:: UPROOTS semantic presentation

REAL is non empty non trivial non finite complex-membered ext-real-membered real-membered V83() V288() V289() V291() set
NAT is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() V286() V288() Element of bool REAL
bool REAL is non empty non trivial non finite set
F_Complex is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() algebraic-closed doubleLoopStr
the carrier of F_Complex is non empty non trivial set
RAT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered V83() set
{} is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() set
omega is epsilon-transitive epsilon-connected ordinal non empty non trivial non finite cardinal limit_cardinal complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() V286() V288() set
bool omega is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
K102(NAT) is V31() set
COMPLEX is non empty non trivial non finite complex-membered V83() set
INT is non empty non trivial non finite complex-membered ext-real-membered real-membered rational-membered integer-membered V83() set
[:REAL,REAL:] is Relation-like non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:REAL,REAL:] is non empty non trivial non finite set
1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
{{},1} is non empty finite V41() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
K413() is set
bool K413() is non empty set
K414() is Element of bool K413()
2 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
[:2,2:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
[:[:2,2:],2:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:2,2:],2:] is non empty finite V41() set
[:1,1:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:1,1:] is non empty finite V41() set
[:[:1,1:],1:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:1,1:],1:] is non empty finite V41() set
[:[:1,1:],REAL:] is Relation-like non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:1,1:],REAL:] is non empty non trivial non finite set
[:[:REAL,REAL:],REAL:] is Relation-like non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:REAL,REAL:],REAL:] is non empty non trivial non finite set
[:[:2,2:],REAL:] is Relation-like non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:2,2:],REAL:] is non empty non trivial non finite set
K645(2) is V253() L21()
the carrier of K645(2) is set
the carrier of K645(2) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K645(2)
[:NAT,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:NAT,NAT:] is non empty non trivial non finite set
[:NAT,REAL:] is Relation-like non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:NAT,REAL:] is non empty non trivial non finite set
1 -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
NAT * is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = 1 } is set
[:COMPLEX,COMPLEX:] is Relation-like non empty non trivial non finite complex-valued set
bool [:COMPLEX,COMPLEX:] is non empty non trivial non finite set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like non empty non trivial non finite complex-valued set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty non trivial non finite set
[:RAT,RAT:] is Relation-like RAT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:RAT,RAT:] is non empty non trivial non finite set
[:[:RAT,RAT:],RAT:] is Relation-like RAT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:RAT,RAT:],RAT:] is non empty non trivial non finite set
[:INT,INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:INT,INT:] is non empty non trivial non finite set
[:[:INT,INT:],INT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued set
bool [:[:INT,INT:],INT:] is non empty non trivial non finite set
[:[:NAT,NAT:],NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
3 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative V54() complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() Element of NAT
card {} is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() set
<*> REAL is Relation-like non-empty empty-yielding NAT -defined REAL -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty proper V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() FinSequence of REAL
Sum (<*> REAL) is V33() real ext-real Element of REAL
K786() is Relation-like [:REAL,REAL:] -defined REAL -valued Function-like non empty total quasi_total V32( REAL ) complex-valued ext-real-valued real-valued commutative associative Element of bool [:[:REAL,REAL:],REAL:]
K786() "**" (<*> REAL) is V33() real ext-real Element of REAL
Seg 1 is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
L is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
dom L is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Sum L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len L) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite len L -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (len L) -tuples_on NAT
(len L) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = len L } is set
Seg (len L) is finite len L -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (len L)) --> 1 is Relation-like Seg (len L) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (len L)),{1}:]
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:(Seg (len L)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (len L)),{1}:] is non empty finite V41() set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Sum s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite len s -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (len s) -tuples_on NAT
(len s) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = len s } is set
Seg (len s) is finite len s -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (len s)) --> 1 is Relation-like Seg (len s) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (len s)),{1}:]
[:(Seg (len s)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (len s)),{1}:] is non empty finite V41() set
x is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<*f*> is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V71() decreasing non-decreasing non-increasing V263() finite-support Element of NAT *
