 + p) is set
(x,p) . r is set
x . (r + p) is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng (p,x) is finite set
rng p is finite set
q is set
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
y is set
(p,x) . y is set
len (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) -' x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len p) - x is ext-real complex real Element of REAL
r + x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(p,x) . r is set
p . (r + x) is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,0) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (x,0) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x,0) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (x,0) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x,0) . p is set
p + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . (p + 0) is set
x . p is set
(len x) - 0 is ext-real non negative complex real integer rational Element of INT
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) + x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
((p ^ q),((len p) + x)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(q,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len (p ^ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p ^ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) + (len q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ((p ^ q),((len p) + x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p ^ q),((len p) + x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (p ^ q)) - ((len p) + x) is ext-real complex real integer rational Element of INT
(len q) + (len p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len q) + (len p)) - ((len p) + x) is ext-real complex real integer rational Element of INT
(len q) - x is ext-real complex real Element of REAL
len (q,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (q,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x + y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom ((p ^ q),((len p) + x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
((p ^ q),((len p) + x)) . y is set
((len p) + x) + y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p ^ q) . (((len p) + x) + y) is set
(len p) + (x + y) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p ^ q) . ((len p) + (x + y)) is set
q . (x + y) is set
(q,x) . y is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x ^ p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
((x ^ p),(len x)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(len x) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x ^ p),((len x) + 0)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p,0) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
rng p is finite set
(p,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p | x) ^ (p,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p ^ {} is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (p | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) - x is ext-real complex real Element of REAL
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((p | x) ^ (p,x)) . r is set
(p | x) . r is set
p . r is set
r - x is ext-real complex real Element of REAL
max (0,(r - x)) is ext-real complex real set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(len (p | x)) + c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | x) ^ (p,x)) . r is set
(p,x) . c6 is set
c6 + x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . (c6 + x) is set
p . r is set
((p | x) ^ (p,x)) . r is set
p . r is set
((p | x) ^ (p,x)) . r is set
p . r is set
len ((p | x) ^ (p,x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p | x) ^ (p,x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + ((len p) - x) is ext-real complex real Element of REAL
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) - q is ext-real complex real Element of REAL
len (p,q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (p,q) is finite set
rng p is finite Element of K19(x)
K19(x) is set
c6 is set
dom (p,q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p,q) . c is set
r + q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c + q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
p . (c + q) is set
c6 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng c6 is finite set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
<%> x is Relation-like non-empty empty-yielding NAT -defined x -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x | q is Relation-like q -defined NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
rng x is finite set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x | q),(p -' 1)) is Relation-like NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x,p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x | q is Relation-like q -defined NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
rng x is finite set
p -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x | q),(p -' 1)) is Relation-like NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
len (x | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p - 1 is ext-real complex real Element of REAL
q + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(q + 1) - 1 is ext-real complex real integer rational Element of INT
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x,(p -' 1)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x,p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x | q is Relation-like q -defined NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
rng x is finite set
((x | q),(p -' 1)) is Relation-like NAT -defined rng x -valued Function-like T-Sequence-like finite V86() set
q + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(q + 1) -' p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x,(p -' 1)) | ((q + 1) -' p) is Relation-like (q + 1) -' p -defined NAT -defined rng (x,(p -' 1)) -valued Function-like T-Sequence-like finite V86() set
rng (x,(p -' 1)) is finite set
len (x | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p + 1) - 1 is ext-real complex real integer rational Element of INT
p - 1 is ext-real complex real Element of REAL
(q + 1) - p is ext-real complex real Element of REAL
q - (p - 1) is ext-real complex real Element of REAL
len (x,p,q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x,p,q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (x,(p -' 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x,(p -' 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) - (p -' 1) is ext-real complex real integer rational Element of INT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x,p,q) . c6 is set
((x,(p -' 1)) | ((q + 1) -' p)) . c6 is set
c6 + (p -' 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(q - (p - 1)) + (p -' 1) is ext-real complex real Element of REAL
c6 + (p - 1) is ext-real complex real Element of REAL
((x | q),(p -' 1)) . c6 is set
(x | q) . (c6 + (p -' 1)) is set
(x,(p -' 1)) . c6 is set
x . (c6 + (p -' 1)) is set
len ((x,(p -' 1)) | ((q + 1) -' p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((x,(p -' 1)) | ((q + 1) -' p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p,1,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
rng p is finite set
1 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | x),(1 -' 1)) is Relation-like NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,1,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
rng p is finite set
1 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | x),(1 -' 1)) is Relation-like NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p,0,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
rng p is finite set
0 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | x),(0 -' 1)) is Relation-like NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
(p,1,x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
1 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | x),(1 -' 1)) is Relation-like NAT -defined rng p -valued Function-like T-Sequence-like finite V86() set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x ^ p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) + (len p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x ^ p),((len x) + 1),((len x) + (len p))) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x ^ p) | ((len x) + (len p)) is Relation-like (len x) + (len p) -defined NAT -defined rng (x ^ p) -valued Function-like T-Sequence-like finite V86() set
rng (x ^ p) is finite set
((len x) + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(((x ^ p) | ((len x) + (len p))),(((len x) + 1) -' 1)) is Relation-like NAT -defined rng (x ^ p) -valued Function-like T-Sequence-like finite V86() set
len (x ^ p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x ^ p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,q,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p | y is Relation-like y -defined NAT -defined x -valued rng p -valued Function-like T-Sequence-like finite V86() set
rng p is finite set
q -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p | y),(q -' 1)) is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is set
{p} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
q \ {p} is Element of K19(q)
K19(q) is set
r is set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng c is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
<%b%> is Relation-like NAT -defined NAT -valued Function-like constant non empty V12() T-Sequence-like complex-valued ext-real-valued real-valued natural-valued V37() decreasing non-decreasing non-increasing finite 1 -element V86() set
c ^ <%b%> is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
kp is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len kp) - 1 is ext-real complex real integer rational Element of INT
len (c ^ <%b%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (c ^ <%b%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (c ^ <%b%>)) - 1 is ext-real complex real integer rational Element of INT
len <%b%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%b%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len c) + (len <%b%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len c) + (len <%b%>)) - 1 is ext-real complex real integer rational Element of INT
(len c) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len c) + 1) - 1 is ext-real complex real integer rational Element of INT
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(len kp) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len (c ^ <%b%>)) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len c) + (len <%b%>)) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len c) + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(c ^ <%b%>) . (((len c) + (len <%b%>)) -' 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(c ^ <%b%>) . (((len c) + 1) -' 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(c ^ <%b%>) . (len c) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{b} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
rng kp is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng <%b%> is non empty V12() finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element Element of K19(NAT)
(rng c) \/ (rng <%b%>) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(q \ {p}) \/ {b} is non empty set
q \/ {b} is non empty set
y is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is set
<%> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
rng (<%> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty V12() ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued V37() decreasing non-decreasing non-increasing finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V82() V86() Element of K19(NAT)
len (<%> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom (<%> NAT) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(<%> NAT) . q is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(<%> NAT) . y is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng y is finite set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is set
<%r%> is Relation-like NAT -defined Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
y ^ <%r%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng (y ^ <%r%>) is finite set
len (y ^ <%r%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (y ^ <%r%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng b is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len <%r%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len y) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + (len y) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng <%r%> is non empty V12() finite 1 -element set
{r} is non empty V12() finite 1 -element set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len <%r%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len y) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len y) + (len <%r%>)) - 1 is ext-real complex real integer rational Element of INT
(y ^ <%r%>) . (((len y) + (len <%r%>)) - 1) is set
(len (y ^ <%r%>)) - 1 is ext-real complex real integer rational Element of INT
(y ^ <%r%>) . ((len (y ^ <%r%>)) - 1) is set
(len (y ^ <%r%>)) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (y ^ <%r%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x is set
(y ^ <%r%>) . x is set
