:: COMPUT_1 semantic presentation

REAL is V78() V79() V80() V84() V93() V94() V96() set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite V41() V42() countable denumerable V78() V79() V80() V81() V82() V83() V84() V91() V93() Element of K32(REAL)
K32(REAL) is non empty set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite V41() V42() countable denumerable V78() V79() V80() V81() V82() V83() V84() V91() V93() set
K32(NAT) is non empty non trivial non finite set
K33(NAT,REAL) is Relation-like complex-valued ext-real-valued real-valued set
K32(K33(NAT,REAL)) is non empty set
K32(K32(REAL)) is non empty set
K32(NAT) is non empty non trivial non finite set
COMPLEX is V78() V84() set
RAT is V78() V79() V80() V81() V84() set
INT is V78() V79() V80() V81() V82() V84() set
{} is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
{{},1} is non empty finite V40() countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K542() is non empty set
0 is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered V65() V67() complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() Element of NAT
{0,1} is non empty finite V40() countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
3 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom {} is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() set
rng {} is empty trivial Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() with_non-empty_elements ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() set
Seg 1 is non empty trivial finite V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
{1} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
Seg 2 is non empty finite V43(2) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
{1,2} is non empty finite V40() countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
<:{}:> is Relation-like Function-like set
Frege {} is Relation-like Function-like set
{} .--> {} is non empty Relation-like {{}} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() set
{{}} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
{{}} --> {} is non empty Relation-like {{}} -defined RAT -valued {{}} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33({{}},{{}}))
K33({{}},{{}}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({{}},{{}})) is non empty finite V40() countable set
{} * is non empty functional FinSequence-membered FinSequenceSet of {}
meet {} is set
len {} is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() set
0 ! is V28() V29() ext-real Element of REAL
i is set
j is set
<*i,j*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like set
F is set
<*i,j*> +* (1,F) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
<*F,j*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like set
<*i,j*> +* (2,F) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
<*i,F*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like set
<*i,j*> . 1 is set
(<*i,j*> +* (2,F)) . 1 is set
<*i,j*> . 2 is set
(<*i,j*> +* (1,F)) . 2 is set
len <*i,j*> is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom <*i,j*> is non empty finite V43(2) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
(<*i,j*> +* (1,F)) . 1 is set
len (<*i,j*> +* (1,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(<*i,j*> +* (2,F)) . 2 is set
len (<*i,j*> +* (2,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is set
j is set
F is set
g is set
F is Relation-like Function-like set
F +* (i,j) is Relation-like Function-like set
F +* (i,g) is Relation-like Function-like set
ii is Relation-like Function-like set
ii +* (i,F) is Relation-like Function-like set
ii +* (i,g) is Relation-like Function-like set
dom (ii +* (i,g)) is set
dom ii is set
dom (ii +* (i,F)) is set
dom (F +* (i,j)) is set
dom F is set
dom (F +* (i,g)) is set
k is set
(F +* (i,g)) . k is set
(ii +* (i,g)) . k is set
(F +* (i,g)) . k is set
F . k is set
(ii +* (i,F)) . k is set
ii . k is set
(ii +* (i,g)) . k is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
F +* (i,j) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Del ((F +* (i,j)),i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Del (F,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom (F +* (i,j)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (F +* (i,j)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (Del ((F +* (i,j)),i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (Del ((F +* (i,j)),i)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Seg ii is finite V43(ii) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (Del (F,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
(Del ((F +* (i,j)),i)) . k is set
(F +* (i,j)) . k is set
F . k is set
(Del (F,i)) . k is set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(Del ((F +* (i,j)),i)) . k is set
(F +* (i,j)) . (k + 1) is set
F . (k + 1) is set
(Del (F,i)) . k is set
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
F +* (i,j) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Del (F,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
g is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
g +* (i,j) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
Del (g,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom (F +* (i,j)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom (g +* (i,j)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom g is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Seg (len g) is finite V43( len g) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Seg (len F) is finite V43( len F) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom (Del (F,i)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (Del (F,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Seg (len (Del (F,i))) is finite V43( len (Del (F,i))) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
(Del (F,i)) . k is set
F . k is set
(g +* (i,j)) . k is set
g . k is set
(Del (g,i)) . k is set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(Del (F,i)) . k is set
F . (k + 1) is set
(g +* (i,j)) . (k + 1) is set
g . (k + 1) is set
(Del (g,i)) . k is set
i is set
0 -tuples_on i is functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
i * is non empty functional FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 0 } is set
F is set
g is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i *
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*> (i *) is empty proper Relation-like non-empty empty-yielding NAT -defined i * -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() FinSequence of i *
K33(NAT,(i *)) is non empty non trivial Relation-like non finite set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on {} is functional finite with_common_domain product-like countable FinSequence-membered FinSequenceSet of {}
{ b₁ where b₁ is Relation-like NAT -defined {} -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of {} * : len b₁ = i } is set
j is set
F is empty Relation-like non-empty empty-yielding NAT -defined {} -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() Element of {} *
len F is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered V65() V67() complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() Element of NAT
i is Relation-like Function-like set
rng i is set
<:i:> is Relation-like Function-like set
dom <:i:> is set
doms i is Relation-like Function-like set
meet (doms i) is set
rng (doms i) is set
meet (rng (doms i)) is set
dom i is set
j is set
i . j is set
dom (doms i) is set
SubFuncs (rng i) is set
i " (SubFuncs (rng i)) is set
(doms i) . j is set
i is non empty set
1 -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
i * is non empty functional FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 1 } is set
j is Relation-like Function-like set
rng j is set
<*j*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() set
<:<*j*>:> is Relation-like Function-like set
rng <:<*j*>:> is set
dom <:<*j*>:> is set
dom j is set
g is set
F is set
<:<*j*>:> . F is set
j . F is set
ii is Element of i
<*ii*> is non empty trivial Relation-like NAT -defined i -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like FinSequence of i
<*(j . F)*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like set
F is Element of i
<*F*> is non empty trivial Relation-like NAT -defined i -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like FinSequence of i
ii is set
j . ii is set
<:<*j*>:> . ii is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty set
(j + 1) -tuples_on F is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of F
F * is non empty functional FinSequence-membered FinSequenceSet of F
{ b₁ where b₁ is Relation-like NAT -defined F -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of F * : len b₁ = j + 1 } is set
j -tuples_on F is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of F
{ b₁ where b₁ is Relation-like NAT -defined F -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of F * : len b₁ = j } is set
F is Relation-like NAT -defined F -valued Function-like finite V43(j + 1) countable FinSequence-like FinSubsequence-like Element of (j + 1) -tuples_on F
Del (F,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is finite V43(j + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (Del (F,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
j is functional FinSequence-membered FinSequenceSet of i
F is set
j is Relation-like Function-like set
F is Relation-like Function-like set
i is functional () set
union i is set
j is set
F is set
[j,F] is set
{j,F} is non empty finite countable set
{j} is non empty trivial finite V43(1) countable set
{{j,F},{j}} is non empty finite V40() countable set
g is set
[j,g] is set
{j,g} is non empty finite countable set
{{j,g},{j}} is non empty finite V40() countable set
F is set
ii is set
k is Relation-like Function-like set
dom k is set
k . j is set
k is Relation-like Function-like set
dom k is set
k . j is set
(dom k) /\ (dom k) is set
j is set
F is set
i is set
union i is set
j is set
F is set
g is set
[F,g] is set
{F,g} is non empty finite countable set
{F} is non empty trivial finite V43(1) countable set
{{F,g},{F}} is non empty finite V40() countable set
F is set
[F,F] is set
{F,F} is non empty finite countable set
{{F,F},{F}} is non empty finite V40() countable set
F is set
j is Relation-like Function-like set
F is Relation-like Function-like set
g is set
dom j is set
dom F is set
(dom j) /\ (dom F) is set
j . g is set
F . g is set
F is set
[g,F] is set
{g,F} is non empty finite countable set
{g} is non empty trivial finite V43(1) countable set
{{g,F},{g}} is non empty finite V40() countable set
ii is set
[g,ii] is set
{g,ii} is non empty finite countable set
{{g,ii},{g}} is non empty finite V40() countable set
i is set
j is set
g is Relation-like Function-like set
F is Relation-like Function-like set
g is Relation-like Function-like epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() Element of {{}}
K33(i,j) is Relation-like set
K32(K33(i,j)) is non empty set
i is non empty functional () set
union i is Relation-like Function-like set
dom (union i) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of i : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of i : verum } is set
F is set
g is set
[F,g] is set
{F,g} is non empty finite countable set
{F} is non empty trivial finite V43(1) countable set
{{F,g},{F}} is non empty finite V40() countable set
F is set
ii is Relation-like Function-like Element of i
dom ii is set
g is set
F is Relation-like Function-like Element of i
dom F is set
ii is set
[F,ii] is set
{F,ii} is non empty finite countable set
{{F,ii},{F}} is non empty finite V40() countable set
i is functional () set
union i is Relation-like Function-like set
dom (union i) is set
j is Relation-like Function-like set
dom j is set
F is set
g is set
[F,g] is set
{F,g} is non empty finite countable set
{F} is non empty trivial finite V43(1) countable set
{{F,g},{F}} is non empty finite V40() countable set
F is set
(union i) . F is set
j . F is set
g is set
[F,g] is set
{F,g} is non empty finite countable set
{F} is non empty trivial finite V43(1) countable set
{{F,g},{F}} is non empty finite V40() countable set
i is non empty functional () set
union i is Relation-like Function-like set
rng (union i) is set
{ (rng b₁) where b₁ is Relation-like Function-like Element of i : verum } is set
union { (rng b₁) where b₁ is Relation-like Function-like Element of i : verum } is set
F is set
g is set
[g,F] is set
{g,F} is non empty finite countable set
{g} is non empty trivial finite V43(1) countable set
{{g,F},{g}} is non empty finite V40() countable set
F is set
ii is Relation-like Function-like Element of i
rng ii is set
g is set
F is Relation-like Function-like Element of i
rng F is set
ii is set
[ii,F] is set
{ii,F} is non empty finite countable set
{ii} is non empty trivial finite V43(1) countable set
{{ii,F},{ii}} is non empty finite V40() countable set
i is set
j is set
F is non empty PartFunc-set of i,j
g is set
i is set
j is set
K33(i,j) is Relation-like set
K32(K33(i,j)) is non empty set
F is non empty functional () PartFunc-set of i,j
union F is Relation-like Function-like set
{ (dom b₁) where b₁ is Relation-like i -defined j -valued Function-like Element of F : verum } is set
{ (rng b₁) where b₁ is Relation-like i -defined j -valued Function-like Element of F : verum } is set
ii is Relation-like Function-like set
rng ii is set
k is set
union { (rng b₁) where b₁ is Relation-like i -defined j -valued Function-like Element of F : verum } is set
k is set
ij is Relation-like i -defined j -valued Function-like Element of F
rng ij is set
dom ii is set
k is set
union { (dom b₁) where b₁ is Relation-like i -defined j -valued Function-like Element of F : verum } is set
k is set
ij is Relation-like i -defined j -valued Function-like Element of F
dom ij is Element of K32(i)
K32(i) is non empty set
NAT * is non empty functional FinSequence-membered FinSequenceSet of NAT
i is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() set
dom i is empty proper Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() Element of K32(NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty set
i -tuples_on j is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of j
j * is non empty functional FinSequence-membered FinSequenceSet of j
{ b₁ where b₁ is Relation-like NAT -defined j -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of j * : len b₁ = i } is set
F is Relation-like set
dom F is set
i is Relation-like set
dom i is set
i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
j is set
i .--> j is non empty Relation-like {i} -defined Function-like one-to-one finite countable set
{i} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{i} --> j is non empty Relation-like {i} -defined {j} -valued Function-like constant total quasi_total finite countable Element of K32(K33({i},{j}))
{j} is non empty trivial finite V43(1) countable set
K33({i},{j}) is non empty Relation-like finite countable set
K32(K33({i},{j})) is non empty finite V40() countable set
dom (i .--> j) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({i})
K32({i}) is non empty finite V40() countable set
the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set .--> 0 is non empty Relation-like { the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() homogeneous set
{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } --> 0 is non empty Relation-like { the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set } -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33({ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set },{0}))
{0} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set },{0}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({ the Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set },{0})) is non empty finite V40() countable set
i is Relation-like Function-like homogeneous set
j is Relation-like Function-like set
i * j is Relation-like Function-like set
dom (i * j) is set
dom i is with_common_domain set
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
j is set
K33((i *),j) is Relation-like set
K32(K33((i *),j)) is non empty set
g is Relation-like i * -defined j -valued Function-like Element of K32(K33((i *),j))
dom g is functional FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
j is non empty set
K33((i *),j) is non empty Relation-like set
K32(K33((i *),j)) is non empty set
the Element of j is Element of j
K32((i *)) is non empty set
0 -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 0 } is set
g is non empty functional FinSequence-membered Element of K32((i *))
g --> the Element of j is non empty Relation-like i * -defined g -defined j -valued Function-like constant total Element of PFuncs ((i *),j)
PFuncs ((i *),j) is non empty functional PartFunc-set of i * ,j
{ the Element of j} is non empty trivial finite V43(1) countable set
K33(g,{ the Element of j}) is non empty Relation-like set
F is Relation-like i * -defined j -valued Function-like Element of K32(K33((i *),j))
dom F is functional FinSequence-membered Element of K32((i *))
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
the epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
the Element of i is Element of i
K32((i *)) is non empty set
the epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = the epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT } is set
g is non empty functional FinSequence-membered Element of K32((i *))
g --> the Element of i is non empty Relation-like i * -defined g -defined i -valued Function-like constant total Element of PFuncs ((i *),i)
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ the Element of i} is non empty trivial finite V43(1) countable set
K33(g,{ the Element of i}) is non empty Relation-like set
F is Relation-like i * -defined i -valued Function-like Element of K32(K33((i *),i))
dom F is functional FinSequence-membered Element of K32((i *))
ii is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional FinSequence-membered set
K32((NAT *)) is non empty set
0 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 0 } is set
i is non empty functional FinSequence-membered Element of K32((NAT *))
i --> 0 is non empty Relation-like NAT * -defined i -defined NAT -valued RAT -valued INT -valued Function-like constant total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of PFuncs ((NAT *),NAT)
PFuncs ((NAT *),NAT) is non empty functional PartFunc-set of NAT * , NAT
{0} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33(i,{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
dom (i --> 0) is non empty functional FinSequence-membered Element of K32(i)
K32(i) is non empty set
F is Relation-like NAT * -defined Function-like set
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional FinSequence-membered Element of K32((NAT *))
(len F) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len F } is set
dom F is functional FinSequence-membered Element of K32((NAT *))
K33((NAT *),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((NAT *),NAT)) is non empty non trivial non finite set
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
j is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is functional FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued () set
rng i is V78() V79() V80() V81() V82() V83() V93() set
dom i is functional FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
arity {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom {} is empty proper Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous Element of K32(NAT)
i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like homogeneous set
dom i is with_common_domain set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
j is set
K33((i *),j) is Relation-like set
K32(K33((i *),j)) is non empty set
F is Relation-like i * -defined j -valued Function-like homogeneous Element of K32(K33((i *),j))
dom F is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) -tuples_on i is functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity F } is set
g is set
<*> (i *) is empty proper Relation-like non-empty empty-yielding NAT -defined i * -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous FinSequence of i *
K33(NAT,(i *)) is non empty non trivial Relation-like non finite set
len (<*> (i *)) is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered V65() V67() complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous Element of NAT
0 -tuples_on i is functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 0 } is set
F is non empty set
ii is Relation-like NAT -defined F -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of F
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) -tuples_on F is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of F
F * is non empty functional FinSequence-membered FinSequenceSet of F
{ b₁ where b₁ is Relation-like NAT -defined F -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of F * : len b₁ = arity F } is set
i is Relation-like NAT * -defined Function-like homogeneous set
dom i is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity i } is set
j is set
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
F is Relation-like i * -defined i -valued Function-like homogeneous Element of K32(K33((i *),i))
dom F is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity F } is set
g is non empty Relation-like i * -defined i -valued Function-like homogeneous quasi_total Element of K32(K33((i *),i))
dom g is non empty functional with_common_domain FinSequence-membered Element of K32((i *))
F is set
ii is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is set
k is Relation-like NAT -defined i -valued Function-like finite V43( arity F) countable FinSequence-like FinSubsequence-like Element of (arity F) -tuples_on i
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional with_common_domain FinSequence-membered set
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
K33((i *),i) is Relation-like set
K32(K33((i *),i)) is non empty set
j is Relation-like i * -defined i -valued Function-like homogeneous Element of K32(K33((i *),i))
dom j is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) -tuples_on i is functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity j } is set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom i is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity i } is set
j is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () Element of K32(K33((NAT *),NAT))
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is set
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is set
ii is Relation-like NAT -defined NAT -valued Function-like finite V43( arity i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of (arity i) -tuples_on NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
j is non empty Relation-like i * -defined i -valued Function-like homogeneous Element of K32(K33((i *),i))
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = F } is set
g is set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined i -valued Function-like finite V43(F) countable FinSequence-like FinSubsequence-like Element of F -tuples_on i
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
j is Relation-like i * -defined i -valued Function-like homogeneous Element of K32(K33((i *),i))
dom j is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = F } is set
g is set
ii is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined i -valued Function-like finite V43(F) countable FinSequence-like FinSubsequence-like Element of F -tuples_on i
i is Relation-like set
j is Relation-like Function-like set
rng i is set
F is Relation-like Function-like set
dom F is set
dom j is set
i is set
<*> i is empty Relation-like non-empty empty-yielding NAT -defined i -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of i
i * is non empty functional FinSequence-membered FinSequenceSet of i
<*> i is empty Relation-like non-empty empty-yielding NAT -defined i -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of i
j is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i *
i is Relation-like () set
rng i is set
j is Relation-like Function-like homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like Function-like homogeneous set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom g is with_common_domain set
dom g is with_common_domain set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is with_common_domain set
dom j is with_common_domain set
dom j is with_common_domain set
the Element of dom j is Element of dom j
dom g is with_common_domain set
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is non empty set
ii -tuples_on k is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of k
k * is non empty functional FinSequence-membered FinSequenceSet of k
{ b₁ where b₁ is Relation-like NAT -defined k -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of k * : len b₁ = ii } is set
k is Relation-like NAT -defined k -valued Function-like finite V43(ii) countable FinSequence-like FinSubsequence-like Element of ii -tuples_on k
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
the Element of dom g is Element of dom g
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is Relation-like NAT -defined k -valued Function-like finite V43(ii) countable FinSequence-like FinSubsequence-like Element of ii -tuples_on k
len k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like Function-like homogeneous set
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like () set
len i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng i is finite countable set
j is Relation-like Function-like homogeneous set