x ^ <*f*> is Relation-like NAT -defined REAL -valued Function-like non empty finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V263() finite-support M24( REAL , NAT )
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len <*f*> is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len x) + (len <*f*>) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len x) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s . (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len x) |-> 1 is Relation-like NAT -defined REAL -valued Function-like finite len x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support Element of (len x) -tuples_on REAL
(len x) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of REAL * : len b₁ = len x } is set
Seg (len x) is finite len x -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (len x)) --> 1 is Relation-like Seg (len x) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (len x)),{1}:]
[:(Seg (len x)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (len x)),{1}:] is non empty finite V41() set
cs is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
len cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cs) -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of REAL * : len b₁ = len cs } is set
Sum x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum x) + f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((Sum x) + f) - f is V33() real V49() ext-real Element of REAL
(cL + 1) - f is V33() real V49() ext-real Element of REAL
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is Relation-like NAT -defined REAL -valued Function-like finite len cs -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support Element of (len cs) -tuples_on REAL
cs1 . cs1a is V33() real ext-real set
g is Relation-like NAT -defined REAL -valued Function-like finite len x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support Element of (len x) -tuples_on REAL
g . cs1a is V33() real ext-real set
Sum cs1 is V33() real ext-real Element of REAL
K786() "**" cs1 is V33() real ext-real Element of REAL
Sum g is V33() real ext-real Element of REAL
K786() "**" g is V33() real ext-real Element of REAL
Sum cs is V33() real ext-real Element of REAL
K786() "**" cs is V33() real ext-real Element of REAL
cL * 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((cL + 1) - f) + f is V33() real V49() ext-real Element of REAL
(len x) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite len x -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (len x) -tuples_on NAT
(len x) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = len x } is set
cL |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite cL -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of cL -tuples_on NAT
cL -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = cL } is set
Seg cL is finite cL -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg cL) --> 1 is Relation-like Seg cL -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg cL),{1}:]
[:(Seg cL),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg cL),{1}:] is non empty finite V41() set
1 |-> 1 is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V71() decreasing non-decreasing non-increasing V263() finite-support Element of 1 -tuples_on NAT
(Seg 1) --> 1 is Relation-like Seg 1 -defined RAT -valued INT -valued {1} -valued Function-like non empty total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg 1),{1}:]
[:(Seg 1),{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg 1),{1}:] is non empty finite V41() set
(cL |-> 1) ^ (1 |-> 1) is Relation-like NAT -defined REAL -valued Function-like non empty finite cL + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V263() finite-support M24( REAL , NAT )
cL + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<*1*> is Relation-like NAT -defined NAT -valued Function-like one-to-one constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V71() decreasing non-decreasing non-increasing V263() finite-support Element of NAT *
(cL |-> 1) ^ <*1*> is Relation-like NAT -defined REAL -valued Function-like non empty finite cL + 1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued V263() finite-support M24( REAL , NAT )
cL is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cL is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Sum cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite len cL -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (len cL) -tuples_on NAT
(len cL) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = len cL } is set
Seg (len cL) is finite len cL -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (len cL)) --> 1 is Relation-like Seg (len cL) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (len cL)),{1}:]
[:(Seg (len cL)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (len cL)),{1}:] is non empty finite V41() set
F₁() is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
1 - 1 is V33() real V49() ext-real Element of REAL
L - 1 is V33() real V49() ext-real Element of REAL
(L - 1) + 1 is V33() real V49() ext-real Element of REAL
L + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
L is non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed V184() V185() V186() V187() addLoopStr
the carrier of L is non empty set
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
cL is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cL is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Sum cL is Element of the carrier of L
cL /. 1 is Element of the carrier of L
cL /. 2 is Element of the carrier of L
(cL /. 1) + (cL /. 2) is Element of the carrier of L
cL . 1 is set
s is Element of the carrier of L
<*s*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
x is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
<*s*> ^ x is Relation-like NAT -defined the carrier of L -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len <*s*> is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cL . 2 is set
2 - 1 is V33() real V49() ext-real Element of REAL