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len y) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len y) + 1) - 1 is ext-real complex real integer rational Element of INT
(len (y ^ <%r%>)) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len y) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len <%r%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len y) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(y ^ <%r%>) . (len y) is set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x9 is set
y . x9 is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(y ^ <%r%>) . x is set
{r} is non empty V12() finite 1 -element set
(rng y) \/ {r} is non empty finite set
((rng y) \/ {r}) \ {r} is finite Element of K19(((rng y) \/ {r}))
K19(((rng y) \/ {r})) is finite V48() set
x9 is set
x9 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is set
rng x9 is finite set
dom x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x1 is set
x9 . x1 is set
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element Element of K19(NAT)
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x9) + x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . ((len x9) + x1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
<%r%> . x1 is set
len <%r%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{0} is functional non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
0 \/ {0} is non empty finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
rng <%r%> is non empty V12() finite 1 -element set
(rng y) \/ (rng <%r%>) is non empty finite set
dom b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
X19 is set
b . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len b) - 1 is ext-real complex real integer rational Element of INT
(len b) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . ((len b) - 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
nx20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
nx20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len <%r%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%r%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len x9) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 ^ <%r%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x ^ <%r%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len (x ^ <%r%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x ^ <%r%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b . (len x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x1 is set
x . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(rng x) \/ {r} is non empty finite set
((rng x) \/ {r}) \ {r} is finite Element of K19(((rng x) \/ {r}))
K19(((rng x) \/ {r})) is finite V48() set
x1 is set
rng <%r%> is non empty V12() finite 1 -element set
(rng y) \/ (rng <%r%>) is non empty finite set
c6 \ {r} is Element of K19(c6)
K19(c6) is set
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . x1 is set
x20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . X19 is set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(y ^ <%r%>) . x1 is set
(len y) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len y) + (len <%r%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(y ^ <%r%>) . X19 is set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(rng x) \/ (rng <%r%>) is non empty finite set
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
y is set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng c6 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(x) is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
p is set
q is set
(x) . p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
card (len (x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(p) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is finite set
card q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng (p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
y is set
(p) . y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
{x} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
({x}) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
({x}) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ({x}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ({x}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card {x} is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ({x}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng ({x}) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is set
p is set
x /\ p is set
(x /\ p) /\ NAT is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
the ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of (x /\ p) /\ NAT is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of (x /\ p) /\ NAT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
x /\ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
the ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of x /\ p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of x /\ p
p is set
q is set
x is set
x /\ p is set
x /\ q is set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(p) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q is set
{q} is non empty V12() finite 1 -element set
q is set
{q} is non empty V12() finite 1 -element set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng (p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{y} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
c6 is set
({y}) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
({y}) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) | (card p) is Relation-like card p -defined NAT -defined NAT -valued RAT -valued rng ((p \/ q)) -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng (((p \/ q)) | (card p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((rng ((p \/ q))))
K19((rng ((p \/ q)))) is finite V48() set
(((p \/ q)) | (card p)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{ b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)) | (card p)) . b2 & b2 in card p ) } is set
len ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (p \/ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19((card p))
K19((card p)) is finite V48() set
(((p \/ q)) | (card p)) .: (dom (((p \/ q)) | (card p))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
b is set
kp is set
(((p \/ q)) | (card p)) . kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card { b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)) | (card p)) . b2 & b2 in card p ) } is epsilon-transitive epsilon-connected ordinal cardinal set
card (len (((p \/ q)) | (card p))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is set
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
x9 is set
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x is set
((p \/ q)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is finite set
c is finite set
c is finite set
c is finite set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((p \/ q)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) | (card p) is Relation-like card p -defined NAT -defined NAT -valued RAT -valued rng ((p \/ q)) -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
{ b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)) | (card p)) . b2 & b2 in card p ) } is set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (p \/ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (((p \/ q)) | (card p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((rng ((p \/ q))))
K19((rng ((p \/ q)))) is finite V48() set
c is set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is set
dom (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19((card p))
K19((card p)) is finite V48() set
kp is set
(((p \/ q)) | (card p)) . kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is set
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
x9 is set
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x is set
((p \/ q)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)) | (card p)) . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (((p \/ q)) | (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19((card p))
K19((card p)) is finite V48() set
(((p \/ q)) | (card p)) .: (dom (((p \/ q)) | (card p))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card { b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)) | (card p)) . b2 & b2 in card p ) } is epsilon-transitive epsilon-connected ordinal cardinal set
card (len (((p \/ q)) | (card p))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is finite set
c is finite set
c is finite set
c is finite set
(((p \/ q)) | (card p)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((p \/ q)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) | (len (p)) is Relation-like len (p) -defined NAT -defined NAT -valued RAT -valued rng ((p \/ q)) -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
y . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card (p \/ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng y is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
card q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(((p \/ q)),(card p)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
rng (((p \/ q)),(card p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
(((p \/ q)),(card p)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (x + (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p /\ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q /\ NAT is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
p /\ (q /\ NAT) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
(p /\ q) /\ NAT is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
{ b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)),(card p)) . b2 & b2 in card q ) } is set
dom (((p \/ q)),(card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(((p \/ q)),(card p)) .: (dom (((p \/ q)),(card p))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
rng ((p \/ q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (p \/ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (((p \/ q)),(card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (((p \/ q)),(card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len ((p \/ q))) -' (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len ((p \/ q))) - (card p) is ext-real complex real integer rational Element of INT
(p \/ q) \ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((p \/ q))
K19((p \/ q)) is finite V48() set
p \ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(p)
K19(p) is finite V48() set
q \ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(q)
K19(q) is finite V48() set
(p \ p) \/ (q \ p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
{} \/ (q \ p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
b is set
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)),(card p)) . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp is set
k1 is set
(((p \/ q)),(card p)) . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)),(card p)) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card { b1 where b1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q : ex b2 being ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set st
( b1 = (((p \/ q)),(card p)) . b2 & b2 in card q ) } is epsilon-transitive epsilon-connected ordinal cardinal set
card (len (((p \/ q)),(card p))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(card p) + (card q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp is set
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
k1 is set
((p \/ q)) . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of p \/ q
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)),(card p)) . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)),(card p)) . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (X29 + (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x20 -' (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x20 - (card p) is ext-real complex real Element of REAL
((card p) + (card q)) - (card p) is ext-real complex real integer rational Element of INT
(x20 -' (card p)) + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(((p \/ q)),(card p)) . (x20 -' (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . x20 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(card p) + X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b is finite set
b is finite set
b is finite set
b is finite set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(q) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(q) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p \/ q)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + (len (p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (x + (len (p))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(((p \/ q)),(len (p))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
r . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (x + (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p \/ q)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (p \/ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p /\ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q /\ NAT is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
p /\ (q /\ NAT) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
(p /\ q) /\ NAT is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