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
K33((i *),i) is Relation-like set
K32(K33((i *),i)) is non empty set
F is set
g is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i)
F is non empty functional set
g is Relation-like Function-like Element of F
K33((i *),i) is Relation-like set
K32(K33((i *),i)) is non empty set
F is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i)
i is set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
K33((i *),i) is Relation-like set
K32(K33((i *),i)) is non empty set
F is Relation-like i * -defined i -valued Function-like Element of K32(K33((i *),i))
g is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i)
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
<*> i is empty proper Relation-like non-empty empty-yielding NAT -defined i -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of i
K33(NAT,i) is non empty non trivial Relation-like non finite set
the Element of i is Element of i
K32((i *)) is non empty set
{(<*> i)} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
g is functional FinSequence-membered Element of K32((i *))
g --> the Element of i is Relation-like i * -defined g -defined i -valued Function-like constant total Element of PFuncs ((i *),i)
{ the Element of i} is non empty trivial finite V43(1) countable set
K33(g,{ the Element of i}) is Relation-like set
(<*> i) .--> the Element of i is non empty Relation-like {(<*> i)} -defined K32(K33(NAT,i)) -defined {(<*> i)} -defined i -valued Function-like one-to-one finite countable homogeneous set
K32(K33(NAT,i)) is non empty non trivial non finite set
{(<*> i)} --> the Element of i is non empty Relation-like {(<*> i)} -defined i -valued { the Element of i} -valued Function-like constant total quasi_total finite countable Element of K32(K33({(<*> i)},{ the Element of i}))
K33({(<*> i)},{ the Element of i}) is non empty Relation-like finite countable set
K32(K33({(<*> i)},{ the Element of i})) is non empty finite V40() countable set
{(<*> i)} --> the Element of i is non empty Relation-like {(<*> i)} -defined i -valued Function-like constant total quasi_total finite countable Element of K32(K33({(<*> i)},i))
K33({(<*> i)},i) is non empty Relation-like set
K32(K33({(<*> i)},i)) is non empty set
F is Relation-like i * -defined i -valued Function-like Element of (i)
dom F is functional FinSequence-membered Element of K32((i *))
ii is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional FinSequence-membered set
i is set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
j is Relation-like i * -defined i -valued Function-like Element of (i)
F is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i)
i is non empty set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
K32((i)) is non empty set
j is non empty functional Element of K32((i))
F is Relation-like Function-like Element of j
(NAT) is non empty functional PartFunc-set of NAT * , NAT
PFuncs ((NAT *),NAT) is non empty functional PartFunc-set of NAT * , NAT
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like Element of PFuncs ((NAT *),NAT) : b₁ is homogeneous } is set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
rng i is V78() V79() V80() V81() V82() V83() V93() set
dom i is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of PFuncs ((NAT *),NAT)
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is functional with_common_domain FinSequence-membered set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is non empty set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
j is Relation-like set
rng j is set
F is Relation-like Function-like set
g is Relation-like Function-like set
dom g is set
dom F is set
F is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
arity ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
0 -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 0 } is set
<*> i is empty proper Relation-like non-empty empty-yielding NAT -defined i -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of i
K33(NAT,i) is non empty non trivial Relation-like non finite set
{(<*> i)} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
k is non empty Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
arity k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity k) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity k } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
dom ((i -tuples_on NAT) --> j) is non empty functional with_common_domain FinSequence-membered Element of K32((i -tuples_on NAT))
K32((i -tuples_on NAT)) is non empty set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
rng (i,j) is V78() V79() V80() V81() V82() V83() V93() set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
arity (i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
the Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
dom (i,j) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
dom F is functional FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
g is set
F is set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of PFuncs ((NAT *),NAT)
dom g is functional FinSequence-membered Element of K32((NAT *))
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
F . ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii /. j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii /. j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of NAT *
F /. k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
g is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom g is functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is set
F . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
ii /. j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii /. j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
rng (i,j) is V78() V79() V80() V81() V82() V83() V93() set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity (i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (i,j) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
F is set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
K32(REAL) * is non empty functional FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
{NAT} is non empty trivial finite V43(1) countable set
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
dom (proj ((i |-> NAT),j)) is functional with_common_domain Element of K32((product (i |-> NAT)))
K32((product (i |-> NAT))) is non empty set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
rng (proj ((i |-> NAT),j)) is set
g is set
F is set
(proj ((i |-> NAT),j)) . F is set
ii is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(proj ((i |-> NAT),j)) . k is set
k . j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
rng (i,j) is V78() V79() V80() V81() V82() V83() V93() set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
rng (i,j) is V78() V79() V80() V81() V82() V83() V93() set
F is set
g is set
(i,j) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |-> g is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
(Seg i) --> g is Relation-like Seg i -defined Seg i -defined NAT -valued RAT -valued INT -valued {g} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((Seg i),{g}))
{g} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((Seg i),{g}) is Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33((Seg i),{g})) is non empty finite V40() countable set
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
dom (i,j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
F is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
arity (i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
F is set
dom (i,j) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
F is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),j) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
F is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
(i,j) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (i,j) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
i is set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty set
F is non empty set
(F) is non empty functional PartFunc-set of F * ,F
F * is non empty functional FinSequence-membered FinSequenceSet of F
PFuncs ((F *),F) is non empty functional PartFunc-set of F * ,F
{ b₁ where b₁ is Relation-like F * -defined F -valued Function-like Element of PFuncs ((F *),F) : b₁ is homogeneous } is set
K33(j,(F)) is non empty Relation-like set
K32(K33(j,(F))) is non empty set
i -tuples_on F is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of F
{ b₁ where b₁ is Relation-like NAT -defined F -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of F * : len b₁ = i } is set
ii is non empty Relation-like j -defined (F) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(j,(F)))
rng ii is non empty set
Union ii is set
union (rng ii) is set
k is set
k is non empty functional () set
{ (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
ij is Relation-like Function-like set
rng ij is set
k1 is set
dom ij is set
k1 is set
ij . k1 is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
k is set
k is Relation-like Function-like Element of k
dom k is set
x is Relation-like F * -defined F -valued Function-like homogeneous Element of (F)
x . k1 is set
rng x is set
dom ij is set
k1 is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
k1 is set
k is Relation-like Function-like Element of k
dom k is set
dom ii is non empty Element of K32(j)
K32(j) is non empty set
k is set
ii . k is Relation-like Function-like set
x is Element of j
ii . x is Relation-like Function-like homogeneous Element of (F)
dom (ii . x) is with_common_domain set
K33((F *),F) is non empty Relation-like set
K32(K33((F *),F)) is non empty set
k1 is Relation-like F * -defined F -valued Function-like Element of K32(K33((F *),F))
k1 is Relation-like F * -defined F -valued Function-like Element of PFuncs ((F *),F)
dom k1 is functional FinSequence-membered Element of K32((F *))
K32((F *)) is non empty set
k is Relation-like F * -defined F -valued Function-like homogeneous Element of (F)
dom k is functional with_common_domain FinSequence-membered Element of K32((F *))
K32((F *)) is non empty set
i is non empty set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
K33(NAT,(i)) is non empty non trivial Relation-like non finite set
K32(K33(NAT,(i))) is non empty non trivial non finite set
F is non empty Relation-like NAT -defined (i) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(i)))
Union F is set
rng F is non empty set
union (rng F) is set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . g is Relation-like Function-like homogeneous Element of (i)
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g + ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . (g + ii) is Relation-like Function-like homogeneous Element of (i)
(g + ii) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F . ((g + ii) + 1) is Relation-like Function-like homogeneous Element of (i)
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g + (ii + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F . (g + (ii + 1)) is Relation-like Function-like homogeneous Element of (i)
g + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . (g + 0) is Relation-like Function-like homogeneous Element of (i)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
g + ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . F is Relation-like Function-like homogeneous Element of (i)
K32((i)) is non empty set
g is non empty functional Element of K32((i))
F is Relation-like Function-like set
ii is Relation-like Function-like set
dom F is non empty countable V78() V79() V80() V81() V82() V83() V91() V93() Element of K32(NAT)
k is set
F . k is Relation-like Function-like set
k is set
F . k is Relation-like Function-like set
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like Function-like homogeneous Element of g
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
F is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i)
dom F is functional FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
{ (dom b₁) where b₁ is Relation-like Function-like homogeneous Element of g : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like homogeneous Element of g : verum } is set
ii is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is set
ij is Relation-like Function-like homogeneous Element of g
dom ij is with_common_domain set
k1 is set
k1 is Relation-like Function-like homogeneous Element of g
dom k1 is with_common_domain set
dom F is non empty countable V78() V79() V80() V81() V82() V83() V91() V93() Element of K32(NAT)
k is set
F . k is Relation-like Function-like set
k is set
F . k is Relation-like Function-like set
x is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
i is non empty set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
F is Relation-like NAT -defined (i) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (i)
<:F:> is Relation-like Function-like set
dom <:F:> is set
(F) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = (F) } is set
doms F is Relation-like Function-like set
meet (doms F) is set
rng (doms F) is set
meet (rng (doms F)) is set
g is set
F is set
dom (doms F) is set
ii is set
(doms F) . ii is set
rng F is finite countable set
SubFuncs (rng F) is set
F " (SubFuncs (rng F)) is finite countable set
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
F . ii is Relation-like Function-like set
k is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
dom k is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
arity k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity k) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity k } is set
i is non empty set
(i) is non empty functional PartFunc-set of i * ,i
i * is non empty functional FinSequence-membered FinSequenceSet of i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
j is Relation-like NAT -defined (i) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (i)
<:j:> is Relation-like Function-like set
dom <:j:> is set
(j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = (j) } is set
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
F is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
g is Relation-like NAT -defined (i) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (i)
<:g:> is Relation-like Function-like homogeneous set
<:g:> * F is Relation-like i -valued Function-like homogeneous set
dom (<:g:> * F) is with_common_domain set
(g) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = (g) } is set
rng (<:g:> * F) is set
dom <:g:> is with_common_domain set
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
i is non empty set
j is non empty set
F is non empty functional PartFunc-set of i,j
K32(F) is non empty set
g is non empty functional Element of K32(F)
F is Relation-like Function-like Element of g
i is Relation-like NAT * -defined Function-like homogeneous set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() set
j is Relation-like Function-like set
rng <*i*> is non empty trivial finite V43(1) countable set
F is Relation-like Function-like set
dom F is set
dom j is set
{i} is non empty trivial functional finite V43(1) with_common_domain countable set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity i } is set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
<*i*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
(<*i*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng <*i*> is non empty trivial finite V43(1) countable set
{i} is non empty trivial functional finite V43(1) with_common_domain countable set
j is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
<:F:> is Relation-like Function-like homogeneous set
<:F:> * i is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(F) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
(F) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (F) } is set
g is set
F is Relation-like NAT -defined NAT -valued Function-like finite V43((F)) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of (F) -tuples_on NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
F is non empty Relation-like i * -defined i -valued Function-like homogeneous quasi_total Element of (i)
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined (i) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (i)
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng g is finite countable set
<:g:> is Relation-like Function-like homogeneous set
<:g:> * F is Relation-like i -valued Function-like homogeneous set
dom (<:g:> * F) is with_common_domain set
(g) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = (g) } is set
k is set
rngs g is Relation-like Function-like set
product (rngs g) is functional with_common_domain product-like set
(len g) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = len g } is set
k is set
dom (rngs g) is set
SubFuncs (rng g) is set
g " (SubFuncs (rng g)) is finite countable set
ij is Relation-like Function-like set
dom ij is set
k1 is set
dom g is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Seg (len g) is finite V43( len g) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
g . k1 is Relation-like Function-like set
k1 is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
rng k1 is finite countable set
k1 is set
dom k1 is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k is set
k1 . k is set
(rngs g) . k is set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . k is Relation-like Function-like set
x is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
rng x is set
(rngs g) . k is set
k1 is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng <:g:> is set
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
(arity F) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity F } is set
dom <:g:> is with_common_domain set
k is set
doms g is Relation-like Function-like set
dom (doms g) is set
SubFuncs (rng g) is set
g " (SubFuncs (rng g)) is finite countable set
ij is set
rng (doms g) is set
k1 is set
(doms g) . k1 is set
dom g is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . k1 is Relation-like Function-like set
(doms g) . k1 is set
k is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
dom k is functional with_common_domain FinSequence-membered Element of K32((i *))
arity k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity k) -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = arity k } is set
ij is set
g . ij is Relation-like Function-like set
meet (doms g) is set
meet (rng (doms g)) is set
<:g:> . k is set
rng (<:g:> * F) is set
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
k is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
k is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom k is functional with_common_domain FinSequence-membered set
dom k is functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
F is Relation-like i * -defined i -valued Function-like homogeneous quasi_total Element of (i)
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined (i) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (i)
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng g is finite countable set
<:g:> is Relation-like Function-like homogeneous set
<:g:> * F is Relation-like i -valued Function-like homogeneous set
K33((i *),i) is non empty Relation-like set
K32(K33((i *),i)) is non empty set
F is Relation-like i * -defined i -valued Function-like Element of K32(K33((i *),i))
ii is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like FinSequence of i
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional FinSequence-membered set
dom F is functional FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
ii is set
ii is Relation-like i * -defined i -valued Function-like homogeneous Element of (i)
ii is set
i is non empty set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(i) is non empty functional PartFunc-set of i * ,i
PFuncs ((i *),i) is non empty functional PartFunc-set of i * ,i
{ b₁ where b₁ is Relation-like i * -defined i -valued Function-like Element of PFuncs ((i *),i) : b₁ is homogeneous } is set
F is non empty Relation-like i * -defined i -valued Function-like homogeneous quasi_total Element of (i)
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is non empty Relation-like i * -defined i -valued Function-like homogeneous quasi_total Element of (i)
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . {} is set
g . {} is set
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((i *))
K32((i *)) is non empty set
0 -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = 0 } is set
<*> i is empty proper Relation-like non-empty empty-yielding NAT -defined i -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of i
K33(NAT,i) is non empty non trivial Relation-like non finite set
{(<*> i)} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
dom g is non empty functional with_common_domain FinSequence-membered Element of K32((i *))
F is set
F . F is set
g . F is set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
K33(NAT,(NAT)) is non empty non trivial Relation-like non finite set
K32(K33(NAT,(NAT))) is non empty non trivial non finite set
g is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g /. F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom g is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del (g,F) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom i is functional with_common_domain FinSequence-membered Element of K32((NAT *))
K32((NAT *)) is non empty set
g +* (F,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
i . (Del (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g +* (F,0)) .--> (i . (Del (g,F))) is non empty Relation-like {(g +* (F,0))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
K33(NAT,NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33(NAT,NAT)) is non empty non trivial non finite set
{(g +* (F,0))} --> (i . (Del (g,F))) is non empty Relation-like {(g +* (F,0))} -defined NAT -valued RAT -valued INT -valued {(i . (Del (g,F)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,0))},{(i . (Del (g,F)))}))
{(i . (Del (g,F)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,0))},{(i . (Del (g,F)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,0))},{(i . (Del (g,F)))})) is non empty finite V40() countable set
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F . (Del (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom ii is functional with_common_domain FinSequence-membered set
ij is non empty functional FinSequence-membered Element of K32((NAT *))
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij --> k is non empty Relation-like NAT * -defined ij -defined NAT -valued RAT -valued INT -valued Function-like constant total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of PFuncs ((NAT *),NAT)
{k} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33(ij,{k}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(g +* (F,0)) .--> k is non empty Relation-like {(g +* (F,0))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,0))} --> k is non empty Relation-like {(g +* (F,0))} -defined NAT -valued RAT -valued INT -valued {k} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,0))},{k}))
K33({(g +* (F,0))},{k}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,0))},{k})) is non empty finite V40() countable set
k1 is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g +* (F,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g +* (F,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
x is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom x is functional with_common_domain FinSequence-membered Element of K32((NAT *))
x . (g +* (F,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(x . (g +* (F,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(g +* (F,k)) ^ <*(x . (g +* (F,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
(g +* (F,(k + 1))) .--> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)) is non empty Relation-like {(g +* (F,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,(k + 1)))} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(g +* (F,(k + 1)))} --> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)) is non empty Relation-like {(g +* (F,(k + 1)))} -defined RAT -valued {(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))}))
{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))})) is non empty finite V40() countable set
x +* ((g +* (F,(k + 1))) .--> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{(g +* (F,(k + 1)))} --> (j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)) is non empty Relation-like {(g +* (F,(k + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(k + 1)))},NAT))
K33({(g +* (F,(k + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(k + 1)))},NAT)) is non empty non trivial non finite set
{(j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(k + 1)))},{(j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
x +* ({(g +* (F,(k + 1)))} --> (j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is Relation-like Function-like set
m is functional FinSequence-membered Element of K32((NAT *))
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m --> m is Relation-like NAT * -defined m -defined NAT -valued RAT -valued INT -valued Function-like constant total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of PFuncs ((NAT *),NAT)
{m} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33(m,{m}) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
dom (m --> m) is functional FinSequence-membered Element of K32(m)
K32(m) is non empty set
dom (x +* ({(g +* (F,(k + 1)))} --> (j . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)))) is set
(dom x) \/ m is functional FinSequence-membered Element of K32((NAT *))