x . (2 - 1) is set
x . 1 is set
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + (len x) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
f is Element of the carrier of L
<*f*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
cs is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
<*f*> ^ cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len <*f*> is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
1 + cs1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
1 + (1 + cs1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs . cs1 is set
x . (1 + cs1) is set
cL . (1 + (1 + cs1)) is set
Sum x is Element of the carrier of L
s + (Sum x) is Element of the carrier of L
Sum cs is Element of the carrier of L
f + (Sum cs) is Element of the carrier of L
s + (f + (Sum cs)) is Element of the carrier of L
f + (0. L) is Element of the carrier of L
s + (f + (0. L)) is Element of the carrier of L
s + f is Element of the carrier of L
(cL /. 1) + f is Element of the carrier of L
L is finite set
card L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is set
x `2 is set
choose (x `2) is Element of x `2
{(choose (x `2))} is non empty trivial finite 1 -element set
(x `2) \ {(choose (x `2))} is Element of bool (x `2)
bool (x `2) is non empty set
[(choose (x `2)),((x `2) \ {(choose (x `2))})] is V22() set
{(choose (x `2)),((x `2) \ {(choose (x `2))})} is non empty finite set
{{(choose (x `2)),((x `2) \ {(choose (x `2))})},{(choose (x `2))}} is non empty finite V41() set
choose L is Element of L
{(choose L)} is non empty trivial finite 1 -element set
L \ {(choose L)} is finite Element of bool L
bool L is non empty finite V41() set
[(choose L),(L \ {(choose L)})] is V22() set
{(choose L),(L \ {(choose L)})} is non empty finite set
{{(choose L),(L \ {(choose L)})},{(choose L)}} is non empty finite V41() set
s is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . 1 is set
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
[:L,(bool L):] is Relation-like finite set
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . x is set
(s . x) `2 is set
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s . (x + 1) is set
(s . (x + 1)) `2 is set
choose ((s . x) `2) is Element of (s . x) `2
{(choose ((s . x) `2))} is non empty trivial finite 1 -element set
((s . x) `2) \ {(choose ((s . x) `2))} is Element of bool ((s . x) `2)
bool ((s . x) `2) is non empty set
[(choose ((s . x) `2)),(((s . x) `2) \ {(choose ((s . x) `2))})] is V22() set
{(choose ((s . x) `2)),(((s . x) `2) \ {(choose ((s . x) `2))})} is non empty finite set
{{(choose ((s . x) `2)),(((s . x) `2) \ {(choose ((s . x) `2))})},{(choose ((s . x) `2))}} is non empty finite V41() set
f is finite set
card f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card f) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
cs is set
[f,cs] is V22() set
{f,cs} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,cs},{f}} is non empty finite V41() set
cs1 is finite set
card cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card cs1) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is finite set
card cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card cs1) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is finite set
card g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card g) + (x + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
{(choose ((s . x) `2))} \/ (((s . x) `2) \ {(choose ((s . x) `2))}) is non empty set
(card g) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(s . 1) `2 is set
x is finite set
card x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card x) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
{(choose L)} \/ (L \ {(choose L)}) is non empty finite set
proj2 s is finite set
x is set
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s . f is set
x is Relation-like NAT -defined [:L,(bool L):] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of [:L,(bool L):]
f is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
proj2 f is finite set
cs is set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
f . cs1 is set
x . cs1 is set
(x . cs1) `1 is set
cs is Relation-like NAT -defined L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
rng cs is finite Element of bool L
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Seg g is finite g -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs | (Seg g) is Relation-like NAT -defined Seg g -defined NAT -defined L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,L:]
[:NAT,L:] is Relation-like set
bool [:NAT,L:] is non empty set
rng (cs | (Seg g)) is finite Element of bool L
x . g is set
(x . g) `2 is set
(rng (cs | (Seg g))) \/ ((x . g) `2) is set
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
Seg (g + 1) is non empty finite g + 1 -element g + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs | (Seg (g + 1)) is Relation-like NAT -defined Seg (g + 1) -defined NAT -defined L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,L:]
rng (cs | (Seg (g + 1))) is finite Element of bool L
x . (g + 1) is set
(x . (g + 1)) `2 is set
(rng (cs | (Seg (g + 1)))) \/ ((x . (g + 1)) `2) is set
choose ((x . g) `2) is Element of (x . g) `2
{(choose ((x . g) `2))} is non empty trivial finite 1 -element set
((x . g) `2) \ {(choose ((x . g) `2))} is Element of bool ((x . g) `2)
bool ((x . g) `2) is non empty set
[(choose ((x . g) `2)),(((x . g) `2) \ {(choose ((x . g) `2))})] is V22() set
{(choose ((x . g) `2)),(((x . g) `2) \ {(choose ((x . g) `2))})} is non empty finite set
{{(choose ((x . g) `2)),(((x . g) `2) \ {(choose ((x . g) `2))})},{(choose ((x . g) `2))}} is non empty finite V41() set
cs1a is finite set
card cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card cs1a) + g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is finite set
card cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card cs1a) + g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
{(choose ((x . g) `2))} \/ (((x . g) `2) \ {(choose ((x . g) `2))}) is non empty set
len cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s1 is Relation-like NAT -defined L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
t is Relation-like NAT -defined L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
p1 is Element of L
<*p1*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
t ^ <*p1*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
rng s1 is finite Element of bool L
rng t is finite Element of bool L
proj2 <*p1*> is non empty trivial finite 1 -element set
(rng t) \/ (proj2 <*p1*>) is non empty finite set
{p1} is non empty trivial finite 1 -element set
(rng t) \/ {p1} is non empty finite set
cs . (g + 1) is set
s1 . (g + 1) is set
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(x . (g + 1)) `1 is set
((x . g) `2) \ {p1} is Element of bool ((x . g) `2)
f1 is finite set
card f1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card f1) + g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is finite set
card cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card cs1) + (g + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(card cs1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,L:]
[:NAT,L:] is Relation-like set
bool [:NAT,L:] is non empty set
rng (cs | (Seg 1)) is finite Element of bool L
x . 1 is set
(x . 1) `2 is set
(rng (cs | (Seg 1))) \/ ((x . 1) `2) is set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
Seg (0 + 1) is non empty finite 0 + 1 -element 0 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1a is Relation-like NAT -defined L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
cs1a . 1 is set
cs . 1 is set
(x . 1) `1 is set
<*(choose L)*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
cs1 is Element of L
<*cs1*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
g is finite set
card g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card g) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
{(choose L)} \/ (L \ {(choose L)}) is non empty finite set
x . (len cs) is set
(x . (len cs)) `2 is set
g is finite set
card g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card g) + (len x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Seg (len cs) is finite len cs -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs | (Seg (len cs)) is Relation-like NAT -defined Seg (len cs) -defined NAT -defined L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,L:]
(rng cs) \/ ((x . (len cs)) `2) is set
L is finite set
the Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
L is finite set
(L) is Relation-like NAT -defined L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
the Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
L is finite set
(L) is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
the Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
len (L) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
card L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
rng (L) is finite Element of bool L
bool L is non empty finite V41() set
L is non empty finite set
(L) is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
the Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
len (L) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
card L is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
L is set
{L} is non empty trivial finite 1 -element set
({L}) is Relation-like NAT -defined {L} -valued Function-like one-to-one non empty onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of {L}
the Relation-like NAT -defined {L} -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of {L} is Relation-like NAT -defined {L} -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of {L}
<*L*> is Relation-like NAT -defined Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support set
rng ({L}) is non empty trivial finite 1 -element Element of bool {L}
bool {L} is non empty finite V41() set
len ({L}) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
card {L} is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
L is finite set
(L) is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
the Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L is Relation-like NAT -defined L -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of L
(L) " is Relation-like Function-like one-to-one set
card L is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
Seg (card L) is finite card L -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
[:L,(Seg (card L)):] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:L,(Seg (card L)):] is non empty finite V41() set
len (L) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (L) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
proj2 ((L) ") is set
rng (L) is finite Element of bool L
bool L is non empty finite V41() set
proj1 ((L) ") is set
L is set
bool L is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
cL is finite Element of bool L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL --> s is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{s} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{s}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> s) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
L --> 0 is Relation-like L -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [:L,NAT:]
[:L,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:L,NAT:] is non empty set
{0} is functional non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:L,{0}:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
dom (cL --> s) is finite Element of bool cL
bool cL is non empty finite V41() set
f is set
((EmptyBag L) +* (cL --> s)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(EmptyBag L) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
((EmptyBag L) +* (cL --> s)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL --> s) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
support ((EmptyBag L) +* (cL --> s)) is set
f is set
((EmptyBag L) +* (cL --> s)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
((EmptyBag L) +* (cL --> s)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL --> s) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