(card q) + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len ((p \/ q))) -' (len (p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(card p) + (card q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((card p) + (card q)) -' (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 + (card p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (c6 + (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((p \/ q)) . (c + (card p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(p) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x \/ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((x \/ p)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((x \/ p)) . (len (x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
0 + (len (x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . (0 + (len (x))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(r) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
len (r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(r) . p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r) . q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x \/ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((x \/ p)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((x \/ p)) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
x \/ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((x \/ p)) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(p) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) ^ (p) is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x /\ p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
the ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of x /\ p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of x /\ p
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is finite set
c is finite set
c6 \/ c is finite set
card (c6 \/ c) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
card c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(card c6) + (card c) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (x)) + (card c) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (x)) + (len (p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((x \/ p)) . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . (b + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . (b + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng ((x \/ p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x1 is set
(x) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
x1 is set
((x \/ p)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (x)) + b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . ((len (x)) + b) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p) . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len (x)) + (b + 1) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . ((len (x)) + (b + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p) . (b + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng (p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng ((x \/ p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is set
(x) . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len (x)) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is set
(p) . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len (x)) + b) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 is set
((x \/ p)) . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len (x)) + b) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len (x)) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . ((len (x)) + 0) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p) . 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
b is set
(p) . b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len (x)) + kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . ((len (x)) + kp) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (x)) + (len (p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((x \/ p)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
k1 is set
(x) . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((x \/ p)) . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p /\ (len x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((p /\ (len x))) (#) x is Relation-like NAT -defined Function-like finite set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng ((p /\ (len x))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is set
(x,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p /\ (len x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((p /\ (len x))) (#) x is Relation-like NAT -defined Function-like finite set
len (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (p /\ (len x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng ((p /\ (len x))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (x,p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len ((p /\ (len x))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((p /\ (len x))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x,p) . q is set
((p /\ (len x))) . q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . (((p /\ (len x))) . q) is set
x is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
q is set
(p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q /\ (len p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((q /\ (len p))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((q /\ (len p))) (#) p is Relation-like NAT -defined x -valued Function-like finite set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,{}) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
{} /\ (len x) is Relation-like ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(({} /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(({} /\ (len x))) (#) x is Relation-like NAT -defined Function-like finite set
len (x,{}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (x,{}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
x is set
({},x) is Relation-like NAT -defined RAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
len {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
x /\ (len {}) is Relation-like finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((x /\ (len {}))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
((x /\ (len {}))) (#) {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is set
x --> p is Relation-like x -defined {p} -valued Function-like V14(x) quasi_total T-Sequence-like finite V86() Element of K19(K20(x,{p}))
{p} is non empty V12() finite 1 -element set
K20(x,{p}) is Relation-like finite set
K19(K20(x,{p})) is finite V48() set
F1() is set
<%> F1() is Relation-like non-empty empty-yielding NAT -defined F1() -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is Relation-like NAT -defined F1() -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
y is set
<%y%> is Relation-like NAT -defined Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
q ^ <%y%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
c6 is Relation-like NAT -defined F1() -valued Function-like T-Sequence-like finite V86() set
rng c6 is finite Element of K19(F1())
K19(F1()) is set
{y} is non empty V12() finite 1 -element set
len c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is Relation-like NAT -defined F1() -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len r) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is Relation-like NAT -defined F1() -valued Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is set
card p is epsilon-transitive epsilon-connected ordinal cardinal set
p is Relation-like NAT -defined F1() -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is non empty set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
q is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
the_unity_wrt q is Element of x
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
p . 0 is set
(len p) - 1 is ext-real complex real integer rational Element of INT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . 0 is set
(len r) - 1 is ext-real complex real integer rational Element of INT
r | y is Relation-like y -defined NAT -defined x -valued rng r -valued Function-like T-Sequence-like finite V86() set
rng r is finite set
len (r | y) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (r | y) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng r is finite Element of K19(x)
K19(x) is set
NAT --> (r . 0) is Relation-like NAT -defined {(r . 0)} -valued Function-like non empty V14( NAT ) quasi_total T-Sequence-like Element of K19(K20(NAT,{(r . 0)}))
{(r . 0)} is non empty V12() finite 1 -element set
K20(NAT,{(r . 0)}) is Relation-like V12() non finite set
K19(K20(NAT,{(r . 0)})) is V12() non finite set
c is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c . 0 is Element of x
kp is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
kp . 0 is Element of x
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
k1 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp . (k1 + 1) is Element of x
kp . k1 is Element of x
r . (k1 + 1) is set
q . ((kp . k1),(r . (k1 + 1))) is set
b is Element of x
kp . ((len r) - 1) is set
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
r . y is set
rng r is finite Element of K19(x)
K19(x) is set
(r | y) . 0 is set
(len (r | y)) - 1 is ext-real complex real integer rational Element of INT
kp is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
kp . 0 is Element of x
b is Element of x
kp . ((len (r | y)) - 1) is set
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is Element of x
q . (b,c) is Element of x
x9 is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
x9 . k1 is Element of x
x9 . y is Element of x
x1 is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
x1 . 0 is Element of x
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
X19 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 . (X19 + 1) is Element of x
x1 . X19 is Element of x
r . (X19 + 1) is set
q . ((x1 . X19),(r . (X19 + 1))) is set
dom (r | y) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(y)
K19(y) is finite V48() set
(r | y) . (X19 + 1) is set
kp . X19 is Element of x
kp . (X19 + 1) is Element of x
x is Element of x
x1 . ((len r) - 1) is set
b is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
b . 0 is Element of x
c is Element of x
b . ((len r) - 1) is set
y is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . 0 is set
(len y) - 1 is ext-real complex real integer rational Element of INT
y is Element of x
c6 is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c6 . 0 is Element of x
r is Element of x
c6 . ((len p) - 1) is set
y is Element of x
r is Element of x
((len p) - 1) + 1 is ext-real complex real integer rational Element of INT
c6 is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c6 . 0 is Element of x
c6 . ((len p) - 1) is set
c is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c . 0 is Element of x
c . ((len p) - 1) is set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . b is Element of x
c . b is Element of x
(b + 1) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . (b + 1) is Element of x
c . (b + 1) is Element of x
c . (b + 1) is Element of x
p . (b + 1) is set
q . ((c . b),(p . (b + 1))) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . 0 is Element of x
c . 0 is Element of x
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
q is Element of x
<%q%> is Relation-like NAT -defined x -valued Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
(x,<%q%>,p) is Element of x
len <%q%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%q%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
<%q%> . 0 is set
1 - 1 is ext-real complex real integer rational Element of INT
y is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
y . 0 is Element of x
y . (1 - 1) is set
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
q is Element of x
y is Element of x
<%q,y%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite 2 -element V86() set
(x,<%q,y%>,p) is Element of x
p . (q,y) is Element of x
<%q,y%> . 0 is set
len <%q,y%> is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%q,y%> is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 2 -element set
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
2 - 1 is ext-real complex real integer rational Element of INT
r is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
r . 0 is Element of x
r . (2 - 1) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . (0 + 1) is Element of x
<%q,y%> . (0 + 1) is set
p . ((r . 0),(<%q,y%> . (0 + 1))) is set
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
q is Element of x
y is Element of x
p . (q,y) is Element of x
r is Element of x
<%q,y,r%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite 3 -element V86() set
(x,<%q,y,r%>,p) is Element of x
p . ((p . (q,y)),r) is Element of x
<%q,y,r%> . 0 is set
<%q,y,r%> . 1 is set
len <%q,y,r%> is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%q,y,r%> is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 3 -element set
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
3 - 1 is ext-real complex real integer rational Element of INT
c is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c . 0 is Element of x
c . (3 - 1) is set
1 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . (1 + 1) is Element of x
c . 1 is Element of x
<%q,y,r%> . (1 + 1) is set
p . ((c . 1),(<%q,y,r%> . (1 + 1))) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . (0 + 1) is Element of x
<%q,y,r%> . (0 + 1) is set
p . ((c . 0),(<%q,y,r%> . (0 + 1))) is set