dom (g +* (F,(k + 1))) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (g +* (F,(k + 1))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(len g) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len g } is set
x +* (m --> m) is Relation-like NAT * -defined NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
y is set
dom (g +* (F,k)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
c₂₀ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of NAT *
len c₂₀ is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (g +* (F,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
y is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom ii is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(g +* (F,(k + 1))) .--> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)) is non empty Relation-like {(g +* (F,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,(k + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,(k + 1)))} --> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>)) is non empty Relation-like {(g +* (F,(k + 1)))} -defined NAT -valued RAT -valued INT -valued {(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))}))
{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(k + 1)))},{(ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))})) is non empty finite V40() countable set
x +* ((g +* (F,(k + 1))) .--> (ii . ((g +* (F,k)) ^ <*(x . (g +* (F,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . (g /. F) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g +* (F,x) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k . x is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . x) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . x) . (g +* (F,x)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . x) . (g +* (F,x)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
x + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (x + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g +* (F,(x + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g +* (F,(x + 1))) .--> (j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>)) is non empty Relation-like {(g +* (F,(x + 1)))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,(x + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,(x + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(g +* (F,(x + 1)))} --> (j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>)) is non empty Relation-like {(g +* (F,(x + 1)))} -defined NAT -valued RAT -valued INT -valued {(j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(x + 1)))},{(j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))}))
{(j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(x + 1)))},{(j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(x + 1)))},{(j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))})) is non empty finite V40() countable set
(k . x) +* ((g +* (F,(x + 1))) .--> (j . ((g +* (F,x)) ^ <*((k . x) . (g +* (F,x)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g +* (F,d) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k . d is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . d) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . d) . (g +* (F,d)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . d) . (g +* (F,d)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(g +* (F,d)) ^ <*((k . d) . (g +* (F,d)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
d + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (d + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom j is with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . (g /. F) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
ij . (g /. F) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ij . (k1 + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g +* (F,k1) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (k . k1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . k1) . (g +* (F,k1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . k1) . (g +* (F,k1)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered set
g +* (F,(k1 + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
(g +* (F,(k1 + 1))) .--> (F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>)) is non empty Relation-like {(g +* (F,(k1 + 1)))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,(k1 + 1)))} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,(k1 + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(g +* (F,(k1 + 1)))} --> (F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>)) is non empty Relation-like {(g +* (F,(k1 + 1)))} -defined RAT -valued {(F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(k1 + 1)))},{(F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))}))
{(F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(k1 + 1)))},{(F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(k1 + 1)))},{(F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))})) is non empty finite V40() countable set
(k . k1) +* ((g +* (F,(k1 + 1))) .--> (F . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
k . (k1 + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V67() set
(g +* (F,(k1 + 1))) .--> (j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>)) is non empty Relation-like {(g +* (F,(k1 + 1)))} -defined K32(K33(NAT,NAT)) -defined {(g +* (F,(k1 + 1)))} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (F,(k1 + 1)))} --> (j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>)) is non empty Relation-like {(g +* (F,(k1 + 1)))} -defined RAT -valued {(j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (F,(k1 + 1)))},{(j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))}))
{(j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (F,(k1 + 1)))},{(j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (F,(k1 + 1)))},{(j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))})) is non empty finite V40() countable set
(k . k1) +* ((g +* (F,(k1 + 1))) .--> (j . ((g +* (F,k1)) ^ <*((k . k1) . (g +* (F,k1)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i,ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,0)) .--> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,0))} --> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (ii,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))}))
{(j . (Del (ii,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,0))},{(j . (Del (ii,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))})) is non empty finite V40() countable set
dom F is with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . (ii /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . ij is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . ij) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (ij + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (ij + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,ij) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k . ij) . (ii +* (i,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . (ii +* (i,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,(ij + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,(ij + 1))) .--> (F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>)) is non empty Relation-like {(ii +* (i,(ij + 1)))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,(ij + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,(ij + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,(ij + 1)))} --> (F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>)) is non empty Relation-like {(ii +* (i,(ij + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,(ij + 1)))},{(F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))}))
{(F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,(ij + 1)))},{(F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,(ij + 1)))},{(F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))})) is non empty finite V40() countable set
(k . ij) +* ((ii +* (i,(ij + 1))) .--> (F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom ((ii +* (i,(ij + 1))) .--> (F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(ii +* (i,(ij + 1)))})
K32({(ii +* (i,(ij + 1)))}) is non empty finite V40() countable set
(dom (k . ij)) \/ (dom ((ii +* (i,(ij + 1))) .--> (F . ((ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*>)))) is non empty set
(dom (k . ij)) \/ {(ii +* (i,(ij + 1)))} is non empty set
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,k1) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii +* (i,ij) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k . ij) . (ii +* (i,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . (ii +* (i,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,k1) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii +* (i,ij) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k . ij) . (ii +* (i,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . (ii +* (i,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,ij)) ^ <*((k . ij) . (ii +* (i,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,0)) .--> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,0))} --> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (ii,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))}))
{(j . (Del (ii,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,0))},{(j . (Del (ii,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))})) is non empty finite V40() countable set
dom F is with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . (ii /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i,k) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (k,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (k,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k +* (i,0)) .--> (j . (Del (k,i))) is non empty Relation-like {(k +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(k +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(k +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(k +* (i,0))} --> (j . (Del (k,i))) is non empty Relation-like {(k +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (k,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(k +* (i,0))},{(j . (Del (k,i)))}))
{(j . (Del (k,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(k +* (i,0))},{(j . (Del (k,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(k +* (i,0))},{(j . (Del (k,i)))})) is non empty finite V40() countable set
dom F is with_common_domain set
k1 is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k1 . (k /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,0)) .--> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,0))} --> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (ii,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))}))
{(j . (Del (ii,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,0))},{(j . (Del (ii,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))})) is non empty finite V40() countable set
k1 is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k1 . (ii /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . k) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . (k + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k1 . k) . (ii +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . k) . (ii +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
x is set
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,(k + 1)))} --> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))}))
{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))})) is non empty finite V40() countable set
(k1 . k) +* ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(ii +* (i,(k + 1)))})
K32({(ii +* (i,(k + 1)))}) is non empty finite V40() countable set
(dom (k1 . k)) \/ (dom ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>)))) is non empty set
(dom (k1 . k)) \/ {(ii +* (i,(k + 1)))} is non empty set
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,d) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,d) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (k1 . k) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k1 . k) . (ii +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . k) . (ii +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V67() set
(ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,(k + 1)))} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,(k + 1)))} --> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined RAT -valued {(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))}))
{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))})) is non empty finite V40() countable set
(k1 . k) +* ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k1 . k) . (ii +* (i,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
k1 . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (k1 . k) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k1 . k) . (k +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . k) . (k +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V67() set
(k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>)) is non empty Relation-like {(k +* (i,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(k +* (i,(k + 1)))} -defined RAT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(k +* (i,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(k +* (i,(k + 1)))} --> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>)) is non empty Relation-like {(k +* (i,(k + 1)))} -defined RAT -valued {(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))}))
{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))})) is non empty finite V40() countable set
(k1 . k) +* ((k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k1 . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . (k + k) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(k + k) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . ((k + k) + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii +* (i,(k + k)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k1 . (k + k)) . (ii +* (i,(k + k))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . (k + k)) . (ii +* (i,(k + k))))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
m is set
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,((k + k) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
{(ii +* (i,((k + k) + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,((k + k) + 1))) .--> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>)) is non empty Relation-like {(ii +* (i,((k + k) + 1)))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,((k + k) + 1)))} --> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>)) is non empty Relation-like {(ii +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,((k + k) + 1)))},{(F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))}))
{(F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,((k + k) + 1)))},{(F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,((k + k) + 1)))},{(F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))})) is non empty finite V40() countable set
dom ((ii +* (i,((k + k) + 1))) .--> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(ii +* (i,((k + k) + 1)))})
K32({(ii +* (i,((k + k) + 1)))}) is non empty finite V40() countable set
{(ii +* (i,((k + k) + 1)))} --> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>)) is non empty Relation-like {(ii +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,((k + k) + 1)))},NAT))
K33({(ii +* (i,((k + k) + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,((k + k) + 1)))},NAT)) is non empty non trivial non finite set
dom ({(ii +* (i,((k + k) + 1)))} --> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(ii +* (i,((k + k) + 1)))})
(dom (k1 . (k + k))) /\ (dom ({(ii +* (i,((k + k) + 1)))} --> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>)))) is trivial functional finite V40() with_common_domain countable Element of K32({(ii +* (i,((k + k) + 1)))})
m is set
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,m) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(ii +* (i,((k + k) + 1))) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,m)) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k1 . (k + k)) +* ((ii +* (i,((k + k) + 1))) .--> (F . ((ii +* (i,(k + k))) ^ <*((k1 . (k + k)) . (ii +* (i,(k + k))))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k + (k + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . (k + (k + 1)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . (k + 0) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . k) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . (k + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k1 . k) . (k +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . k) . (k +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
x is set
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>)) is non empty Relation-like {(k +* (i,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(k +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(k +* (i,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(k +* (i,(k + 1)))} --> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>)) is non empty Relation-like {(k +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))}))
{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(k +* (i,(k + 1)))},{(F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))})) is non empty finite V40() countable set
(k1 . k) +* ((k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom ((k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(k +* (i,(k + 1)))})
K32({(k +* (i,(k + 1)))}) is non empty finite V40() countable set
(dom (k1 . k)) \/ (dom ((k +* (i,(k + 1))) .--> (F . ((k +* (i,k)) ^ <*((k1 . k) . (k +* (i,k)))*>)))) is non empty set
(dom (k1 . k)) \/ {(k +* (i,(k + 1)))} is non empty set
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (i,d) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (i,d) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k1 . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . (k + k) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(k + k) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . ((k + k) + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k +* (i,(k + k)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k1 . (k + k)) . (k +* (i,(k + k))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k1 . (k + k)) . (k +* (i,(k + k))))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
m is set
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k +* (i,((k + k) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
{(k +* (i,((k + k) + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k +* (i,((k + k) + 1))) .--> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>)) is non empty Relation-like {(k +* (i,((k + k) + 1)))} -defined K32(K33(NAT,NAT)) -defined {(k +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(k +* (i,((k + k) + 1)))} --> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>)) is non empty Relation-like {(k +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(k +* (i,((k + k) + 1)))},{(F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))}))
{(F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(k +* (i,((k + k) + 1)))},{(F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(k +* (i,((k + k) + 1)))},{(F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))})) is non empty finite V40() countable set
dom ((k +* (i,((k + k) + 1))) .--> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(k +* (i,((k + k) + 1)))})
K32({(k +* (i,((k + k) + 1)))}) is non empty finite V40() countable set
{(k +* (i,((k + k) + 1)))} --> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>)) is non empty Relation-like {(k +* (i,((k + k) + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(k +* (i,((k + k) + 1)))},NAT))
K33({(k +* (i,((k + k) + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(k +* (i,((k + k) + 1)))},NAT)) is non empty non trivial non finite set
dom ({(k +* (i,((k + k) + 1)))} --> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(k +* (i,((k + k) + 1)))})
(dom (k1 . (k + k))) /\ (dom ({(k +* (i,((k + k) + 1)))} --> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>)))) is trivial functional finite V40() with_common_domain countable Element of K32({(k +* (i,((k + k) + 1)))})
m is set
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (i,m) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k +* (i,((k + k) + 1))) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k +* (i,m)) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k1 . (k + k)) +* ((k +* (i,((k + k) + 1))) .--> (F . ((k +* (i,(k + k))) ^ <*((k1 . (k + k)) . (k +* (i,(k + k))))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k1 . (k + k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k + (k + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 . (k + (k + 1)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . (k + 0) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . ij is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 . ij is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 . m is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . m) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k1 . m is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k1 . m) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom (k1 . m)) /\ (dom (k1 . m)) is functional FinSequence-membered Element of K32((NAT *))
m is set
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,m) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
y is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (i,y) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
k + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
d is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
d is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom d is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom k) /\ (dom d) is functional FinSequence-membered Element of K32((NAT *))
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
d is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom d is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom k) /\ (dom d) is functional FinSequence-membered Element of K32((NAT *))
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
ij + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
x is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom x is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom k) /\ (dom x) is functional FinSequence-membered Element of K32((NAT *))
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom x is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom k) /\ (dom x) is functional FinSequence-membered Element of K32((NAT *))
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
1 + (arity j) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(1 + (arity j)) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 1 + (arity j) } is set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i,ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,0)) .--> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,0))} --> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (ii,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))}))
{(j . (Del (ii,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,0))},{(j . (Del (ii,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))})) is non empty finite V40() countable set
dom F is with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . (ii /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . k) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(k . k) . (ii +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . k) . (ii +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (ii +* (i,k)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ii +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (ii +* (i,(k + 1))) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (ii +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (ii +* (i,(k + 1))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{(ii +* (i,(k + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
k . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,(k + 1)))} --> (F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>)) is non empty Relation-like {(ii +* (i,(k + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))}))
{(F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,(k + 1)))},{(F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))})) is non empty finite V40() countable set
(k . k) +* ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k . (k + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(ii +* (i,(k + 1)))})
K32({(ii +* (i,(k + 1)))}) is non empty finite V40() countable set
(dom (k . k)) \/ (dom ((ii +* (i,(k + 1))) .--> (F . ((ii +* (i,k)) ^ <*((k . k) . (ii +* (i,k)))*>)))) is non empty set
(dom (k . k)) \/ {(ii +* (i,(k + 1)))} is non empty set
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (k + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(arity j) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity j } is set
len (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (ii +* (i,0)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
<*> NAT is empty proper Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
len (ii +* (i,0)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii +* (i,0)) .--> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined K32(K33(NAT,NAT)) -defined {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(ii +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(ii +* (i,0))} --> (j . (Del (ii,i))) is non empty Relation-like {(ii +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (ii,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))}))