((EmptyBag L) +* (cL --> s)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is set
((EmptyBag L) +* (cL --> s)) . cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
proj1 ((EmptyBag L) +* (cL --> s)) is set
dom (EmptyBag L) is Element of bool L
(dom (EmptyBag L)) \/ (dom (cL --> s)) is set
L \/ (dom (cL --> s)) is set
L \/ cL is set
L is set
bool L is non empty set
cL is finite Element of bool L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> s is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{s} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{s}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> s) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
x is set
(L,cL,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (cL --> s) is finite Element of bool cL
bool cL is non empty finite V41() set
(EmptyBag L) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
cL is finite Element of bool L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> s is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{s} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{s}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> s) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
x is set
(L,cL,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (cL --> s) is finite Element of bool cL
bool cL is non empty finite V41() set
(cL --> s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
cL is finite Element of bool L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> s is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{s} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{s}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> s) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
support (L,cL,s) is finite set
x is set
(L,cL,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
(L,cL,s) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
cL is finite Element of bool L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> s is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{s} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{s}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> s) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
x is set
(L,cL,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(EmptyBag L) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(EmptyBag L) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
cL is finite Element of bool L
s is finite Element of bool L
cL \/ s is finite Element of bool L
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cL \/ s),x) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
(cL \/ s) --> x is Relation-like cL \/ s -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(cL \/ s),NAT:]
[:(cL \/ s),NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:(cL \/ s),NAT:] is non empty set
{x} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:(cL \/ s),{x}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* ((cL \/ s) --> x) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,cL,x) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> x is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
[:cL,{x}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> x) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,s,x) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
s --> x is Relation-like s -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s,NAT:]
[:s,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:s,NAT:] is non empty set
[:s,{x}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (s --> x) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,cL,x) + (L,s,x) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
cL /\ s is finite Element of bool L
f is set
(L,(cL \/ s),x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) . f) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) . f) + ((L,s,x) . f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) + (L,s,x)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cL \/ s),x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) . f) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) . f) + ((L,s,x) . f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) + (L,s,x)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cL \/ s),x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,s,x) . f) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,x) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) . f) + ((L,s,x) . f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL,x) + (L,s,x)) . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
cL is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
support cL is finite set
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
((support cL)) * cL is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
rng (((support cL)) * cL) is finite complex-membered ext-real-membered real-membered V288() V289() V290() Element of bool REAL
dom cL is Element of bool L
bool L is non empty set
rng ((support cL)) is finite Element of bool (support cL)
bool (support cL) is non empty finite V41() set
dom (((support cL)) * cL) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom ((support cL)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len ((support cL)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Seg (len ((support cL))) is finite len ((support cL)) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum f is V33() real ext-real Element of REAL
K786() "**" f is V33() real ext-real Element of REAL
s is V33() real ext-real set
f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum f is V33() real ext-real Element of REAL
K786() "**" f is V33() real ext-real Element of REAL
x is V33() real ext-real set
cs is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum cs is V33() real ext-real Element of REAL
K786() "**" cs is V33() real ext-real Element of REAL
L is set
L is set
cL is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) is V33() real ext-real set
support cL is finite set
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