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,p,q) is Element of x
y is Element of x
<%y%> is Relation-like NAT -defined x -valued Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
p ^ <%y%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(x,(p ^ <%y%>),q) is Element of x
q . ((x,p,q),y) is Element of x
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
{} ^ <%y%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
the_unity_wrt q is Element of x
(len p) - 1 is ext-real complex real integer rational Element of INT
(p ^ <%y%>) . (len p) is set
len (p ^ <%y%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p ^ <%y%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len <%y%> is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom <%y%> is non empty V12() ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal 1 -element set
(len p) + (len <%y%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len p) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
(p ^ <%y%>) . 0 is set
(len (p ^ <%y%>)) - 1 is ext-real complex real integer rational Element of INT
c is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c . 0 is Element of x
c . ((len (p ^ <%y%>)) - 1) is set
p . 0 is set
b is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
b . 0 is Element of x
b . ((len p) - 1) is set
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . kp is Element of x
c . kp is Element of x
(kp + 1) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . (kp + 1) is Element of x
c . (kp + 1) is Element of x
c . (kp + 1) is Element of x
(p ^ <%y%>) . (kp + 1) is set
q . ((c . kp),((p ^ <%y%>) . (kp + 1))) is set
- 1 is ext-real non positive complex real integer rational Element of INT
((len p) + 1) + (- 1) is ext-real complex real integer rational Element of INT
((kp + 1) + 1) + (- 1) is ext-real complex real integer rational Element of INT
b . (kp + 1) is Element of x
p . (kp + 1) is set
q . ((b . kp),(p . (kp + 1))) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b . 0 is Element of x
c . 0 is Element of x
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . c6 is Element of x
(p ^ <%y%>) . (c6 + 1) is set
q . ((c . c6),((p ^ <%y%>) . (c6 + 1))) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p ^ q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
y is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(p ^ q),y) is Element of x
(x,p,y) is Element of x
(x,q,y) is Element of x
y . ((x,p,y),(x,q,y)) is Element of x
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is Element of x
<%c6%> is Relation-like NAT -defined x -valued Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
r ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len (r ^ <%c6%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (r ^ <%c6%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c ^ (r ^ <%c6%>) is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
b is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(c ^ (r ^ <%c6%>)),b) is Element of x
(x,c,b) is Element of x
(x,(r ^ <%c6%>),b) is Element of x
b . ((x,c,b),(x,(r ^ <%c6%>),b)) is Element of x
c ^ r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len (c ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (c ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len c) + (len r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len c) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x,(c ^ r),b) is Element of x
b . ((x,(c ^ r),b),c6) is Element of x
(c ^ r) ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(x,((c ^ r) ^ <%c6%>),b) is Element of x
(x,r,b) is Element of x
b . ((x,c,b),(x,r,b)) is Element of x
b . ((b . ((x,c,b),(x,r,b))),c6) is Element of x
b . ((x,r,b),c6) is Element of x
b . ((x,c,b),(b . ((x,r,b),c6))) is Element of x
{} ^ <%c6%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
c ^ ({} ^ <%c6%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,(c ^ ({} ^ <%c6%>)),b) is Element of x
c ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(x,(c ^ <%c6%>),b) is Element of x
b . ((x,c,b),c6) is Element of x
(x,({} ^ <%c6%>),b) is Element of x
b . ((x,c,b),(x,({} ^ <%c6%>),b)) is Element of x
<%> x is Relation-like non-empty empty-yielding NAT -defined x -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
len (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r ^ (<%> x) is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
c6 is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(r ^ (<%> x)),c6) is Element of x
(x,r,c6) is Element of x
(x,(<%> x),c6) is Element of x
c6 . ((x,r,c6),(x,(<%> x),c6)) is Element of x
len (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
r ^ {} is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,(r ^ {}),c6) is Element of x
the_unity_wrt c6 is Element of x
c6 . ((x,r,c6),(the_unity_wrt c6)) is Element of x
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p ^ q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
y is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(p ^ q),y) is Element of x
(x,p,y) is Element of x
(x,q,y) is Element of x
y . ((x,p,y),(x,q,y)) is Element of x
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is Element of x
<%c6%> is Relation-like NAT -defined x -valued Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
r ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len (r ^ <%c6%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (r ^ <%c6%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c ^ (r ^ <%c6%>) is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
b is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(c ^ (r ^ <%c6%>)),b) is Element of x
(x,c,b) is Element of x
(x,(r ^ <%c6%>),b) is Element of x
b . ((x,c,b),(x,(r ^ <%c6%>),b)) is Element of x
c ^ r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len (c ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (c ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len c) + (len r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len c) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(x,(c ^ r),b) is Element of x
b . ((x,(c ^ r),b),c6) is Element of x
(c ^ r) ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(x,((c ^ r) ^ <%c6%>),b) is Element of x
(x,r,b) is Element of x
b . ((x,c,b),(x,r,b)) is Element of x
b . ((b . ((x,c,b),(x,r,b))),c6) is Element of x
b . ((x,r,b),c6) is Element of x
b . ((x,c,b),(b . ((x,r,b),c6))) is Element of x
{} ^ <%c6%> is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
c ^ ({} ^ <%c6%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,(c ^ ({} ^ <%c6%>)),b) is Element of x
c ^ <%c6%> is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
(x,(c ^ <%c6%>),b) is Element of x
b . ((x,c,b),c6) is Element of x
(x,({} ^ <%c6%>),b) is Element of x
b . ((x,c,b),(x,({} ^ <%c6%>),b)) is Element of x
<%> x is Relation-like non-empty empty-yielding NAT -defined x -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
len (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r ^ (<%> x) is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
c6 is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(r ^ (<%> x)),c6) is Element of x
(x,r,c6) is Element of x
(x,(<%> x),c6) is Element of x
c6 . ((x,r,c6),(x,(<%> x),c6)) is Element of x
len (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
r ^ {} is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x,(r ^ {}),c6) is Element of x
the_unity_wrt c6 is Element of x
c6 . ((x,r,c6),(the_unity_wrt c6)) is Element of x
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is non empty set
K20(p,p) is Relation-like set
K20(K20(p,p),p) is Relation-like set
K19(K20(K20(p,p),p)) is set
q is Relation-like NAT -defined p -valued Function-like T-Sequence-like finite V86() set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
q | x is Relation-like x -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
rng q is finite set
q . x is set
q | (x + 1) is Relation-like x + 1 -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
y is Relation-like K20(p,p) -defined p -valued Function-like V14(K20(p,p)) quasi_total Element of K19(K20(K20(p,p),p))
(p,(q | x),y) is Element of p
y . ((p,(q | x),y),(q . x)) is set
(p,(q | (x + 1)),y) is Element of p
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (q | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q | r is Relation-like r -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
len (q | r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q | r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,(q | r),y) is Element of p
q . r is set
y . ((p,(q | r),y),(q . r)) is set
r + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q | (r + 1) is Relation-like r + 1 -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
(p,(q | (r + 1)),y) is Element of p
len (q | (r + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q | (r + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q . (r + 1) is set
y . ((p,(q | (r + 1)),y),(q . (r + 1))) is set
(r + 1) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q | ((r + 1) + 1) is Relation-like (r + 1) + 1 -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
(p,(q | ((r + 1) + 1)),y) is Element of p
(dom q) /\ (r + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom (q | (r + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19((r + 1))
K19((r + 1)) is finite V48() set
(q | (r + 1)) . 0 is set
q . 0 is set
(dom q) /\ ((r + 1) + 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
K20(NAT,p) is Relation-like V12() non finite set
K19(K20(NAT,p)) is V12() non finite set
(r + 1) - 1 is ext-real complex real integer rational Element of INT
kp is Relation-like NAT -defined p -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,p))
kp . 0 is Element of p
kp . ((r + 1) - 1) is set
len (q | ((r + 1) + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q | ((r + 1) + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(q | ((r + 1) + 1)) . 0 is set
((r + 1) + 1) - 1 is ext-real complex real integer rational Element of INT
k1 is Relation-like NAT -defined p -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,p))
k1 . 0 is Element of p
k1 . (((r + 1) + 1) - 1) is set
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
kp . x9 is Element of p
k1 . x9 is Element of p
x9 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp . (x9 + 1) is Element of p
k1 . (x9 + 1) is Element of p
dom (q | ((r + 1) + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(((r + 1) + 1))
K19(((r + 1) + 1)) is finite V48() set
(q | ((r + 1) + 1)) . (x9 + 1) is set
q . (x9 + 1) is set
(q | (r + 1)) . (x9 + 1) is set
k1 . (x9 + 1) is Element of p
y . ((kp . x9),((q | (r + 1)) . (x9 + 1))) is set
dom (q | ((r + 1) + 1)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(((r + 1) + 1))
K19(((r + 1) + 1)) is finite V48() set
kp . 0 is Element of p
k1 . 0 is Element of p
kp . r is Element of p
k1 . r is Element of p
k1 . (r + 1) is Element of p
(q | ((r + 1) + 1)) . (r + 1) is set
y . ((k1 . r),((q | ((r + 1) + 1)) . (r + 1))) is set
q | 0 is Relation-like non-empty empty-yielding NAT -defined 0 -defined NAT -defined RAT -valued p -valued rng q -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
len (q | 0) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom (q | 0) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
(p,(q | 0),y) is Element of p
q . 0 is set
y . ((p,(q | 0),y),(q . 0)) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
q | (0 + 1) is Relation-like 0 + 1 -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
(p,(q | (0 + 1)),y) is Element of p
len (q | 0) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
rng q is finite Element of K19(p)
K19(p) is set
q | 1 is Relation-like 1 -defined NAT -defined p -valued rng q -valued Function-like T-Sequence-like finite V86() set
len (q | 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q | 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(q | 1) . 0 is set
<%((q | 1) . 0)%> is Relation-like NAT -defined Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
the_unity_wrt y is Element of p
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
XFS2FS p is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
q is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,p,q) is Element of x
q "**" (XFS2FS p) is Element of x
len (XFS2FS p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
K20(NAT,x) is Relation-like V12() non finite set
K19(K20(NAT,x)) is V12() non finite set
(XFS2FS p) . 1 is set
r is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
r . 1 is Element of x
r . (len p) is Element of x
p . 0 is set
(len p) - 1 is ext-real complex real integer rational Element of INT
c6 is Relation-like NAT -defined x -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,x))
c6 . 0 is Element of x
c6 . ((len p) - 1) is set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 . c is Element of x
c + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . (c + 1) is Element of x
c6 . (c + 1) is Element of x
(c + 1) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . ((c + 1) + 1) is Element of x
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((c + 1) + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . (((c + 1) + 1) -' 1) is set
(XFS2FS p) . ((c + 1) + 1) is set
c6 . (c + 1) is Element of x
p . (c + 1) is set
q . ((c6 . c),(p . (c + 1))) is set