{(j . (Del (ii,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(ii +* (i,0))},{(j . (Del (ii,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(ii +* (i,0))},{(j . (Del (ii,i)))})) is non empty finite V40() countable set
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . (ii /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (k + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
(arity j) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
1 + (arity j) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
(j,F,i,g) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i,g) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom g is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
g /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (g,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
g +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
j . (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g +* (i,0)) .--> (j . (Del (g,i))) is non empty Relation-like ((arity j) + 1) -tuples_on NAT -defined {(g +* (i,0))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (i,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(g +* (i,0))} --> (j . (Del (g,i))) is non empty Relation-like {(g +* (i,0))} -defined NAT -valued RAT -valued INT -valued {(j . (Del (g,i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (i,0))},{(j . (Del (g,i)))}))
{(j . (Del (g,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (i,0))},{(j . (Del (g,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (i,0))},{(j . (Del (g,i)))})) is non empty finite V40() countable set
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
F . (g /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g +* (i,(g . i)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g +* (i,ii) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F . ii is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (F . ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F . ii) . (g +* (i,ii)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F . ii) . (g +* (i,ii)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity j) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity j) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*k*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
ij is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(g +* (i,ii)) ^ ij is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (g +* (i,ii)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((arity j) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(arity F) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity F } is set
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g +* (i,(ii + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
{(g +* (i,(ii + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{(g +* (i,(ii + 1)))} --> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>)) is non empty Relation-like {(g +* (i,(ii + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (i,(ii + 1)))},NAT))
K33({(g +* (i,(ii + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(g +* (i,(ii + 1)))},NAT)) is non empty non trivial non finite set
{(F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(g +* (i,(ii + 1)))},{(F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
dom ({(g +* (i,(ii + 1)))} --> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(g +* (i,(ii + 1)))})
K32({(g +* (i,(ii + 1)))}) is non empty finite V40() countable set
F . (ii + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(g +* (i,(ii + 1))) .--> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>)) is non empty Relation-like ((arity j) + 1) -tuples_on NAT -defined {(g +* (i,(ii + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(g +* (i,(ii + 1)))} --> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>)) is non empty Relation-like {(g +* (i,(ii + 1)))} -defined NAT -valued RAT -valued INT -valued {(F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(g +* (i,(ii + 1)))},{(F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))}))
K32(K33({(g +* (i,(ii + 1)))},{(F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))})) is non empty finite V40() countable set
(F . ii) +* ((g +* (i,(ii + 1))) .--> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (F . (ii + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom (F . ii)) \/ (dom ({(g +* (i,(ii + 1)))} --> (F . ((g +* (i,ii)) ^ <*((F . ii) . (g +* (i,ii)))*>)))) is non empty set
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(len ii) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len ii } is set
(arity j) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity j } is set
len (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined NAT -valued Function-like finite V43( len ii) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of (len ii) -tuples_on NAT
dom (F . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
K33((((arity i) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity i) + 1) -tuples_on NAT),(NAT))) is non empty set
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity i) + 1) -tuples_on NAT),(NAT)))
K32((NAT)) is non empty set
rng k is non empty set
k is non empty functional Element of K32((NAT))
ij is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),k)
ij is non empty functional PartFunc-set of NAT * , NAT
k1 is Relation-like Function-like set
k1 is Relation-like Function-like set
dom k is non empty functional with_common_domain FinSequence-membered Element of K32((((arity i) + 1) -tuples_on NAT))
K32((((arity i) + 1) -tuples_on NAT)) is non empty set
k is set
k . k is Relation-like Function-like set
k is set
k . k is Relation-like Function-like set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
d is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(g,F,F,d) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(g,F,F,x) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
k . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(g,F,F,k1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . k1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
Union k is set
union (rng k) is set
k1 is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom k1 is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k1 is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
k . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(i,j,F,k1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity i) + 1) -tuples_on NAT),(NAT)))
Union ii is set
rng ii is non empty set
union (rng ii) is set
k is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity i) + 1) -tuples_on NAT),(NAT)))
Union k is set
rng k is non empty set
union (rng k) is set
k is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(i,j,F,k) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . k is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
K33((((arity j) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity j) + 1) -tuples_on NAT),(NAT))) is non empty set
ii is non empty Relation-like ((arity j) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity j) + 1) -tuples_on NAT),(NAT)))
Union ii is set
rng ii is non empty set
union (rng ii) is set
dom ii is non empty functional with_common_domain FinSequence-membered Element of K32((((arity j) + 1) -tuples_on NAT))
K32((((arity j) + 1) -tuples_on NAT)) is non empty set
the Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
k is set
ij is set
ii . ij is Relation-like Function-like set
k1 is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
ii . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,i,k1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii . the Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,i, the Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
g is set
K33((((arity j) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity j) + 1) -tuples_on NAT),(NAT))) is non empty set
F is non empty Relation-like ((arity j) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity j) + 1) -tuples_on NAT),(NAT)))
Union F is set
rng F is non empty set
union (rng F) is set
ii is Relation-like Function-like set
k is Relation-like Function-like set
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((((arity j) + 1) -tuples_on NAT))
K32((((arity j) + 1) -tuples_on NAT)) is non empty set
k is set
F . k is Relation-like Function-like set
k1 is set
F . k1 is Relation-like Function-like set
k1 is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,i,k1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F . ij is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,i,ij) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F . ii is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (F . ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom ii is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
1 + (arity j) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity (j,F,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is set
K33((((arity j) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity j) + 1) -tuples_on NAT),(NAT))) is non empty set
k is non empty Relation-like ((arity j) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity j) + 1) -tuples_on NAT),(NAT)))
Union k is set
rng k is non empty set
union (rng k) is set
k is non empty functional () set
union k is Relation-like Function-like set
dom (union k) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
dom k is non empty functional with_common_domain FinSequence-membered Element of K32((((arity j) + 1) -tuples_on NAT))
K32((((arity j) + 1) -tuples_on NAT)) is non empty set
ii is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
k . ii is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(j,F,i,ii) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i,ii) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F +* (g,(F + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
F +* (g,F) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (F +* (g,(F + 1))) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((F +* (g,(F + 1))),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom i is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F +* (g,(F + 1))) +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
{((F +* (g,(F + 1))) +* (g,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
i . (Del ((F +* (g,(F + 1))),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{((F +* (g,(F + 1))) +* (g,0))} --> (i . (Del ((F +* (g,(F + 1))),g))) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,0))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,0))},NAT))
K33({((F +* (g,(F + 1))) +* (g,0))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,(F + 1))) +* (g,0))},NAT)) is non empty non trivial non finite set
{(i . (Del ((F +* (g,(F + 1))),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,(F + 1))) +* (g,0))},{(i . (Del ((F +* (g,(F + 1))),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom j is non empty with_common_domain set
dom (F +* (g,F)) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((F +* (g,F)),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
ij is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
ij . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F +* (g,F)) +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
{((F +* (g,F)) +* (g,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
i . (Del ((F +* (g,F)),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{((F +* (g,F)) +* (g,0))} --> (i . (Del ((F +* (g,F)),g))) is non empty Relation-like {((F +* (g,F)) +* (g,0))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,F)) +* (g,0))},NAT))
K33({((F +* (g,F)) +* (g,0))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,F)) +* (g,0))},NAT)) is non empty non trivial non finite set
{(i . (Del ((F +* (g,F)),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,F)) +* (g,0))},{(i . (Del ((F +* (g,F)),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
dom F is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
F +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
Del (F,g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k1 is set
ij . k1 is Relation-like Function-like set
k . k1 is Relation-like Function-like set
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . k1 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ij . (k1 + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . (k1 + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F +* (g,F)) +* (g,(k1 + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
F +* (g,(k1 + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,(F + 1))) +* (g,(k1 + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,F)) +* (g,k1) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
F +* (g,k1) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,(F + 1))) +* (g,k1) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (ij . k1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(ij . k1) . ((F +* (g,F)) +* (g,k1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((F +* (g,F)) +* (g,(k1 + 1))) .--> (j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>)) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {((F +* (g,F)) +* (g,(k1 + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{((F +* (g,F)) +* (g,(k1 + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{((F +* (g,F)) +* (g,(k1 + 1)))} --> (j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>)) is non empty Relation-like {((F +* (g,F)) +* (g,(k1 + 1)))} -defined NAT -valued RAT -valued INT -valued {(j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,F)) +* (g,(k1 + 1)))},{(j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))}))
{(j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,F)) +* (g,(k1 + 1)))},{(j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,F)) +* (g,(k1 + 1)))},{(j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))})) is non empty finite V40() countable set
(ij . k1) +* (((F +* (g,F)) +* (g,(k1 + 1))) .--> (j . (((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (ij . k1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(ij . k1) . ((F +* (g,F)) +* (g,k1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (ij . k1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(ij . k1) . ((F +* (g,F)) +* (g,k1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,F)) +* (g,k1)) ^ <*((ij . k1) . ((F +* (g,F)) +* (g,k1)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom F is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F +* (g,F) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F +* (g,(F + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (F +* (g,(F + 1))) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((F +* (g,(F + 1))),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom i is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F +* (g,(F + 1))) +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
{((F +* (g,(F + 1))) +* (g,0))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
i . (Del ((F +* (g,(F + 1))),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{((F +* (g,(F + 1))) +* (g,0))} --> (i . (Del ((F +* (g,(F + 1))),g))) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,0))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,0))},NAT))
K33({((F +* (g,(F + 1))) +* (g,0))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,(F + 1))) +* (g,0))},NAT)) is non empty non trivial non finite set
{(i . (Del ((F +* (g,(F + 1))),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,(F + 1))) +* (g,0))},{(i . (Del ((F +* (g,(F + 1))),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
dom j is non empty with_common_domain set
k . (F + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(k . (F + 1)) . (F +* (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(k . F) . (F +* (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F +* (g,(F + 1))) +* (g,F) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . F) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . F) . ((F +* (g,(F + 1))) +* (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F +* (g,(F + 1))) +* (g,(F + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,(F + 1))) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F +* (g,F)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{((F +* (g,(F + 1))) +* (g,(F + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{((F +* (g,(F + 1))) +* (g,(F + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>)) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,(F + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},NAT))
K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},NAT)) is non empty non trivial non finite set
{(j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
dom ({((F +* (g,(F + 1))) +* (g,(F + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({((F +* (g,(F + 1))) +* (g,(F + 1)))})
K32({((F +* (g,(F + 1))) +* (g,(F + 1)))}) is non empty finite V40() countable set
((F +* (g,(F + 1))) +* (g,(F + 1))) .--> (j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>)) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {((F +* (g,(F + 1))) +* (g,(F + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{((F +* (g,(F + 1))) +* (g,(F + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>)) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,(F + 1)))} -defined NAT -valued RAT -valued INT -valued {(j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))}))
K32(K33({((F +* (g,(F + 1))) +* (g,(F + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))})) is non empty finite V40() countable set
(k . F) +* (((F +* (g,(F + 1))) +* (g,(F + 1))) .--> (j . (((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(F +* (g,(F + 1))) +* (g,F) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . F) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . F) . ((F +* (g,(F + 1))) +* (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F +* (g,(F + 1))) +* (g,F) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . F) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . F) . ((F +* (g,(F + 1))) +* (g,F)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,F)) ^ <*((k . F) . ((F +* (g,(F + 1))) +* (g,F)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . ij is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . ij) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k . (ij + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (ij + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F +* (g,k1) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,(F + 1))) +* (g,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . ij) . ((F +* (g,(F + 1))) +* (g,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F +* (g,(F + 1))) +* (g,(ij + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((F +* (g,(F + 1))) +* (g,(ij + 1))) .--> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>)) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {((F +* (g,(F + 1))) +* (g,(ij + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{((F +* (g,(F + 1))) +* (g,(ij + 1)))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{((F +* (g,(F + 1))) +* (g,(ij + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>)) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,(ij + 1)))} -defined NAT -valued RAT -valued INT -valued {(j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))}))
{(j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},{(j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))})) is non empty finite V40() countable set
(k . ij) +* (((F +* (g,(F + 1))) +* (g,(ij + 1))) .--> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
{((F +* (g,(F + 1))) +* (g,(ij + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>)) is non empty Relation-like {((F +* (g,(F + 1))) +* (g,(ij + 1)))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},NAT))
K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({((F +* (g,(F + 1))) +* (g,(ij + 1)))},NAT)) is non empty non trivial non finite set
dom ({((F +* (g,(F + 1))) +* (g,(ij + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({((F +* (g,(F + 1))) +* (g,(ij + 1)))})
K32({((F +* (g,(F + 1))) +* (g,(ij + 1)))}) is non empty finite V40() countable set
(dom (k . ij)) \/ (dom ({((F +* (g,(F + 1))) +* (g,(ij + 1)))} --> (j . (((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*>)))) is non empty set
(dom (k . ij)) \/ {((F +* (g,(F + 1))) +* (g,(ij + 1)))} is non empty set
F +* (g,(ij + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,k1)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F +* (g,(F + 1))) +* (g,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . ij) . ((F +* (g,(F + 1))) +* (g,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F +* (g,(F + 1))) +* (g,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . ij) . ((F +* (g,(F + 1))) +* (g,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
((F +* (g,(F + 1))) +* (g,ij)) ^ <*((k . ij) . ((F +* (g,(F + 1))) +* (g,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F +* (g,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
F +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(F +* (g,ij)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (k . F) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
j is Relation-like NAT -defined NAT -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of i -tuples_on NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j +* (F,g) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued set
j +* (F,g) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (j +* (F,g)) is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom j is finite V43(i) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
len (j +* (F,g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom F is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
K33((((arity i) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity i) + 1) -tuples_on NAT),(NAT))) is non empty set
ii is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity i) + 1) -tuples_on NAT),(NAT)))
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,k) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,k) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + ij)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,(k + ij))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k + ij) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,((k + ij) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(i,j,g,(((arity i) + 1),F,g,(k + ij))) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,(k + ij)) /. g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (((arity i) + 1),F,g,(k + ij)) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((((arity i) + 1),F,g,(k + ij)),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom i is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
i . (Del ((((arity i) + 1),F,g,(k + ij)),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0) .--> (i . (Del ((((arity i) + 1),F,g,(k + ij)),g))) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} --> (i . (Del ((((arity i) + 1),F,g,(k + ij)),g))) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} -defined NAT -valued RAT -valued INT -valued {(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))}))
{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))})) is non empty finite V40() countable set
dom j is non empty with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . ((((arity i) + 1),F,g,(k + ij)) /. g) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (((arity i) + 1),F,g,((k + ij) + 1)) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity i) + 1),F,g,((k + ij) + 1)) /. g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,((k + ij) + 1)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j,g,(((arity i) + 1),F,g,((k + ij) + 1))) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
Del ((((arity i) + 1),F,g,((k + ij) + 1)),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0) .--> (i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g))) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} --> (i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g))) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} -defined NAT -valued RAT -valued INT -valued {(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))}))
{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))})) is non empty finite V40() countable set
x is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
x . ((((arity i) + 1),F,g,((k + ij) + 1)) /. g) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k + (ij + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + (ij + 1))) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + (ij + 1))) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . ((k + ij) + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,(k + ij)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . (k + ij) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . (k + ij)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) .--> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued {(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))}))
{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))})) is non empty finite V40() countable set
(k . (k + ij)) +* ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) .--> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k . ((k + ij) + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT))
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT)) is non empty non trivial non finite set
dom ({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))})
K32({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))}) is non empty finite V40() countable set