q . ((r . (c + 1)),((XFS2FS p) . ((c + 1) + 1))) is set
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . 0 is Element of x
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . (0 + 1) is Element of x
1 -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . (1 -' 1) is set
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (XFS2FS p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
the_unity_wrt q is Element of x
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
K20((dom p),(dom p)) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
K19(K20((dom p),(dom p))) is finite V48() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
y is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,p,y) is Element of x
(x,q,y) is Element of x
r is Relation-like dom p -defined dom p -valued Function-like one-to-one V14( dom p) quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued finite Element of K19(K20((dom p),(dom p)))
r (#) p is Relation-like dom p -defined x -valued Function-like finite set
XFS2FS p is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
len (XFS2FS p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
XFS2FS q is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
Seg b is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered b -element Element of K19(NAT)
kp is set
1 + 0 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 - 1 is ext-real complex real integer rational Element of INT
x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x9 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . x9) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom p))
K19((dom p)) is finite V48() set
rng r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom p))
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x - 1 is ext-real complex real Element of REAL
r . (x - 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . (x - 1)) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
K20((Seg b),(Seg b)) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
K19(K20((Seg b),(Seg b))) is finite V48() set
kp is Relation-like Seg b -defined Seg b -valued Function-like V14( Seg b) quasi_total complex-valued ext-real-valued real-valued natural-valued finite Element of K19(K20((Seg b),(Seg b)))
k1 is set
dom kp is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((Seg b))
K19((Seg b)) is finite V48() set
x9 is set
kp . k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp . x9 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + 0 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x - 1 is ext-real complex real integer rational Element of INT
x1 - 1 is ext-real complex real integer rational Element of INT
X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X19 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom p))
K19((dom p)) is finite V48() set
X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
X29 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . X19 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . X19) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((r . X19) + 1) - 1 is ext-real complex real integer rational Element of INT
r . X29 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . X29) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((r . X29) + 1) - 1 is ext-real complex real integer rational Element of INT
(x - 1) + 1 is ext-real complex real integer rational Element of INT
(x1 - 1) + 1 is ext-real complex real integer rational Element of INT
card (Seg b) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (XFS2FS p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
Seg (len p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered len p -element Element of K19(NAT)
K20((dom (XFS2FS p)),(dom (XFS2FS p))) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite set
K19(K20((dom (XFS2FS p)),(dom (XFS2FS p)))) is finite V48() set
k1 is Relation-like dom (XFS2FS p) -defined dom (XFS2FS p) -valued Function-like one-to-one V14( dom (XFS2FS p)) quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued finite Element of K19(K20((dom (XFS2FS p)),(dom (XFS2FS p))))
dom k1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom (XFS2FS p)))
K19((dom (XFS2FS p))) is finite V48() set
dom (XFS2FS q) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
len (XFS2FS q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
Seg (len (XFS2FS q)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered len (XFS2FS q) -element Element of K19(NAT)
rng k1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom (XFS2FS p)))
(XFS2FS p) * k1 is Relation-like dom (XFS2FS p) -defined x -valued Function-like finite Element of K19(K20((dom (XFS2FS p)),x))
K20((dom (XFS2FS p)),x) is Relation-like set
K19(K20((dom (XFS2FS p)),x)) is set
dom ((XFS2FS p) * k1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom (XFS2FS p)))
rng r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom p))
dom (r (#) p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom p))
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x9 is set
(XFS2FS q) . x9 is set
((XFS2FS p) * k1) . x9 is set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x - 1 is ext-real complex real integer rational Element of INT
Seg (len (XFS2FS p)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered len (XFS2FS p) -element Element of K19(NAT)
r . (x - 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . (x - 1)) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 . x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(XFS2FS q) . x is set
(r (#) p) . (x -' 1) is set
x1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x1 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . (x -' 1) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(r . (x -' 1)) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((r . (x -' 1)) + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . (r . (x -' 1)) is set
(XFS2FS p) . ((r . (x - 1)) + 1) is set
y "**" (XFS2FS q) is Element of x
y "**" (XFS2FS p) is Element of x
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p ^ q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
y is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(p ^ q),y) is Element of x
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x,r,y) is Element of x
XFS2FS q is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
XFS2FS p is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
(x,p,y) is Element of x
y "**" (XFS2FS p) is Element of x
(x,q,y) is Element of x
y "**" (XFS2FS q) is Element of x
XFS2FS r is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of x
len (XFS2FS r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (XFS2FS r) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(XFS2FS r) . kp is set
(XFS2FS p) . kp is set
(XFS2FS q) . kp is set
y . (((XFS2FS p) . kp),((XFS2FS q) . kp)) is set
Seg (len (XFS2FS r)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered len (XFS2FS r) -element Element of K19(NAT)
kp - 1 is ext-real complex real integer rational Element of INT
k1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
k1 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . k1 is set
p . k1 is set
q . k1 is set
y . ((p . k1),(q . k1)) is set
r . (kp -' 1) is set
p . (kp -' 1) is set
len (XFS2FS p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len (XFS2FS q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y "**" (XFS2FS r) is Element of x
y . ((y "**" (XFS2FS p)),(y "**" (XFS2FS q))) is Element of x
y . ((x,p,y),(x,q,y)) is Element of x
x is non empty set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is non empty set
K20(q,q) is Relation-like set
K20(K20(q,q),q) is Relation-like set
K19(K20(K20(q,q),q)) is set
y is non empty set
K20(y,y) is Relation-like set
K20(K20(y,y),y) is Relation-like set
K19(K20(K20(y,y),y)) is set
q /\ y is set
r is Relation-like K20(q,q) -defined q -valued Function-like V14(K20(q,q)) quasi_total Element of K19(K20(K20(q,q),q))
(q,p,r) is Element of q
c6 is Relation-like K20(y,y) -defined y -valued Function-like V14(K20(y,y)) quasi_total Element of K19(K20(K20(y,y),y))
(y,p,c6) is Element of y
rng p is finite Element of K19(x)
K19(x) is set
K20(NAT,q) is Relation-like V12() non finite set
K19(K20(NAT,q)) is V12() non finite set
p . 0 is set
(len p) - 1 is ext-real complex real integer rational Element of INT
c is Relation-like NAT -defined q -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,q))
c . 0 is Element of q
c . ((len p) - 1) is set
K20(NAT,y) is Relation-like V12() non finite set
K19(K20(NAT,y)) is V12() non finite set
b is Relation-like NAT -defined y -valued Function-like non empty V14( NAT ) quasi_total Element of K19(K20(NAT,y))
b . 0 is Element of y
b . ((len p) - 1) is set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c . 0 is Element of q
b . 0 is Element of y
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c . kp is Element of q
b . kp is Element of y
kp + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . (kp + 1) is Element of q
b . (kp + 1) is Element of y
p . (kp + 1) is set
c . (kp + 1) is Element of q
r . ((c . kp),(p . (kp + 1))) is set
c6 . ((b . kp),(p . (kp + 1))) is set
b . (kp + 1) is Element of y
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
kp + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
rng x is finite complex-membered set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
rng x is finite complex-membered ext-real-membered real-membered set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
x is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
len x is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom x is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(REAL,x,addreal) is ext-real complex real Element of REAL
rng x is finite set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
REAL /\ COMPLEX is complex-membered ext-real-membered real-membered set
p is set
q is set
addreal . (p,q) is set
y is ext-real complex real Element of REAL
r is ext-real complex real Element of REAL
y + r is ext-real complex real Element of REAL
addcomplex . (p,q) is set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(RAT,x,addrat) is ext-real complex real rational Element of RAT
rng x is finite set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
RAT /\ COMPLEX is complex-membered ext-real-membered real-membered rational-membered set
p is set
q is set
addrat . (p,q) is set
y is ext-real complex real rational Element of RAT
r is ext-real complex real rational Element of RAT
y + r is ext-real complex real rational Element of RAT
addcomplex . (p,q) is set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(INT,x,addint) is ext-real complex real integer rational Element of INT
rng x is finite set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
INT /\ COMPLEX is complex-membered ext-real-membered real-membered rational-membered integer-membered set
p is set
q is set
addint . (p,q) is set
y is ext-real complex real integer rational Element of INT
r is ext-real complex real integer rational Element of INT
y + r is ext-real complex real integer rational Element of INT
addcomplex . (p,q) is set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(NAT,x,addnat) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng x is finite set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
NAT /\ COMPLEX is complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(REAL)
p is set
q is set
addnat . (p,q) is set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y + r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
addcomplex . (p,q) is set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(REAL,x,addreal) is ext-real complex real Element of REAL
x is Relation-like NAT -defined RAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(RAT,x,addrat) is ext-real complex real rational Element of RAT
x is Relation-like NAT -defined INT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(INT,x,addint) is ext-real complex real integer rational Element of INT
x is Relation-like NAT -defined RAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite V86() set
(x) is ext-real complex real rational Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(NAT,x,addnat) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x is Relation-like set
rng x is set
p is ext-real complex real set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
x is complex set
<%x%> is Relation-like NAT -defined Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
(<%x%>) is complex Element of COMPLEX
(COMPLEX,<%x%>,addcomplex) is complex Element of COMPLEX
x is complex set
p is complex set
<%x,p%> is Relation-like NAT -defined Function-like T-Sequence-like finite 2 -element V86() set
(<%x,p%>) is complex Element of COMPLEX
(COMPLEX,<%x,p%>,addcomplex) is complex Element of COMPLEX
x + p is complex Element of COMPLEX
addcomplex . (x,p) is set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
x ^ p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
((x ^ p)) is complex Element of COMPLEX
(COMPLEX,(x ^ p),addcomplex) is complex Element of COMPLEX
(p) is complex Element of COMPLEX
(COMPLEX,p,addcomplex) is complex Element of COMPLEX
(x) + (p) is complex Element of COMPLEX
addcomplex . ((x),(p)) is complex Element of COMPLEX
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(p) is ext-real complex real Element of COMPLEX