(dom (k . (k + ij))) \/ (dom ({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)))) is non empty set
dom (ii . (((arity i) + 1),F,g,(k + (ij + 1)))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . (k + ij)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (ii . (((arity i) + 1),F,g,(k + (ij + 1)))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom (k . (k + ij)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + 0)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + 0)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,(k + 0))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
dom F is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
K33((((arity i) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity i) + 1) -tuples_on NAT),(NAT))) is non empty set
ii is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity i) + 1) -tuples_on NAT),(NAT)))
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,k) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,k) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,k)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + ij)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,(k + ij))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k + (ij + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + (ij + 1))) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + (ij + 1))) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,(k + (ij + 1)))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(i,j,g,(((arity i) + 1),F,g,(k + ij))) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,(k + ij)) /. g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (((arity i) + 1),F,g,(k + ij)) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((((arity i) + 1),F,g,(k + ij)),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom i is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
i . (Del ((((arity i) + 1),F,g,(k + ij)),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0) .--> (i . (Del ((((arity i) + 1),F,g,(k + ij)),g))) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} --> (i . (Del ((((arity i) + 1),F,g,(k + ij)),g))) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)} -defined NAT -valued RAT -valued INT -valued {(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))}))
{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,0)},{(i . (Del ((((arity i) + 1),F,g,(k + ij)),g)))})) is non empty finite V40() countable set
dom j is non empty with_common_domain set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . ((((arity i) + 1),F,g,(k + ij)) /. g) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,(k + ij)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . (k + ij) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (k + ij)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k + ij) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,((k + ij) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(i,j,g,(((arity i) + 1),F,g,((k + ij) + 1))) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,((k + ij) + 1)) /. g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (((arity i) + 1),F,g,((k + ij) + 1)) is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
Del ((((arity i) + 1),F,g,((k + ij) + 1)),g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0) .--> (i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g))) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} --> (i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g))) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)} -defined NAT -valued RAT -valued INT -valued {(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))}))
{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,((k + ij) + 1)),g,0)},{(i . (Del ((((arity i) + 1),F,g,((k + ij) + 1)),g)))})) is non empty finite V40() countable set
x is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
x . ((((arity i) + 1),F,g,((k + ij) + 1)) /. g) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),F,g,((k + ij) + 1)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . ((k + ij) + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) .--> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like ((arity i) + 1) -tuples_on NAT -defined {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued {(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))}))
{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},{(j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))})) is non empty finite V40() countable set
(k . (k + ij)) +* ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) .--> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (k . ((k + ij) + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
{(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)) is non empty Relation-like {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT))
K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))},NAT)) is non empty non trivial non finite set
dom ({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))})
K32({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))}) is non empty finite V40() countable set
(dom (k . (k + ij))) \/ (dom ({(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} --> (j . ((((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*>)))) is non empty set
(dom (k . (k + ij))) \/ {(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1))} is non empty set
(((arity i) + 1),F,g,k) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,((k + ij) + 1)) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
(k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)) ^ <*((k . (k + ij)) . (((arity i) + 1),(((arity i) + 1),F,g,(k + ij)),g,(k + ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity i) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity i) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
k + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity i) + 1),F,g,(k + 0)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
ii . (((arity i) + 1),F,g,(k + 0)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (ii . (((arity i) + 1),F,g,(k + 0))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity F) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity F) + 1 } is set
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(F,g,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (F,g,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom g is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
dom F is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity F) + 1),F,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
Del (F,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(F,g,i) . (((arity F) + 1),F,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . (Del (F,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),F,i,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(F,g,i) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity F) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(F,g,i) . (((arity F) + 1),F,i,(j + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . ((((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
K33((((arity F) + 1) -tuples_on NAT),(NAT)) is non empty Relation-like set
K32(K33((((arity F) + 1) -tuples_on NAT),(NAT))) is non empty set
k is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33((((arity F) + 1) -tuples_on NAT),(NAT)))
Union k is set
rng k is non empty set
union (rng k) is set
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
k . (((arity F) + 1),F,i,ij) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (((arity F) + 1),F,i,ij)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is functional () set
union k is Relation-like Function-like set
k1 is non empty functional () set
union k1 is Relation-like Function-like set
dom (union k1) is set
{ (dom b₁) where b₁ is Relation-like Function-like Element of k1 : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k1 : verum } is set
k is set
k is Relation-like Function-like Element of k1
dom k is set
dom k is non empty functional with_common_domain FinSequence-membered Element of K32((((arity F) + 1) -tuples_on NAT))
K32((((arity F) + 1) -tuples_on NAT)) is non empty set
x is set
k . x is Relation-like Function-like set
d is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
k . d is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F,g,i,d) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
d . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,(d . i)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),(((arity F) + 1),F,i,ij),i,(d . i)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),d,i,(d . i)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),d,i,m) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
(d . i) + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(d . i) + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (k . d) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
ij + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij + m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (((arity F) + 1),F,i,0) is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
{(((arity F) + 1),F,i,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
(((arity F) + 1),F,i,0) /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,0) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g,i,(((arity F) + 1),F,i,0)) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
Del ((((arity F) + 1),F,i,0),i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(((arity F) + 1),(((arity F) + 1),F,i,0),i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
F . (Del ((((arity F) + 1),F,i,0),i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),(((arity F) + 1),F,i,0),i,0) .--> (F . (Del ((((arity F) + 1),F,i,0),i))) is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined {(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)} --> (F . (Del ((((arity F) + 1),F,i,0),i))) is non empty Relation-like {(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)} -defined NAT -valued RAT -valued INT -valued {(F . (Del ((((arity F) + 1),F,i,0),i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)},{(F . (Del ((((arity F) + 1),F,i,0),i)))}))
{(F . (Del ((((arity F) + 1),F,i,0),i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)},{(F . (Del ((((arity F) + 1),F,i,0),i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,0),i,0)},{(F . (Del ((((arity F) + 1),F,i,0),i)))})) is non empty finite V40() countable set
dom g is non empty with_common_domain set
ij is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
ij . ((((arity F) + 1),F,i,0) /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . (((arity F) + 1),F,i,0) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (k . (((arity F) + 1),F,i,0)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (ij . 0) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
{(((arity F) + 1),F,i,0)} --> (F . (Del ((((arity F) + 1),F,i,0),i))) is non empty Relation-like {(((arity F) + 1),F,i,0)} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),F,i,0)},NAT))
K33({(((arity F) + 1),F,i,0)},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity F) + 1),F,i,0)},NAT)) is non empty non trivial non finite set
K33({(((arity F) + 1),F,i,0)},{(F . (Del ((((arity F) + 1),F,i,0),i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
{(((arity F) + 1),F,i,0)} --> (F . (Del (F,i))) is non empty Relation-like {(((arity F) + 1),F,i,0)} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),F,i,0)},NAT))
{(F . (Del (F,i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity F) + 1),F,i,0)},{(F . (Del (F,i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
(k . (((arity F) + 1),F,i,0)) . (((arity F) + 1),F,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . (((arity F) + 1),F,i,(j + 1)) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
k is non empty functional () set
{ (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
k is non empty functional () set
union k is Relation-like Function-like set
(union k) . (((arity F) + 1),F,i,0) is set
dom (((arity F) + 1),F,i,(j + 1)) is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
dom (((arity F) + 1),F,i,j) is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity F) + 1),F,i,j) /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,j) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g,i,(((arity F) + 1),F,i,j)) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
Del ((((arity F) + 1),F,i,j),i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(((arity F) + 1),(((arity F) + 1),F,i,j),i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
F . (Del ((((arity F) + 1),F,i,j),i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),(((arity F) + 1),F,i,j),i,0) .--> (F . (Del ((((arity F) + 1),F,i,j),i))) is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined {(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)} --> (F . (Del ((((arity F) + 1),F,i,j),i))) is non empty Relation-like {(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)} -defined NAT -valued RAT -valued INT -valued {(F . (Del ((((arity F) + 1),F,i,j),i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)},{(F . (Del ((((arity F) + 1),F,i,j),i)))}))
{(F . (Del ((((arity F) + 1),F,i,j),i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)},{(F . (Del ((((arity F) + 1),F,i,j),i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,j),i,0)},{(F . (Del ((((arity F) + 1),F,i,j),i)))})) is non empty finite V40() countable set
k is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
k . ((((arity F) + 1),F,i,j) /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F,g,i,(((arity F) + 1),F,i,(j + 1))) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity F) + 1),F,i,(j + 1)) /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del ((((arity F) + 1),F,i,(j + 1)),i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
F . (Del ((((arity F) + 1),F,i,(j + 1)),i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0) .--> (F . (Del ((((arity F) + 1),F,i,(j + 1)),i))) is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined {(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
{(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)} --> (F . (Del ((((arity F) + 1),F,i,(j + 1)),i))) is non empty Relation-like {(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)} -defined NAT -valued RAT -valued INT -valued {(F . (Del ((((arity F) + 1),F,i,(j + 1)),i)))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)},{(F . (Del ((((arity F) + 1),F,i,(j + 1)),i)))}))
{(F . (Del ((((arity F) + 1),F,i,(j + 1)),i)))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)},{(F . (Del ((((arity F) + 1),F,i,(j + 1)),i)))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,0)},{(F . (Del ((((arity F) + 1),F,i,(j + 1)),i)))})) is non empty finite V40() countable set
x is non empty Relation-like NAT -defined (NAT) -valued Function-like total quasi_total Function-yielding V90() Element of K32(K33(NAT,(NAT)))
x . ((((arity F) + 1),F,i,(j + 1)) /. i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x . 0 is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
x . (j + 1) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(x . (j + 1)) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
x . j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(x . j) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{(((arity F) + 1),F,i,(j + 1))} is non empty trivial functional finite V40() V43(1) with_common_domain countable set
<*((x . j) . (((arity F) + 1),F,i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)) is non empty Relation-like {(((arity F) + 1),F,i,(j + 1))} -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),F,i,(j + 1))},NAT))
K33({(((arity F) + 1),F,i,(j + 1))},NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33({(((arity F) + 1),F,i,(j + 1))},NAT)) is non empty non trivial non finite set
{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33({(((arity F) + 1),F,i,(j + 1))},{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))}) is non empty Relation-like RAT -valued INT -valued finite countable complex-valued ext-real-valued real-valued natural-valued set
dom ({(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))) is non empty trivial functional finite V40() V43(1) with_common_domain countable Element of K32({(((arity F) + 1),F,i,(j + 1))})
K32({(((arity F) + 1),F,i,(j + 1))}) is non empty finite V40() countable set
k . (((arity F) + 1),F,i,j) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(((arity F) + 1),F,i,(j + 1)) . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (x . j) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k . (((arity F) + 1),F,i,(j + 1))) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (k . (((arity F) + 1),F,i,j)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity F) + 1),F,i,j) ^ <*((k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (x . (j + 1)) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom (x . j)) \/ (dom ({(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)))) is non empty set
(((arity F) + 1),F,i,(j + 1)) .--> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)) is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined {(((arity F) + 1),F,i,(j + 1))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)) is non empty Relation-like {(((arity F) + 1),F,i,(j + 1))} -defined NAT -valued RAT -valued INT -valued {(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),F,i,(j + 1))},{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))}))
K32(K33({(((arity F) + 1),F,i,(j + 1))},{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))})) is non empty finite V40() countable set
(x . j) +* ((((arity F) + 1),F,i,(j + 1)) .--> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(((arity F) + 1),F,i,(j + 1)) .--> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)) is non empty Relation-like ((arity F) + 1) -tuples_on NAT -defined {(((arity F) + 1),F,i,(j + 1))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued homogeneous set
{(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>)) is non empty Relation-like {(((arity F) + 1),F,i,(j + 1))} -defined NAT -valued RAT -valued INT -valued {(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))} -valued Function-like constant total quasi_total finite Cardinal-yielding countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33({(((arity F) + 1),F,i,(j + 1))},{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))}))
K32(K33({(((arity F) + 1),F,i,(j + 1))},{(g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))})) is non empty finite V40() countable set
(x . j) +* ((((arity F) + 1),F,i,(j + 1)) .--> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),F,i,(j + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
({(((arity F) + 1),F,i,(j + 1))} --> (g . ((((arity F) + 1),F,i,j) ^ <*((x . j) . (((arity F) + 1),F,i,j))*>))) . (((arity F) + 1),F,i,(j + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . ((((arity F) + 1),F,i,j) ^ <*((k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),(((arity F) + 1),F,i,(j + 1)),i,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
{ (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
union { (dom b₁) where b₁ is Relation-like Function-like Element of k : verum } is set
(union k) . (((arity F) + 1),F,i,j) is set
(k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(union k) . (((arity F) + 1),F,i,(j + 1)) is set
(union k) . (((arity F) + 1),F,i,j) is set
(k . (((arity F) + 1),F,i,(j + 1))) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
dom g is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity j) + 1),g,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
Del (g,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
dom g is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity j) + 1),g,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(j,F,i) . (((arity j) + 1),g,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (g,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
j . (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
dom g is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity j) + 1),g,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
(j,F,i) . (((arity j) + 1),g,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (g,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
j . (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len (Del (g,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity j } is set
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity F) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity F) + 1 } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
dom F is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(F,g,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (F,g,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(((arity F) + 1),F,i,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(F,g,i) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity F) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom g is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity F) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity F) + 1 } is set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
dom F is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(F,g,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (F,g,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(F,g,i) . (((arity F) + 1),F,i,(j + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(F,g,i) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity F) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g . ((((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity F) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity F) + 1 } is set
(arity F) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
1 + (arity F) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),F,i,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(F,g,i) . (((arity F) + 1),F,i,(j + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity F) + 1),F,i,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(F,g,i) . (((arity F) + 1),F,i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity F) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g . ((((arity F) + 1),F,i,j) ^ <*((F,g,i) . (((arity F) + 1),F,i,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is finite V43((arity F) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
dom (F,g,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
g is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = g } is set
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
ii is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii +* (i,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Del (ii,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
(j,F,i) . (ii +* (i,0)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j . (Del (ii,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ii +* (i,(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii +* (i,k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i) . (ii +* (i,k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((j,F,i) . (ii +* (i,k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(ii +* (i,k)) ^ <*((j,F,i) . (ii +* (i,k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(j,F,i) . (ii +* (i,(k + 1))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . ((ii +* (i,k)) ^ <*((j,F,i) . (ii +* (i,k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,i) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom g is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = k } is set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
k is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
dom (j,F,i) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity j) + 1),k,i,(k /. i)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom k is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity j) + 1),k,i,k1) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
g . (((arity j) + 1),k,i,k1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) . (((arity j) + 1),k,i,k1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(((arity j) + 1),k,i,(k1 + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
g . (((arity j) + 1),k,i,(k1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) . (((arity j) + 1),k,i,(k1 + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(g . (((arity j) + 1),k,i,k1))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity j) + 1),k,i,k1) ^ <*(g . (((arity j) + 1),k,i,k1))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity j) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity j) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*((j,F,i) . (((arity j) + 1),k,i,k1))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity j) + 1),k,i,k1) ^ <*((j,F,i) . (((arity j) + 1),k,i,k1))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity j) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . ((((arity j) + 1),k,i,k1) ^ <*(g . (((arity j) + 1),k,i,k1))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(((arity j) + 1),k,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
Del (k,i) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
g . (((arity j) + 1),k,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j . (Del (k,i)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) . (((arity j) + 1),k,i,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) . k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is set
g . k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,i) . k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,0) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
0 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 0 } is set
(0 -tuples_on NAT) --> 0 is non empty Relation-like 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33((0 -tuples_on NAT),NAT))
K33((0 -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((0 -tuples_on NAT),NAT)) is non empty non trivial non finite set
{0} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((0 -tuples_on NAT),{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(1,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
j |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like Element of j -tuples_on K32(REAL)
j -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = j } is set
Seg j is finite V43(j) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg j) --> NAT is Relation-like non-empty Seg j -defined Seg j -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg j),{NAT}))
K33((Seg j),{NAT}) is Relation-like finite countable set
K32(K33((Seg j),{NAT})) is non empty finite V40() countable set
proj ((j |-> NAT),F) is Relation-like product (j |-> NAT) -defined Function-like total set
product (j |-> NAT) is functional with_common_domain product-like set
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng F is finite countable set
<:F:> is Relation-like Function-like homogeneous set
<:F:> * j is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
K32((NAT)) is non empty set
i is non empty functional Element of K32((NAT))
i is non empty set
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j -tuples_on i is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of i
i * is non empty functional FinSequence-membered FinSequenceSet of i
{ b₁ where b₁ is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of i * : len b₁ = j } is set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined i -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like Element of j -tuples_on i
g is Element of i
F +* (F,g) is Relation-like NAT -defined i -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like Element of j -tuples_on i
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
F is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
dom ii is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = k } is set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k,k,i,ij) is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(k,k,i,(ij + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
(k,k,i,0) is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
k is set
ij is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
ij . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k,ij,i,(ij . i)) is Relation-like NAT -defined NAT -valued Function-like finite V43(k) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of k -tuples_on NAT
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total Element of (NAT)
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = F } is set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(arity j) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity j } is set
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is functional with_common_domain FinSequence-membered set
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (g,ij) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total Element of (NAT)
dom ii is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(arity F) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = arity F } is set
i . (k +* (g,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(i . (k +* (g,ij)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k +* (g,ij)) ^ <*(i . (k +* (g,ij)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len ((k +* (g,ij)) ^ <*(i . (k +* (g,ij)))*>) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
len (k +* (g,ij)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len <*(i . (k +* (g,ij)))*> is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(len (k +* (g,ij))) + (len <*(i . (k +* (g,ij)))*>) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F + (len <*(i . (k +* (g,ij)))*>) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k +* (g,(ij + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(len k) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = len k } is set
Del (k,g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
k +* (g,0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k /. g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k +* (g,(k /. g)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is set
k is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
Del (k,g) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
F is non empty functional () Element of K32((NAT))
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(0 -tuples_on NAT) --> F is non empty Relation-like 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((0 -tuples_on NAT),NAT))
{F} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((0 -tuples_on NAT),{F}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(0,(F + 1)) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(0 -tuples_on NAT) --> (F + 1) is non empty Relation-like non-empty 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((0 -tuples_on NAT),NAT))
{(F + 1)} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((0 -tuples_on NAT),{(F + 1)}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
<*> NAT is empty proper Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of NAT
(0,F) . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(0,F)*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<*ii*> is non empty trivial Relation-like NAT -defined (NAT) -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () FinSequence of (NAT)
{(0,F)} is non empty trivial functional finite V43(1) with_common_domain countable set
arity (1,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng k is finite countable set
ij is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<:k:> is Relation-like Function-like homogeneous set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total Element of (NAT)
<:k:> * g is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity (0,(F + 1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,(F + 1)) . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (1,1) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
1 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 1 } is set
arity (0,F) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (0,F) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
<:k:> . {} is set
<*F*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
<:<*(0,F)*>:> is Relation-like Function-like set
dom <:<*(0,F)*>:> is set
ij is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total Element of (NAT)
ij . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,1) . (<:k:> . {}) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
k1 /. 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k1 /. 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<:<*(0,F)*>:> . {} is set
dom ij is functional with_common_domain FinSequence-membered Element of K32((NAT *))
(k) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<:<*(0,F)*>:> * (1,1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((g + 1),F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(g + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = g + 1 } is set
((g + 1) -tuples_on NAT) --> F is non empty Relation-like (g + 1) -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33(((g + 1) -tuples_on NAT),NAT))
K33(((g + 1) -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33(((g + 1) -tuples_on NAT),NAT)) is non empty non trivial non finite set
{F} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33(((g + 1) -tuples_on NAT),{F}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(g,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
g -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = g } is set
(g -tuples_on NAT) --> F is non empty Relation-like g -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((g -tuples_on NAT),NAT))
K33((g -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((g -tuples_on NAT),NAT)) is non empty non trivial non finite set
K33((g -tuples_on NAT),{F}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
g + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((g + 2),(g + 2)) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(g + 2) |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(g + 2) countable FinSequence-like FinSubsequence-like Element of (g + 2) -tuples_on K32(REAL)
(g + 2) -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = g + 2 } is set
Seg (g + 2) is non empty finite V43(g + 2) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
(Seg (g + 2)) --> NAT is non empty Relation-like non-empty Seg (g + 2) -defined Seg (g + 2) -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg (g + 2)),{NAT}))
K33((Seg (g + 2)),{NAT}) is non empty Relation-like finite countable set
K32(K33((Seg (g + 2)),{NAT})) is non empty finite V40() countable set
proj (((g + 2) |-> NAT),(g + 2)) is Relation-like product ((g + 2) |-> NAT) -defined Function-like total set
product ((g + 2) |-> NAT) is functional with_common_domain product-like set
dom ((g + 1),F) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(g + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity ((g + 2),(g + 2)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ((g + 2),(g + 2)) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(g + 2) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = g + 2 } is set
arity (g,F) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (g,F) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity ((g + 1),F) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = k1 } is set
(arity (g,F)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 +* ((g + 1),0) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
Del (k1,(g + 1)) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
((g + 1),F) . (k1 +* ((g + 1),0)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g,F) . (Del (k1,(g + 1))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k1 +* ((g + 1),(k + 1)) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
k1 +* ((g + 1),k) is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((g + 1),F) . (k1 +* ((g + 1),k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(((g + 1),F) . (k1 +* ((g + 1),k)))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k1 +* ((g + 1),k)) ^ <*(((g + 1),F) . (k1 +* ((g + 1),k)))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((g + 1),F) . (k1 +* ((g + 1),(k + 1))) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((g + 2),(g + 2)) . ((k1 +* ((g + 1),k)) ^ <*(((g + 1),F) . (k1 +* ((g + 1),k)))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(k1 +* ((g + 1),k)) ^ <*F*> is non empty Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len (k1 +* ((g + 1),k)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((k1 +* ((g + 1),k)) ^ <*F*>) . ((g + 1) + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((g + 2),(g + 2)) . ((k1 +* ((g + 1),k)) ^ <*F*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,g) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(0 -tuples_on NAT) --> g is non empty Relation-like 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((0 -tuples_on NAT),NAT))
{g} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((0 -tuples_on NAT),{g}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
F is non empty functional () Element of K32((NAT))
arity (1,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
j |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like Element of j -tuples_on K32(REAL)
j -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = j } is set
Seg j is finite V43(j) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg j) --> NAT is Relation-like non-empty Seg j -defined Seg j -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg j),{NAT}))
K33((Seg j),{NAT}) is Relation-like finite countable set
K32(K33((Seg j),{NAT})) is non empty finite V40() countable set
proj ((j |-> NAT),i) is Relation-like product (j |-> NAT) -defined Function-like total set
product (j |-> NAT) is functional with_common_domain product-like set
<*(j,i)*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
len <*(j,i)*> is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<*g*> is non empty trivial Relation-like NAT -defined (NAT) -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () FinSequence of (NAT)
rng (j,i) is non empty V78() V79() V80() V81() V82() V83() V91() V93() set
<:<*(j,i)*>:> is Relation-like Function-like set
rng <:<*(j,i)*>:> is set
1 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 1 } is set
dom (j,i) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
j -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = j } is set
dom (j,i) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom <:<*(j,i)*>:> is set
dom (1,1) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
<:<*(j,i)*>:> * (1,1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (<:<*(j,i)*>:> * (1,1)) is set
F is set
ii is Relation-like NAT -defined NAT -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of j -tuples_on NAT
ii . i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is finite V43(j) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(j,i) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii /. i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(ii /. i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(ii . i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(ii . i)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(<:<*(j,i)*>:> * (1,1)) . ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<:<*(j,i)*>:> . ii is set
(1,1) . (<:<*(j,i)*>:> . ii) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,i) . ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((j,i) . ii)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(1,1) . <*((j,i) . ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,1) . <*(ii . i)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
k /. 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(k /. 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(<:<*(j,i)*>:> * (1,1)) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng <*(j,i)*> is non empty trivial finite V43(1) countable set
F is set
{(j,i)} is non empty trivial functional finite V43(1) with_common_domain countable set
i is non empty functional () Element of K32((NAT))
(NAT) * is non empty functional FinSequence-membered FinSequenceSet of (NAT)
arity (0,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) *
<:j:> is Relation-like Function-like homogeneous set
<:j:> * (0,0) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i is non empty functional () Element of K32((NAT))
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),i)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined i -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of i
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<:F:> is Relation-like Function-like set
<:F:> * j is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
rng F is finite countable set
i * is non empty functional FinSequence-membered FinSequenceSet of i
(NAT) * is non empty functional FinSequence-membered FinSequenceSet of (NAT)
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) *
<:F:> is Relation-like Function-like homogeneous set
g is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<:F:> * g is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
i is non empty functional () Element of K32((NAT))
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),i)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),i)
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,g) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
{ b₁ where b₁ is functional Element of K32((NAT)) : b₁ is () } is set
meet { b₁ where b₁ is functional Element of K32((NAT)) : b₁ is () } is set
{(NAT)} is non empty trivial finite V43(1) countable set
bool (NAT) is non empty Element of K32(K32((NAT)))
K32(K32((NAT))) is non empty set
F is set
() is functional Element of K32((NAT))
i is functional Element of K32((NAT))
F is set
{(NAT)} is non empty trivial finite V43(1) countable set
bool (NAT) is non empty Element of K32(K32((NAT)))
K32(K32((NAT))) is non empty set
j is set
F is functional Element of K32((NAT))
j is set
F is functional Element of K32((NAT))
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is set
(j,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
j |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(j) countable FinSequence-like FinSubsequence-like Element of j -tuples_on K32(REAL)
j -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = j } is set
Seg j is finite V43(j) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg j) --> NAT is Relation-like non-empty Seg j -defined Seg j -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg j),{NAT}))
K33((Seg j),{NAT}) is Relation-like finite countable set
K32(K33((Seg j),{NAT})) is non empty finite V40() countable set
proj ((j |-> NAT),F) is Relation-like product (j |-> NAT) -defined Function-like total set
product (j |-> NAT) is functional with_common_domain product-like set
F is functional Element of K32((NAT))
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng F is finite countable set
g is set
F is functional Element of K32((NAT))
<:F:> is Relation-like Function-like homogeneous set
<:F:> * j is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is set
k is functional Element of K32((NAT))
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),())
i is set
i is Relation-like set
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous (NAT * , NAT ,(NAT),())
the Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (NAT * , NAT ,(NAT),()) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (NAT * , NAT ,(NAT),())
the Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (NAT * , NAT ,(NAT),()) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (NAT * , NAT ,(NAT),())
{(0,0),(1,1)} is non empty functional finite countable set
{ (b₁,b₂) where b₁, b₂ is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT : ( 1 <= b₂ & b₂ <= b₁ ) } is set
{(0,0),(1,1)} \/ { (b₁,b₂) where b₁, b₂ is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT : ( 1 <= b₂ & b₂ <= b₁ ) } is non empty set
F is set
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
g |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(g) countable FinSequence-like FinSubsequence-like Element of g -tuples_on K32(REAL)
g -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = g } is set
Seg g is finite V43(g) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg g) --> NAT is Relation-like non-empty Seg g -defined Seg g -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg g),{NAT}))
K33((Seg g),{NAT}) is Relation-like finite countable set
K32(K33((Seg g),{NAT})) is non empty finite V40() countable set
proj ((g |-> NAT),F) is Relation-like product (g |-> NAT) -defined Function-like total set
product (g |-> NAT) is functional with_common_domain product-like set
F is set
i is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i & b₃ in i & (b₁,b₂,b₃,b₄) ) } is set
i \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i & b₃ in i & (b₁,b₂,b₃,b₄) ) } is set
F is set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(NAT) * is non empty functional FinSequence-membered FinSequenceSet of (NAT)
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i & arity b₁ = len b₂ & rng b₂ c= i ) } is set
i \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i & arity b₁ = len b₂ & rng b₂ c= i ) } is set
F is set
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) *
<:F:> is Relation-like Function-like homogeneous set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<:F:> * g is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng F is finite countable set
() is functional Element of K32((NAT))
bool (NAT) is non empty Element of K32(K32((NAT)))
K32(K32((NAT))) is non empty set
K33(NAT,(bool (NAT))) is non empty non trivial Relation-like non finite set
K32(K33(NAT,(bool (NAT)))) is non empty non trivial non finite set
i is non empty Relation-like NAT -defined bool (NAT) -valued Function-like total quasi_total Element of K32(K33(NAT,(bool (NAT))))
i . 0 is functional Element of bool (NAT)
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i . (j + 1) is functional Element of bool (NAT)
i . j is functional Element of bool (NAT)
((i . j)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i . j & b₃ in i . j & (b₁,b₂,b₃,b₄) ) } is set
(i . j) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i . j & b₃ in i . j & (b₁,b₂,b₃,b₄) ) } is set
((i . j)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i . j & arity b₁ = len b₂ & rng b₂ c= i . j ) } is set
(i . j) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i . j & arity b₁ = len b₂ & rng b₂ c= i . j ) } is set
((i . j)) \/ ((i . j)) is functional Element of K32((NAT))
i is non empty Relation-like NAT -defined bool (NAT) -valued Function-like total quasi_total Element of K32(K33(NAT,(bool (NAT))))
i . 0 is functional Element of bool (NAT)
j is non empty Relation-like NAT -defined bool (NAT) -valued Function-like total quasi_total Element of K32(K33(NAT,(bool (NAT))))
j . 0 is functional Element of bool (NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i . (F + 1) is functional Element of bool (NAT)
i . F is functional Element of bool (NAT)
((i . F)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i . F & b₃ in i . F & (b₁,b₂,b₃,b₄) ) } is set
(i . F) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in i . F & b₃ in i . F & (b₁,b₂,b₃,b₄) ) } is set
((i . F)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i . F & arity b₁ = len b₂ & rng b₂ c= i . F ) } is set
(i . F) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in i . F & arity b₁ = len b₂ & rng b₂ c= i . F ) } is set
((i . F)) \/ ((i . F)) is functional Element of K32((NAT))
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
j . (F + 1) is functional Element of bool (NAT)
j . F is functional Element of bool (NAT)
((j . F)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in j . F & b₃ in j . F & (b₁,b₂,b₃,b₄) ) } is set
(j . F) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in j . F & b₃ in j . F & (b₁,b₂,b₃,b₄) ) } is set
((j . F)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in j . F & arity b₁ = len b₂ & rng b₂ c= j . F ) } is set
(j . F) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in j . F & arity b₁ = len b₂ & rng b₂ c= j . F ) } is set
((j . F)) \/ ((j . F)) is functional Element of K32((NAT))
() is non empty Relation-like NAT -defined bool (NAT) -valued Function-like total quasi_total Element of K32(K33(NAT,(bool (NAT))))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . i is functional Element of bool (NAT)
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . j is functional Element of bool (NAT)
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . g is functional Element of bool (NAT)
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (g + 1) is functional Element of bool (NAT)
((() . g)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . g & b₃ in () . g & (b₁,b₂,b₃,b₄) ) } is set
(() . g) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . g & b₃ in () . g & (b₁,b₂,b₃,b₄) ) } is set
((() . g)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . g & arity b₁ = len b₂ & rng b₂ c= () . g ) } is set
(() . g) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . g & arity b₁ = len b₂ & rng b₂ c= () . g ) } is set
((() . g)) \/ ((() . g)) is functional Element of K32((NAT))
() . 0 is functional Element of bool (NAT)
Union () is functional Element of K32((NAT))
rng () is non empty set
union (rng ()) is set
dom () is non empty countable V78() V79() V80() V81() V82() V83() V91() V93() Element of K32(NAT)
() . 0 is functional Element of bool (NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
g |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(g) countable FinSequence-like FinSubsequence-like Element of g -tuples_on K32(REAL)
g -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = g } is set
Seg g is finite V43(g) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg g) --> NAT is Relation-like non-empty Seg g -defined Seg g -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg g),{NAT}))
K33((Seg g),{NAT}) is Relation-like finite countable set
K32(K33((Seg g),{NAT})) is non empty finite V40() countable set
proj ((g |-> NAT),F) is Relation-like product (g |-> NAT) -defined Function-like total set
product (g |-> NAT) is functional with_common_domain product-like set
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . a₁ & arity b₁ = len b₂ & rng b₂ c= () . a₁ ) } is set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng F is finite countable set
ii is set
() . ii is set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . k is functional Element of bool (NAT)
<:F:> is Relation-like Function-like homogeneous set
<:F:> * g is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k & arity b₁ = len b₂ & rng b₂ c= () . k ) } is set
(() . k) \/ H₁(k) is set
((() . k)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k & b₃ in () . k & (b₁,b₂,b₃,b₄) ) } is set
(() . k) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k & b₃ in () . k & (b₁,b₂,b₃,b₄) ) } is set
((() . k)) is functional Element of K32((NAT))
(() . k) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k & arity b₁ = len b₂ & rng b₂ c= () . k ) } is set