(COMPLEX,p,addcomplex) is complex Element of COMPLEX
q is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
q | (x + 1) is Relation-like NAT -defined x + 1 -defined NAT -defined REAL -valued Function-like complex-valued ext-real-valued real-valued finite Element of K19(K20(NAT,REAL))
Partial_Sums q is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
(Partial_Sums q) . x is ext-real complex real Element of REAL
dom q is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(dom q) /\ (x + 1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
p . 0 is ext-real complex real set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(REAL,p,addreal) is ext-real complex real Element of REAL
(len p) - 1 is ext-real complex real integer rational Element of INT
y is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
y . 0 is ext-real complex real Element of REAL
y . ((len p) - 1) is ext-real complex real set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y . r is ext-real complex real Element of REAL
(Partial_Sums q) . r is ext-real complex real Element of REAL
r + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y . (r + 1) is ext-real complex real Element of REAL
(Partial_Sums q) . (r + 1) is ext-real complex real Element of REAL
y . (r + 1) is ext-real complex real Element of REAL
p . (r + 1) is ext-real complex real set
addreal . ((y . r),(p . (r + 1))) is set
(y . r) + (p . (r + 1)) is ext-real complex real Element of REAL
q . (r + 1) is ext-real complex real Element of REAL
(y . r) + (q . (r + 1)) is ext-real complex real Element of REAL
(Partial_Sums q) . (r + 1) is ext-real complex real Element of REAL
(Partial_Sums q) . 0 is ext-real complex real Element of REAL
q . 0 is ext-real complex real Element of REAL
y . 0 is ext-real complex real Element of REAL
(Partial_Sums q) . 0 is ext-real complex real Element of REAL
y . x is ext-real complex real Element of REAL
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p) is ext-real complex real Element of COMPLEX
(COMPLEX,p,addcomplex) is complex Element of COMPLEX
r is Relation-like NAT -defined REAL -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(r) is ext-real complex real Element of COMPLEX
(COMPLEX,r,addcomplex) is complex Element of COMPLEX
(REAL,r,addreal) is ext-real complex real Element of REAL
c6 is Relation-like NAT -defined REAL -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(c6) is ext-real complex real Element of COMPLEX
(COMPLEX,c6,addcomplex) is complex Element of COMPLEX
(REAL,c6,addreal) is ext-real complex real Element of REAL
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . 0 is ext-real complex real set
(len r) - 1 is ext-real complex real integer rational Element of INT
c is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
c . 0 is ext-real complex real Element of REAL
c . ((len r) - 1) is ext-real complex real set
c6 . 0 is ext-real complex real set
len c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len c6) - 1 is ext-real complex real integer rational Element of INT
b is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
b . 0 is ext-real complex real Element of REAL
b . ((len c6) - 1) is ext-real complex real set
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c . kp is ext-real complex real Element of REAL
b . kp is ext-real complex real Element of REAL
kp + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . (kp + 1) is ext-real complex real Element of REAL
b . (kp + 1) is ext-real complex real Element of REAL
r . (kp + 1) is ext-real complex real set
c6 . (kp + 1) is ext-real complex real set
c . (kp + 1) is ext-real complex real Element of REAL
addreal . ((c . kp),(r . (kp + 1))) is set
(c . kp) + (r . (kp + 1)) is ext-real complex real Element of REAL
b . (kp + 1) is ext-real complex real Element of REAL
addreal . ((b . kp),(c6 . (kp + 1))) is set
(b . kp) + (c6 . (kp + 1)) is ext-real complex real Element of REAL
(len r) - 0 is ext-real non negative complex real integer rational Element of INT
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c . 0 is ext-real complex real Element of REAL
b . 0 is ext-real complex real Element of REAL
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is complex set
x --> p is Relation-like x -defined {p} -valued Function-like V14(x) quasi_total T-Sequence-like complex-valued finite V86() Element of K19(K20(x,{p}))
{p} is non empty V12() finite complex-membered 1 -element set
K20(x,{p}) is Relation-like complex-valued finite set
K19(K20(x,{p})) is finite V48() set
((x --> p)) is complex Element of COMPLEX
(COMPLEX,(x --> p),addcomplex) is complex Element of COMPLEX
x * p is complex Element of COMPLEX
y is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
K20(NAT,COMPLEX) is Relation-like V12() complex-valued non finite set
K19(K20(NAT,COMPLEX)) is V12() non finite set
y . 0 is complex set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(y) is complex Element of COMPLEX
(COMPLEX,y,addcomplex) is complex Element of COMPLEX
(len y) - 1 is ext-real complex real integer rational Element of INT
r is Relation-like NAT -defined COMPLEX -valued Function-like non empty V14( NAT ) quasi_total complex-valued Element of K19(K20(NAT,COMPLEX))
r . 0 is complex Element of COMPLEX
r . ((len y) - 1) is complex set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
r . c6 is complex Element of COMPLEX
c6 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(c6 + 1) * p is complex Element of COMPLEX
r . (c6 + 1) is complex Element of COMPLEX
(c6 + 1) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((c6 + 1) + 1) * p is complex Element of COMPLEX
r . (c6 + 1) is complex Element of COMPLEX
y . (c6 + 1) is complex set
addcomplex . ((r . c6),(y . (c6 + 1))) is set
(r . c6) + (y . (c6 + 1)) is complex Element of COMPLEX
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . 0 is complex Element of COMPLEX
0 + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(0 + 1) * p is complex Element of COMPLEX
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is ext-real complex real set
(len x) * p is ext-real complex real Element of REAL
(len x) --> p is Relation-like len x -defined {p} -valued Function-like V14( len x) quasi_total T-Sequence-like complex-valued ext-real-valued real-valued finite V86() Element of K19(K20((len x),{p}))
{p} is non empty V12() finite complex-membered ext-real-membered real-membered 1 -element set
K20((len x),{p}) is Relation-like complex-valued ext-real-valued real-valued finite set
K19(K20((len x),{p})) is finite V48() set
len ((len x) --> p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((len x) --> p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x . y is ext-real complex real set
((len x) --> p) . y is ext-real complex real set
(((len x) --> p)) is ext-real complex real Element of COMPLEX
(COMPLEX,((len x) --> p),addcomplex) is complex Element of COMPLEX
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is ext-real complex real set
(len x) * p is ext-real complex real Element of REAL
(len x) --> p is Relation-like len x -defined {p} -valued Function-like V14( len x) quasi_total T-Sequence-like complex-valued ext-real-valued real-valued finite V86() Element of K19(K20((len x),{p}))
{p} is non empty V12() finite complex-membered ext-real-membered real-membered 1 -element set
K20((len x),{p}) is Relation-like complex-valued ext-real-valued real-valued finite set
K19(K20((len x),{p})) is finite V48() set
len ((len x) --> p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((len x) --> p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x . y is ext-real complex real set
((len x) --> p) . y is ext-real complex real set
(((len x) --> p)) is ext-real complex real Element of COMPLEX
(COMPLEX,((len x) --> p),addcomplex) is complex Element of COMPLEX
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is ext-real complex real set
q is set
x . q is ext-real complex real set
q is set
x . q is ext-real complex real set
(len x) - 1 is ext-real complex real integer rational Element of INT
y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . 0 is ext-real complex real set
(REAL,x,addreal) is ext-real complex real Element of REAL
c6 is Relation-like NAT -defined REAL -valued Function-like non empty V14( NAT ) quasi_total complex-valued ext-real-valued real-valued Element of K19(K20(NAT,REAL))
c6 . 0 is ext-real complex real Element of REAL
c6 . ((len x) - 1) is ext-real complex real set
rng x is finite complex-membered ext-real-membered real-membered set
c6 . 0 is ext-real complex real Element of REAL
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 . c is ext-real complex real Element of REAL
c + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . (c + 1) is ext-real complex real Element of REAL
c6 . (c + 1) is ext-real complex real Element of REAL
x . (c + 1) is ext-real complex real set
addreal . ((c6 . c),(x . (c + 1))) is set
0 + 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
(c6 . c) + (x . (c + 1)) is ext-real complex real Element of REAL
c6 . r is ext-real complex real Element of REAL
r - 1 is ext-real complex real integer rational Element of INT
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . r is ext-real complex real Element of REAL
c6 . c is ext-real complex real Element of REAL
x . r is ext-real complex real set
addreal . ((c6 . c),(x . r)) is set
(c6 . c) + (x . r) is ext-real complex real Element of REAL
p + 0 is ext-real complex real Element of REAL
c6 . r is ext-real complex real Element of REAL
c6 . r is ext-real complex real Element of REAL
c6 . r is ext-real complex real Element of REAL
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 . c is ext-real complex real Element of REAL
c + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . (c + 1) is ext-real complex real Element of REAL
x . (c + 1) is ext-real complex real set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 . (b + 1) is ext-real complex real Element of REAL
c6 . b is ext-real complex real Element of REAL
x . (b + 1) is ext-real complex real set
addreal . ((c6 . b),(x . (b + 1))) is set
(c6 . b) + (x . (b + 1)) is ext-real complex real Element of REAL
p + 0 is ext-real complex real Element of REAL
c6 . (c + 1) is ext-real complex real Element of REAL
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued ext-real-valued real-valued finite V86() set
(x) is ext-real complex real Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) --> 0 is Relation-like len x -defined NAT -valued RAT -valued INT -valued Function-like V14( len x) quasi_total T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() Element of K19(K20((len x),NAT))
K20((len x),NAT) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued nonnegative-yielding set
K19(K20((len x),NAT)) is set
dom ((len x) --> 0) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19((len x))
K19((len x)) is finite V48() set
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((len x) --> 0) . q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
x . q is ext-real complex real set
rng x is finite complex-membered ext-real-membered real-membered set
(len x) * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like complex-valued finite V86() set
rng p is finite complex-membered set
q is complex set
q (#) (p | x) is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
q (#) p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
(q (#) p) | x is Relation-like x -defined NAT -defined rng (q (#) p) -valued Function-like T-Sequence-like complex-valued finite V86() set
rng (q (#) p) is finite complex-membered set
len (q (#) p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q (#) p) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (q (#) (p | x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (q (#) (p | x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (p | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len ((q (#) p) | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom ((q (#) p) | x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom (q (#) (p | x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(x)
K19(x) is finite V48() set
(q (#) (p | x)) . c6 is complex set
(p | x) . c6 is complex set
q * ((p | x) . c6) is complex Element of COMPLEX
p . c6 is complex set
q * (p . c6) is complex Element of COMPLEX
(q (#) p) . c6 is complex set
((q (#) p) | x) . c6 is complex set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is complex set
p * (x) is complex Element of COMPLEX
p (#) x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
((p (#) x)) is complex Element of COMPLEX
(COMPLEX,(p (#) x),addcomplex) is complex Element of COMPLEX
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(y) is complex Element of COMPLEX
(COMPLEX,y,addcomplex) is complex Element of COMPLEX
p * (y) is complex Element of COMPLEX
p (#) y is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
((p (#) y)) is complex Element of COMPLEX
(COMPLEX,(p (#) y),addcomplex) is complex Element of COMPLEX
dom (p (#) y) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c6 is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
c6 | (q + 1) is Relation-like q + 1 -defined NAT -defined COMPLEX -valued rng c6 -valued Function-like T-Sequence-like complex-valued finite V86() set
rng c6 is finite complex-membered set