((() . k)) \/ ((() . k)) is functional Element of K32((NAT))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (k + 1) is functional Element of bool (NAT)
Seg (len F) is finite V43( len F) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
k is set
F . k is Relation-like Function-like set
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
ij is set
() . ij is set
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . k1 is functional Element of bool (NAT)
K33((Seg (len F)),NAT) is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
K32(K33((Seg (len F)),NAT)) is non empty set
k is Relation-like Seg (len F) -defined NAT -valued Function-like total quasi_total finite countable complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((Seg (len F)),NAT))
rng k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() set
k1 is non empty finite countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
max k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
max (k,(max k1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
dom k is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32((Seg (len F)))
K32((Seg (len F))) is non empty finite V40() countable set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . k is functional Element of bool (NAT)
x is set
dom F is finite countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
d is set
F . d is Relation-like Function-like set
k . d is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . m is functional Element of bool (NAT)
() . k is functional Element of bool (NAT)
<:F:> is Relation-like Function-like homogeneous set
<:F:> * g is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k & arity b₁ = len b₂ & rng b₂ c= () . k ) } is set
(() . k) \/ H₁(k) is set
((() . k)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k & b₃ in () . k & (b₁,b₂,b₃,b₄) ) } is set
(() . k) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k & b₃ in () . k & (b₁,b₂,b₃,b₄) ) } is set
((() . k)) is functional Element of K32((NAT))
(() . k) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k & arity b₁ = len b₂ & rng b₂ c= () . k ) } is set
((() . k)) \/ ((() . k)) is functional Element of K32((NAT))
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (k + 1) is functional Element of bool (NAT)
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . a₁ & b₃ in () . a₁ & (b₁,b₂,b₃,b₄) ) } is set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is set
() . k is set
k1 is set
() . k1 is set
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . k1 is functional Element of bool (NAT)
() . ij is functional Element of bool (NAT)
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . ij & b₃ in () . ij & (b₁,b₂,b₃,b₄) ) } is set
(() . ij) \/ H₁(ij) is set
((() . ij)) is functional Element of K32((NAT))
(() . ij) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . ij & b₃ in () . ij & (b₁,b₂,b₃,b₄) ) } is set
((() . ij)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . ij & arity b₁ = len b₂ & rng b₂ c= () . ij ) } is set
(() . ij) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . ij & arity b₁ = len b₂ & rng b₂ c= () . ij ) } is set
((() . ij)) \/ ((() . ij)) is functional Element of K32((NAT))
ij + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (ij + 1) is functional Element of bool (NAT)
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . ij is functional Element of bool (NAT)
() . k1 is functional Element of bool (NAT)
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k1 & b₃ in () . k1 & (b₁,b₂,b₃,b₄) ) } is set
(() . k1) \/ H₁(k1) is set
((() . k1)) is functional Element of K32((NAT))
(() . k1) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . k1 & b₃ in () . k1 & (b₁,b₂,b₃,b₄) ) } is set
((() . k1)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k1 & arity b₁ = len b₂ & rng b₂ c= () . k1 ) } is set
(() . k1) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . k1 & arity b₁ = len b₂ & rng b₂ c= () . k1 ) } is set
((() . k1)) \/ ((() . k1)) is functional Element of K32((NAT))
k1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (k1 + 1) is functional Element of bool (NAT)
k1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . j is functional Element of bool (NAT)
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (j + 1) is functional Element of bool (NAT)
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . j & arity b₁ = len b₂ & rng b₂ c= () . j ) } is set
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . j & b₃ in () . j & (b₁,b₂,b₃,b₄) ) } is set
F is set
((() . j)) is functional Element of K32((NAT))
(() . j) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . j & b₃ in () . j & (b₁,b₂,b₃,b₄) ) } is set
((() . j)) is functional Element of K32((NAT))
(() . j) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . j & arity b₁ = len b₂ & rng b₂ c= () . j ) } is set
((() . j)) \/ ((() . j)) is functional Element of K32((NAT))
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) *
<:k:> is Relation-like Function-like homogeneous set
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<:k:> * ii is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng k is finite countable set
() . 0 is functional Element of bool (NAT)
j is set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
F |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(F) countable FinSequence-like FinSubsequence-like Element of F -tuples_on K32(REAL)
F -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = F } is set
Seg F is finite V43(F) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg F) --> NAT is Relation-like non-empty Seg F -defined Seg F -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg F),{NAT}))
K33((Seg F),{NAT}) is Relation-like finite countable set
K32(K33((Seg F),{NAT})) is non empty finite V40() countable set
proj ((F |-> NAT),g) is Relation-like product (F |-> NAT) -defined Function-like total set
product (F |-> NAT) is functional with_common_domain product-like set
j is set
F is set
dom () is non empty countable V78() V79() V80() V81() V82() V83() V91() V93() Element of K32(NAT)
g is set
() . g is set
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . F is functional Element of bool (NAT)
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . i is functional Element of bool (NAT)
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . F is functional Element of bool (NAT)
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
() . (F + 1) is functional Element of bool (NAT)
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
((() . F)) is functional Element of K32((NAT))
{ b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . F & b₃ in () . F & (b₁,b₂,b₃,b₄) ) } is set
(() . F) \/ { b₁ where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) : ex b₂, b₃ being Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT) ex b₄ being epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT st
( b₂ in () . F & b₃ in () . F & (b₁,b₂,b₃,b₄) ) } is set
((() . F)) is functional Element of K32((NAT))
{ (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . F & arity b₁ = len b₂ & rng b₂ c= () . F ) } is set
(() . F) \/ { (<:b₂:> * b₁) where b₁ is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT), b₂ is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) * : ( b₁ in () . F & arity b₁ = len b₂ & rng b₂ c= () . F ) } is set
((() . F)) \/ ((() . F)) is functional Element of K32((NAT))
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of (NAT) *
<:ii:> is Relation-like Function-like homogeneous set
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
<:ii:> * F is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng ii is finite countable set
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
() . 0 is functional Element of bool (NAT)
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g,F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
g |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(g) countable FinSequence-like FinSubsequence-like Element of g -tuples_on K32(REAL)
g -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = g } is set
Seg g is finite V43(g) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(Seg g) --> NAT is Relation-like non-empty Seg g -defined Seg g -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg g),{NAT}))
K33((Seg g),{NAT}) is Relation-like finite countable set
K32(K33((Seg g),{NAT})) is non empty finite V40() countable set
proj ((g |-> NAT),F) is Relation-like product (g |-> NAT) -defined Function-like total set
product (g |-> NAT) is functional with_common_domain product-like set
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (NAT * , NAT ,(NAT),())
F is set
dom () is non empty countable V78() V79() V80() V81() V82() V83() V91() V93() Element of K32(NAT)
g is set
() . g is set
i is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
i is Relation-like NAT * -defined Function-like set
i is Relation-like Function-like homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is with_common_domain set
j is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like Function-like homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is with_common_domain set
j is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is Relation-like Function-like homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is with_common_domain set
j is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(i,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(i) countable FinSequence-like FinSubsequence-like Element of i -tuples_on K32(REAL)
i -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = i } is set
Seg i is non empty finite V43(i) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
(Seg i) --> NAT is non empty Relation-like non-empty Seg i -defined Seg i -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg i),{NAT}))
K33((Seg i),{NAT}) is non empty Relation-like finite countable set
K32(K33((Seg i),{NAT})) is non empty finite V40() countable set
proj ((i |-> NAT),1) is Relation-like product (i |-> NAT) -defined Function-like total set
product (i |-> NAT) is functional with_common_domain product-like set
(2,2) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
2 |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Element of 2 -tuples_on K32(REAL)
2 -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = 2 } is set
(Seg 2) --> NAT is non empty Relation-like non-empty Seg 2 -defined Seg 2 -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg 2),{NAT}))
K33((Seg 2),{NAT}) is non empty Relation-like finite countable set
K32(K33((Seg 2),{NAT})) is non empty finite V40() countable set
proj ((2 |-> NAT),2) is Relation-like product (2 |-> NAT) -defined Function-like total set
product (2 |-> NAT) is functional with_common_domain product-like set
(3,3) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
(0 -tuples_on NAT) --> i is non empty Relation-like 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((0 -tuples_on NAT),NAT))
{i} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((0 -tuples_on NAT),{i}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
arity (0,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
1 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 1 } is set
(1 -tuples_on NAT) --> i is non empty Relation-like 1 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((1 -tuples_on NAT),NAT))
K33((1 -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((1 -tuples_on NAT),NAT)) is non empty non trivial non finite set
K33((1 -tuples_on NAT),{i}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
arity (1,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(2,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
2 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 2 } is set
(2 -tuples_on NAT) --> i is non empty Relation-like 2 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((2 -tuples_on NAT),NAT))
K33((2 -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((2 -tuples_on NAT),NAT)) is non empty non trivial non finite set
K33((2 -tuples_on NAT),{i}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
arity (2,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
3 -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = 3 } is set
(3 -tuples_on NAT) --> i is non empty Relation-like 3 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((3 -tuples_on NAT),NAT))
K33((3 -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((3 -tuples_on NAT),NAT)) is non empty non trivial non finite set
K33((3 -tuples_on NAT),{i}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
arity (3,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
1 |-> NAT is non empty trivial Relation-like NAT -defined K32(REAL) -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Element of 1 -tuples_on K32(REAL)
1 -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = 1 } is set
(Seg 1) --> NAT is non empty Relation-like non-empty Seg 1 -defined Seg 1 -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg 1),{NAT}))
K33((Seg 1),{NAT}) is non empty Relation-like finite countable set
K32(K33((Seg 1),{NAT})) is non empty finite V40() countable set
proj ((1 |-> NAT),i) is Relation-like product (1 |-> NAT) -defined Function-like total set
product (1 |-> NAT) is functional with_common_domain product-like set
arity (1,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(2,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
proj ((2 |-> NAT),i) is Relation-like product (2 |-> NAT) -defined Function-like total set
arity (2,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
3 |-> NAT is Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like Element of 3 -tuples_on K32(REAL)
3 -tuples_on K32(REAL) is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of K32(REAL)
{ b₁ where b₁ is Relation-like NAT -defined K32(REAL) -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of K32(REAL) * : len b₁ = 3 } is set
Seg 3 is non empty finite V43(3) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
(Seg 3) --> NAT is non empty Relation-like non-empty Seg 3 -defined Seg 3 -defined K32(REAL) -valued {NAT} -valued Function-like constant total total quasi_total finite Cardinal-yielding countable FinSequence-like FinSubsequence-like Element of K32(K33((Seg 3),{NAT}))
K33((Seg 3),{NAT}) is non empty Relation-like finite countable set
K32(K33((Seg 3),{NAT})) is non empty finite V40() countable set
proj ((3 |-> NAT),i) is Relation-like product (3 |-> NAT) -defined Function-like total set
product (3 |-> NAT) is functional with_common_domain product-like set
arity (3,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity (1,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(2,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity (2,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
arity (3,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () set
i -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = i } is set
(i -tuples_on NAT) --> j is non empty Relation-like i -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((i -tuples_on NAT),NAT))
K33((i -tuples_on NAT),NAT) is non empty non trivial Relation-like RAT -valued INT -valued non finite complex-valued ext-real-valued real-valued natural-valued set
K32(K33((i -tuples_on NAT),NAT)) is non empty non trivial non finite set
{j} is non empty trivial finite V40() V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
K33((i -tuples_on NAT),{j}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(2,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
proj ((2 |-> NAT),1) is Relation-like product (2 |-> NAT) -defined Function-like total set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
(3,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
proj ((3 |-> NAT),1) is Relation-like product (3 |-> NAT) -defined Function-like total set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
(1,0) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
(1 -tuples_on NAT) --> 0 is non empty Relation-like 1 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33((1 -tuples_on NAT),NAT))
K33((1 -tuples_on NAT),{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(2,0) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
(2 -tuples_on NAT) --> 0 is non empty Relation-like 2 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33((2 -tuples_on NAT),NAT))
K33((2 -tuples_on NAT),{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(3,0) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
(3 -tuples_on NAT) --> 0 is non empty Relation-like 3 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33((3 -tuples_on NAT),NAT))
K33((3 -tuples_on NAT),{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () ( 0 ) set
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
(i,j,1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom i is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
<*> NAT is empty proper Relation-like non-empty empty-yielding NAT -defined NAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () FinSequence of NAT
{(<*> NAT)} is non empty trivial functional finite V40() V43(1) with_common_domain countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
0 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*0*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Function-yielding V90() V97() FinSequence of NAT
dom <*0*> is non empty trivial finite V43(1) countable V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of K32(NAT)
Del (<*0*>,1) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
F is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
(1,F,1,0) is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
dom (i,j,1) is functional with_common_domain FinSequence-membered Element of K32((NAT *))
len <*0*> is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity (i,j,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
(i,j,1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity (i,j,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i,j,2) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
1 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity (i,j,2) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (1) set
j . <*i*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(j,F,2) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,2) . <*i,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom g is finite V43((arity j) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity j) + 1),g,2,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
Del (g,2) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
j . (Del (g,2)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*0*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Function-yielding V90() V97() FinSequence of NAT
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
(i,j,1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(i,j,1) . <*0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity i) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity i) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity i) + 1 } is set
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
len F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is finite V43((arity i) + 1) countable V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of K32(NAT)
(((arity i) + 1),F,1,0) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity i) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity i) + 1) -tuples_on NAT
Del (F,1) is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
i . (Del (F,1)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*i,(j + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (1) set
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (3) set
(F,g,2) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(F,g,2) . <*i,(j + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g,2) . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j,((F,g,2) . <*i,j*>)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . <*i,j,((F,g,2) . <*i,j*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity F) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity F) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity F) + 1 } is set
F is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),F,2,(j + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(((arity F) + 1),F,2,j) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity F) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity F) + 1) -tuples_on NAT
(arity F) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
arity g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F,g,2) . (((arity F) + 1),F,2,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((F,g,2) . (((arity F) + 1),F,2,j))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity F) + 1),F,2,j) ^ <*((F,g,2) . (((arity F) + 1),F,2,j))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity F) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity F) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g . ((((arity F) + 1),F,2,j) ^ <*((F,g,2) . (((arity F) + 1),F,2,j))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(i + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,1) is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of (NAT)
(j,F,1) . <*(i + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(j,F,1) . <*i*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,((j,F,1) . <*i*>)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F . <*i,((j,F,1) . <*i*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(arity j) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
((arity j) + 1) -tuples_on NAT is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of NAT * : len b₁ = (arity j) + 1 } is set
g is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
(((arity j) + 1),g,1,(i + 1)) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
(((arity j) + 1),g,1,i) is Relation-like NAT -defined NAT -valued Function-like finite V43((arity j) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of ((arity j) + 1) -tuples_on NAT
(arity j) + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(j,F,1) . (((arity j) + 1),g,1,i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((j,F,1) . (((arity j) + 1),g,1,i))*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(((arity j) + 1),g,1,i) ^ <*((j,F,1) . (((arity j) + 1),g,1,i))*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(((arity j) + 1) + 1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((arity j) + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F . ((((arity j) + 1),g,1,i) ^ <*((j,F,1) . (((arity j) + 1),g,1,i))*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*0,0,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() FinSequence of NAT
(3,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
proj ((3 |-> NAT),1) is Relation-like product (3 |-> NAT) -defined Function-like total set
(3,3) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (3) set
proj ((3 |-> NAT),3) is Relation-like product (3 |-> NAT) -defined Function-like total set
<*(3,1),(3,3)*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() set
<:<*(3,1),(3,3)*>:> is Relation-like Function-like set
j is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () Element of K32(K33((NAT *),NAT))
<:<*(3,1),(3,3)*>:> * j is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
rng (<:<*(3,1),(3,3)*>:> * j) is V78() V79() V80() V81() V82() V83() V93() set
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom <:<*(3,1),(3,3)*>:> is set
dom (3,1) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom (3,3) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom (3,1)) /\ (dom (3,3)) is functional FinSequence-membered Element of K32((NAT *))
(3 -tuples_on NAT) /\ (dom (3,3)) is functional FinSequence-membered Element of K32((NAT *))
(3 -tuples_on NAT) /\ (3 -tuples_on NAT) is set
rngs <*(3,1),(3,3)*> is Relation-like Function-like set
product (rngs <*(3,1),(3,3)*>) is functional with_common_domain product-like set
rng (3,1) is non empty V78() V79() V80() V81() V82() V83() V91() V93() set
rng (3,3) is non empty V78() V79() V80() V81() V82() V83() V91() V93() set
<*(rng (3,1)),(rng (3,3))*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like set
product <*(rng (3,1)),(rng (3,3))*> is functional with_common_domain product-like set
<*NAT,(rng (3,3))*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like set
product <*NAT,(rng (3,3))*> is functional with_common_domain product-like set
<*NAT,NAT*> is non empty Relation-like NAT -defined K32(REAL) -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like FinSequence of K32(REAL)
product <*NAT,NAT*> is functional with_common_domain product-like set
rng <:<*(3,1),(3,3)*>:> is set
F is set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*k,k*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*k,0,k*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ij is Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 3 -tuples_on NAT
<:<*(3,1),(3,3)*>:> . ij is set
(3,1) . ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,3) . ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((3,1) . ij),((3,3) . ij)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ij . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ij . 1),((3,3) . ij)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ij . 3 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ij . 1),(ij . 3)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*k,(ij . 3)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
dom (<:<*(3,1),(3,3)*>:> * j) is set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((NAT *),NAT))
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of K32(K33((NAT *),NAT))