len c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
c6 | q is Relation-like q -defined NAT -defined COMPLEX -valued rng c6 -valued Function-like T-Sequence-like complex-valued finite V86() set
len (c6 | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (c6 | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(COMPLEX,(c6 | q),addcomplex) is complex Element of COMPLEX
c6 . q is complex set
addcomplex . ((COMPLEX,(c6 | q),addcomplex),(c6 . q)) is set
(COMPLEX,(c6 | (q + 1)),addcomplex) is complex Element of COMPLEX
(c6) is complex Element of COMPLEX
(COMPLEX,c6,addcomplex) is complex Element of COMPLEX
((c6 | q)) is complex Element of COMPLEX
((c6 | q)) + (c6 . q) is complex Element of COMPLEX
p * ((c6 | q)) is complex Element of COMPLEX
p (#) (c6 | q) is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
((p (#) (c6 | q))) is complex Element of COMPLEX
(COMPLEX,(p (#) (c6 | q)),addcomplex) is complex Element of COMPLEX
c is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
c | q is Relation-like q -defined NAT -defined COMPLEX -valued rng c -valued Function-like T-Sequence-like complex-valued finite V86() set
rng c is finite complex-membered set
((c | q)) is complex Element of COMPLEX
(COMPLEX,(c | q),addcomplex) is complex Element of COMPLEX
p * (c6 . q) is complex Element of COMPLEX
c . q is complex set
c | (q + 1) is Relation-like q + 1 -defined NAT -defined COMPLEX -valued rng c -valued Function-like T-Sequence-like complex-valued finite V86() set
addcomplex . ((COMPLEX,(c | q),addcomplex),(c . q)) is set
(COMPLEX,(c | (q + 1)),addcomplex) is complex Element of COMPLEX
(c) is complex Element of COMPLEX
(COMPLEX,c,addcomplex) is complex Element of COMPLEX
((c | q)) + (c . q) is complex Element of COMPLEX
q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(q) is complex Element of COMPLEX
(COMPLEX,q,addcomplex) is complex Element of COMPLEX
p * (q) is complex Element of COMPLEX
p (#) q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
((p (#) q)) is complex Element of COMPLEX
(COMPLEX,(p (#) q),addcomplex) is complex Element of COMPLEX
r is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(COMPLEX,r,addcomplex) is complex Element of COMPLEX
c6 is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
len c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(q) is complex Element of COMPLEX
(COMPLEX,q,addcomplex) is complex Element of COMPLEX
p * (q) is complex Element of COMPLEX
p (#) q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
((p (#) q)) is complex Element of COMPLEX
(COMPLEX,(p (#) q),addcomplex) is complex Element of COMPLEX
x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
x + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
p | x is Relation-like x -defined NAT -defined rng p -valued Function-like T-Sequence-like complex-valued finite V86() set
rng p is finite complex-membered set
((p | x)) is complex Element of COMPLEX
(COMPLEX,(p | x),addcomplex) is complex Element of COMPLEX
p . x is complex set
((p | x)) + (p . x) is complex Element of COMPLEX
p | (x + 1) is Relation-like x + 1 -defined NAT -defined rng p -valued Function-like T-Sequence-like complex-valued finite V86() set
((p | (x + 1))) is complex Element of COMPLEX
(COMPLEX,(p | (x + 1)),addcomplex) is complex Element of COMPLEX
q is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
q | x is Relation-like x -defined NAT -defined COMPLEX -valued rng q -valued Function-like T-Sequence-like complex-valued finite V86() set
rng q is finite complex-membered set
(COMPLEX,(q | x),addcomplex) is complex Element of COMPLEX
q . x is complex set
addcomplex . ((COMPLEX,(q | x),addcomplex),(q . x)) is set
q | (x + 1) is Relation-like x + 1 -defined NAT -defined COMPLEX -valued rng q -valued Function-like T-Sequence-like complex-valued finite V86() set
(COMPLEX,(q | (x + 1)),addcomplex) is complex Element of COMPLEX
x is set
{x} is non empty V12() finite 1 -element set
p is set
{p} is non empty V12() finite 1 -element set
q is Relation-like Function-like set
q . x is set
dom q is set
(dom q) \ {x} is Element of K19((dom q))
K19((dom q)) is set
q | ((dom q) \ {x}) is Relation-like Function-like set
(q | ((dom q) \ {x})) " {p} is set
{x} \/ ((q | ((dom q) \ {x})) " {p}) is non empty set
q " {p} is set
r is set
(q | ((dom q) \ {x})) . r is set
dom (q | ((dom q) \ {x})) is set
(dom q) /\ ((dom q) \ {x}) is Element of K19((dom q))
q . r is set
r is set
dom (q | ((dom q) \ {x})) is set
(dom q) /\ ((dom q) \ {x}) is Element of K19((dom q))
q . r is set
(q | ((dom q) \ {x})) . r is set
r is set
x is set
{x} is non empty V12() finite 1 -element set
p is set
{p} is non empty V12() finite 1 -element set
q is Relation-like Function-like set
q . x is set
dom q is set
(dom q) \ {x} is Element of K19((dom q))
K19((dom q)) is set
q | ((dom q) \ {x}) is Relation-like Function-like set
(q | ((dom q) \ {x})) " {p} is set
q " {p} is set
dom (q | ((dom q) \ {x})) is set
(dom q) /\ ((dom q) \ {x}) is Element of K19((dom q))
r is set
q . r is set
(q | ((dom q) \ {x})) . r is set
r is set
(q | ((dom q) \ {x})) . r is set
dom (q | ((dom q) \ {x})) is set
(dom q) /\ ((dom q) \ {x}) is Element of K19((dom q))
q . r is set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
rng x is finite complex-membered set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
p is complex set
{0,p} is non empty finite complex-membered set
{p} is non empty V12() finite complex-membered 1 -element set
x " {p} is finite set
card (x " {p}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p * (card (x " {p})) is complex Element of COMPLEX
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng y is finite complex-membered set
(y) is complex Element of COMPLEX
(COMPLEX,y,addcomplex) is complex Element of COMPLEX
r is complex set
{0,r} is non empty finite complex-membered set
{r} is non empty V12() finite complex-membered 1 -element set
y " {r} is finite set
card (y " {r}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r * (card (y " {r})) is complex Element of COMPLEX
{q} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
q \/ {q} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(dom y) \ {q} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
y . q is complex set
y | (q + 1) is Relation-like q + 1 -defined NAT -defined rng y -valued Function-like T-Sequence-like complex-valued finite V86() set
y | q is Relation-like q -defined NAT -defined rng y -valued Function-like T-Sequence-like complex-valued finite V86() set
((y | q)) is complex Element of COMPLEX
(COMPLEX,(y | q),addcomplex) is complex Element of COMPLEX
((y | q)) + 0 is complex Element of COMPLEX
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (y | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (y | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (y | q) is finite complex-membered Element of K19((rng y))
K19((rng y)) is finite V48() set
(y | q) " {r} is finite set
y | q is Relation-like q -defined NAT -defined rng y -valued Function-like T-Sequence-like complex-valued finite V86() set
(y | q) " {r} is finite set
dom (y | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(q)
K19(q) is finite V48() set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
rng (y | q) is finite complex-membered Element of K19((rng y))
K19((rng y)) is finite V48() set
len (y | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (y | q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((y | q)) is complex Element of COMPLEX
(COMPLEX,(y | q),addcomplex) is complex Element of COMPLEX
card ((y | q) " {r}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r * (card ((y | q) " {r})) is complex Element of COMPLEX
y | (q + 1) is Relation-like q + 1 -defined NAT -defined rng y -valued Function-like T-Sequence-like complex-valued finite V86() set
((y | q)) + r is complex Element of COMPLEX
{q} \/ ((y | q) " {r}) is non empty finite set
(card ((y | q) " {r})) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c6 is set
y . c6 is complex set
(dom y) --> 0 is Relation-like dom y -defined NAT -valued RAT -valued INT -valued Function-like V14( dom y) quasi_total T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() Element of K19(K20((dom y),NAT))
K20((dom y),NAT) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued nonnegative-yielding set
K19(K20((dom y),NAT)) is set
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len y) * 0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng q is finite complex-membered set
(q) is complex Element of COMPLEX
(COMPLEX,q,addcomplex) is complex Element of COMPLEX
y is complex set
{0,y} is non empty finite complex-membered set
{y} is non empty V12() finite complex-membered 1 -element set
q " {y} is finite set
card (q " {y}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y * (card (q " {y})) is complex Element of COMPLEX
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
q is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
rng q is finite complex-membered set
y is complex set
{0,y} is non empty finite complex-membered set
(q) is complex Element of COMPLEX
(COMPLEX,q,addcomplex) is complex Element of COMPLEX
{y} is non empty V12() finite complex-membered 1 -element set
q " {y} is finite set
card (q " {y}) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y * (card (q " {y})) is complex Element of COMPLEX
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
(x) is complex Element of COMPLEX
(COMPLEX,x,addcomplex) is complex Element of COMPLEX
(x) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
((x)) is complex Element of COMPLEX
(COMPLEX,(x),addcomplex) is complex Element of COMPLEX
q is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined COMPLEX -valued Function-like T-Sequence-like complex-valued finite V86() set
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
card (len q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y is set
r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + r is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) -' (1 + r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) - (1 + r) is ext-real complex real integer rational Element of INT
(len q) + (1 + r) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) + 0 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len q) + (1 + r)) - (1 + r) is ext-real complex real integer rational Element of INT
c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
1 + c6 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) - (1 + c6) is ext-real complex real integer rational Element of INT
K20((len q),(len q)) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding set
K19(K20((len q),(len q))) is finite V48() set
y is Relation-like len q -defined len q -valued Function-like V14( len q) quasi_total complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding Element of K19(K20((len q),(len q)))
r is set
c6 is set
y . r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
y . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + b is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) - (1 + b) is ext-real complex real integer rational Element of INT
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + c is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) - (1 + c) is ext-real complex real integer rational Element of INT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
K20((dom q),(dom q)) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding set
K19(K20((dom q),(dom q))) is finite V48() set
c6 is set
r is Relation-like dom q -defined dom q -valued Function-like one-to-one V14( dom q) quasi_total onto bijective complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding Element of K19(K20((dom q),(dom q)))
c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r . c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
1 + c is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len q) - (1 + c) is ext-real complex real integer rational Element of INT
q . c6 is complex set
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . (r . c6) is complex set
c6 is set
dom r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom q))
K19((dom q)) is finite V48() set
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
rng r is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19((dom q))
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
c6 is set
r . c6 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
r (#) p is Relation-like dom q -defined COMPLEX -valued Function-like complex-valued finite set
x is Relation-like Function-like set
dom x is set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng p is finite set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
rng q is finite set
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p (#) x is Relation-like NAT -defined Function-like finite set
r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q (#) x is Relation-like NAT -defined Function-like finite set
y ^ r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p ^ q) (#) x is Relation-like NAT -defined Function-like finite set
rng (p ^ q) is finite set
(rng p) \/ (rng q) is finite set
dom ((p ^ q) (#) x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (p ^ q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) + (len q) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (y ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