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of K32(K33((NAT *),NAT))
i is Relation-like NAT -defined NAT -valued Function-like finite countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
len i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is functional with_common_domain FinSequence-membered Element of K32((NAT *))
k is Relation-like NAT -defined Function-like finite countable FinSequence-like FinSubsequence-like set
len k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is Relation-like Function-like set
<:<*(3,1),(3,3)*>:> * i is Relation-like Function-like set
i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(i) is Relation-like Function-like set
<:<*(3,1),(3,3)*>:> * i is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (i) is set
i is Relation-like Function-like homogeneous set
(i) is Relation-like Function-like set
<:<*(3,1),(3,3)*>:> * i is Relation-like Function-like set
dom (i) is set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
(i) is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous set
<:<*(3,1),(3,3)*>:> * i is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous Element of K32(K33((NAT *),NAT))
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () Element of K32(K33((NAT *),NAT))
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j,F*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*i,F*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
(g) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (3) set
<:<*(3,1),(3,3)*>:> * g is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(g) . <*i,j,F*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g . <*i,F*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () Element of K32(K33((NAT *),NAT))
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<:<*(3,1),(3,3)*>:> * F is Relation-like NAT -valued RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
dom (<:<*(3,1),(3,3)*>:> * F) is set
ii is Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 3 -tuples_on NAT
<:<*(3,1),(3,3)*>:> . ii is set
g . (<:<*(3,1),(3,3)*>:> . ii) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,1) . ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(3,3) . ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((3,1) . ii),((3,3) . ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . <*((3,1) . ii),((3,3) . ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ii . 1),((3,3) . ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . <*(ii . 1),((3,3) . ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii . 3 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ii . 1),(ii . 3)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . <*(ii . 1),(ii . 3)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,(ii . 3)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
g . <*i,(ii . 3)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
2 -tuples_on () is non empty functional with_common_domain product-like FinSequence-membered FinSequenceSet of ()
() * is non empty functional FinSequence-membered FinSequenceSet of ()
{ b₁ where b₁ is Relation-like NAT -defined () -valued Function-like finite countable FinSequence-like FinSubsequence-like Element of () * : len b₁ = 2 } is set
i is Relation-like NAT -defined () -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() Element of () *
j is Relation-like Function-like set
rng i is finite countable set
F is Relation-like Function-like set
dom F is set
dom j is set
{(3,1),(3,3)} is non empty functional finite countable set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
(F) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (3) set
<:<*(3,1),(3,3)*>:> * F is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT -defined () -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () Element of () *
len j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
(i) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (3) set
<:<*(3,1),(3,3)*>:> * i is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
arity (i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
proj ((1 |-> NAT),1) is Relation-like product (1 |-> NAT) -defined Function-like total set
((1,1),(3,3),2) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (2) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,g,(i + g)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*i,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*i,(g + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,(g + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 3 -tuples_on NAT
(3,3) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F /. 3 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F /. 3) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
(i + g) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i + (g + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*i,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
(1,1) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,0) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
(1 -tuples_on NAT) --> 0 is non empty Relation-like 1 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() Element of K32(K33((1 -tuples_on NAT),NAT))
K33((1 -tuples_on NAT),{0}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(()) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
<:<*(3,1),(3,3)*>:> * () is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
((1,0),(()),2) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (2) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i * j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*i,(j + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,(j + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j,(i * j)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(()) . <*i,j,(i * j)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,(i * j)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,(i * j)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i * 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i * 1) + (i * j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i * (j + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
<*i,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
(1,0) . j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i * 0 is empty Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V28() V29() ext-real non positive non negative finite finite-yielding V40() V41() V43( {} ) Cardinal-yielding with_common_domain countable FinSequence-like FinSubsequence-like FinSequence-membered V65() V67() complex-valued ext-real-valued real-valued natural-valued V78() V79() V80() V81() V82() V83() V84() Function-yielding V90() V93() V94() V95() V96() V97() homogeneous () Element of NAT
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<*i,j*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() set
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
<*F,g*> is non empty Relation-like NAT -defined () -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() FinSequence of ()
rng <*F,g*> is non empty finite countable set
{F,g} is non empty functional finite countable set
F is Relation-like Function-like homogeneous set
ii is Relation-like Function-like homogeneous set
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<*j,F*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() () set
<:<*j,F*>:> is Relation-like Function-like set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<:<*j,F*>:> * i is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
<*g,F*> is non empty Relation-like NAT -defined () -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() FinSequence of ()
rng <*g,F*> is non empty finite countable set
rng <*j,F*> is non empty finite countable set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
F is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<*j,F*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() () set
<:<*j,F*>:> is Relation-like Function-like set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<:<*j,F*>:> * i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () set
the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
k is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
<*k,k*> is non empty Relation-like NAT -defined () -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() FinSequence of ()
rng <*k,k*> is non empty finite countable set
rng <*j,F*> is non empty finite countable set
ij is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
<:ij:> is Relation-like Function-like homogeneous set
dom <:ij:> is with_common_domain set
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom F is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
(dom j) /\ (dom F) is functional FinSequence-membered Element of K32((NAT *))
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng ij is finite countable set
{j,F} is non empty functional finite countable set
(ij) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom k is functional with_common_domain FinSequence-membered Element of K32((NAT *))
<:ij:> . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is set
k . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(k . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT),(k . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
arity (<:<*j,F*>:> * i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () set
<*j*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
<:<*j*>:> is Relation-like Function-like set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
<:<*j*>:> * i is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
<*F*> is non empty trivial Relation-like NAT -defined () -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () FinSequence of ()
rng <*F*> is non empty trivial finite V43(1) countable set
rng <*j*> is non empty trivial finite V43(1) countable set
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
len g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<*j*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
<:<*j*>:> is Relation-like Function-like set
i is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
<:<*j*>:> * i is Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () set
the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
g is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
ii is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
<*ii*> is non empty trivial Relation-like NAT -defined () -valued Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () FinSequence of ()
rng <*ii*> is non empty trivial finite V43(1) countable set
rng <*j*> is non empty trivial finite V43(1) countable set
k is Relation-like NAT -defined (NAT) -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of (NAT)
<:k:> is Relation-like Function-like homogeneous set
dom <:k:> is with_common_domain set
dom j is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom g is functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng k is finite countable set
{j} is non empty trivial functional finite V43(1) with_common_domain countable set
(k) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom ii is functional with_common_domain FinSequence-membered Element of K32((NAT *))
<:k:> . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is set
ii . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ii . the Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
F is Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (NAT * , NAT ,(NAT),())
arity (<:<*j*>:> * i) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () ( 0 ) set
(0 -tuples_on NAT) --> 1 is non empty Relation-like non-empty 0 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((0 -tuples_on NAT),NAT))
K33((0 -tuples_on NAT),{1}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(2,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
proj ((2 |-> NAT),1) is Relation-like product (2 |-> NAT) -defined Function-like total set
<*(2,1)*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
<:<*(2,1)*>:> is Relation-like Function-like set
<:<*(2,1)*>:> * (1,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
<*(<:<*(2,1)*>:> * (1,1)),(2,2)*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() () set
<:<*(<:<*(2,1)*>:> * (1,1)),(2,2)*>:> is Relation-like Function-like set
<:<*(<:<*(2,1)*>:> * (1,1)),(2,2)*>:> * () is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
((0,1),(<:<*(<:<*(2,1)*>:> * (1,1)),(2,2)*>:> * ()),1) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (1) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
F₁() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (1) set
F₂() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
<*F₂()*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
<:<*F₂()*>:> is Relation-like Function-like set
<:<*F₂()*>:> * F₁() is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*F₂()*>:> * F₁()) . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₄(i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₃(F₄(i,j)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity F₂() is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F₂() is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom <:<*F₂()*>:> is set
<:<*F₂()*>:> . <*i,j*> is set
F₁() . (<:<*F₂()*>:> . <*i,j*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₂() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(F₂() . <*i,j*>)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
F₁() . <*(F₂() . <*i,j*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F₄(i,j)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
F₁() . <*F₄(i,j)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₁() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
F₂() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
F₃() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () (2) set
<*F₂(),F₃()*> is non empty Relation-like NAT -defined Function-like finite V43(2) countable FinSequence-like FinSubsequence-like Function-yielding V90() set
<:<*F₂(),F₃()*>:> is Relation-like Function-like set
<:<*F₂(),F₃()*>:> * F₁() is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*F₂(),F₃()*>:> * F₁()) . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₅(i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₆(i,j) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₄(F₅(i,j),F₆(i,j)) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity F₂() is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F₂() is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
arity F₃() is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom F₃() is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
dom <:<*F₂(),F₃()*>:> is set
(dom F₂()) /\ (dom F₃()) is functional FinSequence-membered Element of K32((NAT *))
<:<*F₂(),F₃()*>:> . <*i,j*> is set
F₁() . (<:<*F₂(),F₃()*>:> . <*i,j*>) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₂() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F₃() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(F₂() . <*i,j*>),(F₃() . <*i,j*>)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F₁() . <*(F₂() . <*i,j*>),(F₃() . <*i,j*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F₅(i,j),(F₃() . <*i,j*>)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F₁() . <*F₅(i,j),(F₃() . <*i,j*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F₅(i,j),F₆(i,j)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
F₁() . <*F₅(i,j),F₆(i,j)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*i*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i ! is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(2,1) . <*F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
F . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
(1,1) . <*F*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
g is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
g /. 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(g /. 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,1)*>:> * (1,1)) . <*F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(2,2) . <*F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
F . 2 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F * g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity (0,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity (<:<*(<:<*(2,1)*>:> * (1,1)),(2,2)*>:> * ()) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F ! is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*F*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(F + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(F + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(<:<*(2,1)*>:> * (1,1)),(2,2)*>:> * ()) . <*F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F + 1) * g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F + 1) ! is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
() . <*0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,1) . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
0 ! is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,1) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
(1 -tuples_on NAT) --> 1 is non empty Relation-like non-empty 1 -tuples_on NAT -defined NAT -valued RAT -valued INT -valued Function-like constant total quasi_total Cardinal-yielding complex-valued ext-real-valued real-valued natural-valued Element of K32(K33((1 -tuples_on NAT),NAT))
K33((1 -tuples_on NAT),{1}) is non empty Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
(()) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
<:<*(3,1),(3,3)*>:> * () is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
((1,1),(()),2) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (2) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |^ j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |^ F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,F*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,F*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*i,(F + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,(F + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,F,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(()) . <*i,F,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,g*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,g*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i * g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |^ (F + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
<*i,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
(1,1) . F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i |^ 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
((0,0),(2,1),1) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (1) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (1) set
() . <*0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(i + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(i + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*0,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Function-yielding V90() FinSequence of NAT
arity (0,0) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
arity (2,1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
F + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(F + 1),F*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
<*(F + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(F + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(F + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*((F + 1) + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*((F + 1) + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
(2,1) . g is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(F + 1),F*> . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(0,0) . {} is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*(0 + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(0 + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
(2,1) . j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*0,0*> . 1 is Relation-like Function-like epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(2,2)*> is non empty trivial Relation-like NAT -defined Function-like constant finite V43(1) countable FinSequence-like FinSubsequence-like Function-yielding V90() V97() () set
<:<*(2,2)*>:> is Relation-like Function-like set
<:<*(2,2)*>:> * () is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
((<:<*(2,2)*>:> * ())) is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (3) set
<:<*(3,1),(3,3)*>:> * (<:<*(2,2)*>:> * ()) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
((1,1),((<:<*(2,2)*>:> * ())),2) is non empty Relation-like NAT * -defined NAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous quasi_total () () (2) Element of (NAT)
() is non empty Relation-like NAT * -defined RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued homogeneous () () (2) set
i is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,j*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,j*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -' j is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
rng <*(2,2)*> is non empty trivial finite V43(1) countable set
F is set
{(2,2)} is non empty trivial functional finite V43(1) with_common_domain countable set
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*ii,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*ii,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
<*ii,(k + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*ii,(k + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
dom (2,2) is non empty functional with_common_domain FinSequence-membered Element of K32((NAT *))
F is Relation-like NAT -defined () -valued Function-like finite countable FinSequence-like FinSubsequence-like Function-yielding V90() () FinSequence of ()
<:F:> is Relation-like Function-like set
dom <:F:> is set
k is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
<:F:> . k is set
() . (<:F:> . k) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(2,2) . k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((2,2) . k)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*((2,2) . k)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k . 2 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(k . 2)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(k . 2)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij is Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued Element of 2 -tuples_on NAT
<:F:> . ij is set
() . (<:F:> . ij) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(2,2) . ij is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*((2,2) . ij)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*((2,2) . ij)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ij . 2 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(ij . 2)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(ij . 2)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*(k + 1)*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
() . <*(k + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,ii*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,ii*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -' ii is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i - (ii + 1) is V28() V29() ext-real set
<*i,(i -' ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*i,(i -' ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -' (ii + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable set
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i - ii is V28() V29() ext-real set
k is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
k + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i - (ii + 1) is V28() V29() ext-real set
<*i,(i -' ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*i,(i -' ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -' (ii + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,(i -' ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*i,(i -' ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i -' (ii + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,(i -' ii)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
(<:<*(2,2)*>:> * ()) . <*i,(i -' ii)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real positive non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V91() V92() V93() V94() V95() Element of NAT
i -' (ii + 1) is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,(ii + 1)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,(ii + 1)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i,ii,(() . <*i,ii*>)*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(3) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
((<:<*(2,2)*>:> * ())) . <*i,ii,(() . <*i,ii*>)*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
<*i*> is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing FinSequence of NAT
<*i,0*> is non empty Relation-like NAT -defined NAT -valued Function-like finite V43(2) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued FinSequence of NAT
() . <*i,0*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(1,1) . <*i*> is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
ii is non empty trivial Relation-like NAT -defined NAT -valued Function-like one-to-one constant finite V43(1) countable FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V72() decreasing non-decreasing non-increasing Element of 1 -tuples_on NAT
ii . 1 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i + 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
(i + 0) -' 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT
i -' 0 is epsilon-transitive epsilon-connected ordinal natural V28() V29() ext-real non negative finite V41() countable V65() V67() V78() V79() V80() V81() V82() V83() V93() V94() V95() Element of NAT