len y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom y is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom r is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len y) + (len r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len (y ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (y ^ r) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
b is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(y ^ r) . b is set
c . b is set
dom c is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(p ^ q) . b is set
p . b is set
y . b is set
x . (p . b) is set
b - (len p) is ext-real complex real Element of REAL
kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len p) + kp is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p ^ q) . b is set
q . kp is set
r . kp is set
x . (q . kp) is set
x is Relation-like NAT -defined Function-like T-Sequence-like complex-valued finite V86() set
p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
p \/ q is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(x,(p \/ q)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p \/ q) /\ (len x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
(((p \/ q) /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() set
(((p \/ q) /\ (len x))) (#) x is Relation-like NAT -defined Function-like complex-valued finite set
((x,(p \/ q))) is complex Element of COMPLEX
(COMPLEX,(x,(p \/ q)),addcomplex) is complex Element of COMPLEX
(x,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p /\ (len x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() set
((p /\ (len x))) (#) x is Relation-like NAT -defined Function-like complex-valued finite set
((x,p)) is complex Element of COMPLEX
(COMPLEX,(x,p),addcomplex) is complex Element of COMPLEX
(x,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q /\ (len x) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((q /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like one-to-one T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() set
((q /\ (len x))) (#) x is Relation-like NAT -defined Function-like complex-valued finite set
((x,q)) is complex Element of COMPLEX
(COMPLEX,(x,q),addcomplex) is complex Element of COMPLEX
((x,p)) + ((x,q)) is complex Element of COMPLEX
(p /\ (len x)) \/ (q /\ (len x)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
((p /\ (len x))) ^ ((q /\ (len x))) is Relation-like NAT -defined NAT -valued Function-like T-Sequence-like complex-valued ext-real-valued real-valued natural-valued finite nonnegative-yielding V86() set
rng ((p /\ (len x))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
rng ((q /\ (len x))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(x,p) ^ (x,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is non empty set
K20(x,x) is Relation-like set
K20(K20(x,x),x) is Relation-like set
K19(K20(K20(x,x),x)) is set
<%> x is Relation-like non-empty empty-yielding NAT -defined x -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
p is Relation-like K20(x,x) -defined x -valued Function-like V14(K20(x,x)) quasi_total Element of K19(K20(K20(x,x),x))
(x,(<%> x),p) is Element of x
the_unity_wrt p is Element of x
len (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued integer finite finite-yielding V48() rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() Element of NAT
dom (<%> x) is Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
x is set
x ^omega is functional non empty set
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
p is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
q is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
q . (y,r) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y ^ r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),p,r) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c6 is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),p,c6) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
b is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r . (c,b) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c ^ b is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c6 . (c,b) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is set
x ^omega is functional non empty set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
<%p%> is Relation-like NAT -defined x ^omega -valued Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
(x,<%p%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
q is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),<%p%>,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is set
x ^omega is functional non empty set
<%> (x ^omega) is Relation-like non-empty empty-yielding NAT -defined x ^omega -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
(x,(<%> (x ^omega))) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
<%> x is Relation-like non-empty empty-yielding NAT -defined x -valued Function-like one-to-one constant functional empty ext-real non positive non negative epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural complex real complex-valued ext-real-valued real-valued natural-valued finite finite-yielding V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V56() nonnegative-yielding cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V86() set
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
p is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),(<%> (x ^omega)),p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
p . ({},y) is set
{} ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p . (y,{}) is set
y ^ {} is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
the_unity_wrt p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is set
x ^omega is functional non empty set
p is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
p ^ q is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
(x,(p ^ q)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,p) ^ (x,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
y is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),(p ^ q),y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c6 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . (c6,c) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . (r,(y . (c6,c))) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r ^ (y . (c6,c)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c6 ^ c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r ^ (c6 ^ c) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r ^ c6 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(r ^ c6) ^ c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . (r,c6) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(y . (r,c6)) ^ c is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . ((y . (r,c6)),c) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
c6 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . ({},c6) is set
{} ^ c6 is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
y . (c6,{}) is set
c6 ^ {} is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
r is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
len q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
((x ^omega),p,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
((x ^omega),q,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . (((x ^omega),p,y),((x ^omega),q,y)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
((x ^omega),p,y) ^ ((x ^omega),q,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,p) ^ ((x ^omega),q,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is set
x ^omega is functional non empty set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
<%p,q%> is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite 2 -element V86() set
(x,<%p,q%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
p ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
y is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),<%p,q%>,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y . (p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is set
x ^omega is functional non empty set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
<%p,q,y%> is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite 3 -element V86() set
(x,<%p,q,y%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
p ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(p ^ q) ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
K20((x ^omega),(x ^omega)) is Relation-like set
K20(K20((x ^omega),(x ^omega)),(x ^omega)) is Relation-like set
K19(K20(K20((x ^omega),(x ^omega)),(x ^omega))) is set
r is Relation-like K20((x ^omega),(x ^omega)) -defined x ^omega -valued Function-like V14(K20((x ^omega),(x ^omega))) quasi_total Element of K19(K20(K20((x ^omega),(x ^omega)),(x ^omega)))
((x ^omega),<%p,q,y%>,r) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r . (p,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
r . ((r . (p,q)),y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(r . (p,q)) ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p,(len x)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x ^ (p,(len x)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len (p,(len x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p,(len x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) + (len (p,(len x))) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) -' (len x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len x) + ((len p) -' (len x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(len p) + (len x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len p) + (len x)) -' (len x) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
p . q is set
x . q is set
dom (p,(len x)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len x) + q is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
p . ((len x) + q) is set
(p,(len x)) . q is set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
len x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom x is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(p,(len x)) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is non empty set
p is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
p ^ r is Relation-like NAT -defined x -valued Function-like T-Sequence-like finite V86() set
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ q is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
(p ^ x) ^ y is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is set
x ^omega is functional non empty set
p is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
q is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
(x,p) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,q) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
y is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
p ^ y is Relation-like NAT -defined x ^omega -valued Function-like T-Sequence-like finite V86() set
(x,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
(x,p) ^ (x,y) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() Element of x ^omega
x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p is Relation-like NAT -defined Function-like non empty T-Sequence-like finite V86() set
x ^ p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ x is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
x is set
<%x%> is Relation-like NAT -defined Function-like constant non empty V12() T-Sequence-like finite 1 -element V86() set
p is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
p ^ <%x%> is Relation-like NAT -defined Function-like non empty T-Sequence-like finite V86() set
CutLastLoc (p ^ <%x%>) is Relation-like NAT -defined Function-like T-Sequence-like finite V86() set
LastLoc (p ^ <%x%>) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
(p ^ <%x%>) . (LastLoc (p ^ <%x%>)) is set
(LastLoc (p ^ <%x%>)) .--> ((p ^ <%x%>) . (LastLoc (p ^ <%x%>))) is finite set
(p ^ <%x%>) \ ((LastLoc (p ^ <%x%>)) .--> ((p ^ <%x%>) . (LastLoc (p ^ <%x%>)))) is Relation-like NAT -defined Function-like finite Element of K19((p ^ <%x%>))
K19((p ^ <%x%>)) is finite V48() set
len (p ^ <%x%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p ^ <%x%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len (p ^ <%x%>)) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
len p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
(len p) + 1 is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
((len p) + 1) -' 1 is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real integer finite rational complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of NAT
dom (p ^ <%x%>) is non empty ext-real positive non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
dom p is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
{(len p)} is non empty V12() finite V48() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered 1 -element set
(dom p) \/ {(len p)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered set
dom (CutLastLoc (p ^ <%x%>)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal Element of K19(NAT)
(dom (p ^ <%x%>)) \ {(len p)} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered Element of K19(NAT)
dom (CutLastLoc (p ^ <%x%>)) is ext-real non negative epsilon-transitive epsilon-connected ordinal natural complex real finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered cardinal set
y is set
(CutLastLoc (p ^ <%x%>)) . y is set
p . y is set
(p ^ <%x%>) . y is set