((support cL)) * cL is Relation-like NAT -defined RAT -valued Function-like finite complex-valued ext-real-valued real-valued natural-valued finite-support set
s is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum s is V33() real ext-real Element of REAL
K786() "**" s is V33() real ext-real Element of REAL
rng s is finite complex-membered ext-real-membered real-membered V288() V289() V290() Element of bool REAL
x is set
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is set
s . f is V33() real ext-real set
((support cL)) . f is set
cL . (((support cL)) . f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
support cL is finite set
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
((support cL)) * cL is Relation-like NAT -defined RAT -valued Function-like finite complex-valued ext-real-valued real-valued natural-valued finite-support set
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum f is V33() real ext-real Element of REAL
K786() "**" f is V33() real ext-real Element of REAL
rng f is finite complex-membered ext-real-membered real-membered V288() V289() V290() Element of bool REAL
cs is set
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1 is set
f . cs1 is V33() real ext-real set
((support cL)) . cs1 is set
cL . (((support cL)) . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
cs1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
rng f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool REAL
cs is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
L is set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
cL is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
(L,cL) is V33() real ext-real set
support cL is finite set
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
((support cL)) * cL is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
<*> NAT is Relation-like non-empty empty-yielding NAT -defined NAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() V263() FinSequence-yielding finite-support V288() V289() V290() V291() Element of NAT *
L is set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
cL is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) is V33() real ext-real set
support cL is finite set
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
(L,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((support cL)) * cL is Relation-like NAT -defined RAT -valued Function-like finite complex-valued ext-real-valued real-valued natural-valued finite-support set
x is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is set
rng ((support cL)) is finite Element of bool (support cL)
bool (support cL) is non empty finite V41() set
dom ((support cL)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
((support cL)) . cs is set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((support cL)) . cs1 is set
cL . (((support cL)) . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cL is Element of bool L
bool L is non empty set
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
cL is finite Element of bool L
card cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(L,cL,1) is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
Bags L is non empty set
Bags L is functional non empty Element of bool (Bags L)
bool (Bags L) is non empty set
EmptyBag L is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags L
cL --> 1 is Relation-like cL -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cL,NAT:]
[:cL,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cL,NAT:] is non empty set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:cL,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag L) +* (cL --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
s is Relation-like L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support s is finite set
(L,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) is Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL
the Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL is Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL
(cL) * s is Relation-like NAT -defined RAT -valued Function-like finite complex-valued ext-real-valued real-valued natural-valued finite-support set
dom s is Element of bool L
(card cL) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite card cL -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (card cL) -tuples_on NAT
(card cL) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = card cL } is set
Seg (card cL) is finite card cL -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (card cL)) --> 1 is Relation-like Seg (card cL) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (card cL)),{1}:]
[:(Seg (card cL)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (card cL)),{1}:] is non empty finite V41() set
len (cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
rng (cL) is finite Element of bool cL
bool cL is non empty finite V41() set
dom ((cL) * s) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom (cL) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Seg (len (cL)) is finite len (cL) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1a . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . cs1 is set
s . ((cL) . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cs1a) |-> 1 is Relation-like NAT -defined NAT -valued Function-like finite len cs1a -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support Element of (len cs1a) -tuples_on NAT
(len cs1a) -tuples_on NAT is functional non empty FinSequence-membered FinSequenceSet of NAT
{ b₁ where b₁ is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of NAT * : len b₁ = len cs1a } is set
Seg (len cs1a) is finite len cs1a -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg (len cs1a)) --> 1 is Relation-like Seg (len cs1a) -defined RAT -valued INT -valued {1} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg (len cs1a)),{1}:]
[:(Seg (len cs1a)),{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg (len cs1a)),{1}:] is non empty finite V41() set
support (L,cL,1) is finite set
dom (L,cL,1) is Element of bool L
cs1 is set
(cL) " is Relation-like Function-like one-to-one set
((cL) ") . cs1 is set
s . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . (((cL) ") . cs1) is set
s . ((cL) . (((cL) ") . cs1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . (((cL) ") . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,1) . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,1) . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
proj2 cs1 is finite set
g is set
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1a is set
cs1 . cs1a is set
(cL) . cs1a is set
s . ((cL) . cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len ((card cL) |-> 1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom g is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
g . cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . cs1a is set
s . ((cL) . cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((card cL) |-> 1) . cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(card cL) * 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is set
bool L is non empty set
cL is finite Element of bool L
(cL) is Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL
the Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL is Relation-like NAT -defined cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cL
s is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
support s is finite set
(cL) * s is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
(L,s) is V33() real ext-real set
card (support s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
card cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support s)) is Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s
the Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s is Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s
rng ((support s)) is finite Element of bool (support s)
bool (support s) is non empty finite V41() set
len (cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (cL) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Seg (card cL) is finite card cL -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len ((support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom ((support s)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Seg (card (support s)) is finite card (support s) -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
((support s)) * s is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
cs1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum cs1 is V33() real ext-real Element of REAL
K786() "**" cs1 is V33() real ext-real Element of REAL
dom s is Element of bool L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
rng (cL) is finite Element of bool cL
bool cL is non empty finite V41() set
dom ((cL) * s) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
proj2 g is finite set
rng s is complex-membered ext-real-membered real-membered Element of bool REAL
cs1a is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum cs1a is V33() real ext-real Element of REAL
K786() "**" cs1a is V33() real ext-real Element of REAL
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ( b₁ in dom cs1a & cs1a . b₁ = 0 ) } is set
t is set
s1 is set
(cL) . s1 is set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . p1 is V33() real ext-real set
s . t is V33() real ext-real set
s1 is set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . p1 is V33() real ext-real set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(card (support s)) + p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p1 |-> 0 is Relation-like empty-yielding NAT -defined REAL -valued Function-like Function-yielding finite p1 -element FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued FinSequence-yielding finite-support Element of p1 -tuples_on REAL
p1 -tuples_on REAL is functional non empty FinSequence-membered FinSequenceSet of REAL
REAL * is functional non empty FinSequence-membered FinSequenceSet of REAL
{ b₁ where b₁ is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of REAL * : len b₁ = p1 } is set
Seg p1 is finite p1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg p1) --> 0 is Relation-like Seg p1 -defined RAT -valued INT -valued {0} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(Seg p1),{0}:]
{0} is functional non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:(Seg p1),{0}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(Seg p1),{0}:] is non empty finite V41() set
t is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
t ^ (p1 |-> 0) is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
s1 is non empty finite set
(s1) is Relation-like NAT -defined s1 -valued Function-like one-to-one non empty onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s1
the Relation-like NAT -defined s1 -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s1 is Relation-like NAT -defined s1 -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s1
rng (s1) is non empty finite Element of bool s1
bool s1 is non empty finite V41() set
t1 is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
t1 " is Relation-like Function-like set
card s1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
Seg (card s1) is non empty finite card s1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
[:s1,(Seg (card s1)):] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
bool [:s1,(Seg (card s1)):] is non empty finite V41() set
[:s1,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:s1,NAT:] is non empty non trivial non finite set
x is Relation-like s1 -defined Seg (card s1) -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s1,(Seg (card s1)):]
i is Relation-like s1 -defined NAT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s1,NAT:]
i is Relation-like Function-like set
proj1 i is set
((support s)) " is Relation-like Function-like one-to-one set
(cL) * (((support s)) ") is Relation-like NAT -defined Function-like one-to-one finite finite-support set
((cL) * (((support s)) ")) +* i is Relation-like Function-like set
dom i is non empty finite Element of bool s1
(dom cs1a) \ s1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(cL) | ((dom cs1a) \ s1) is Relation-like NAT -defined (dom cs1a) \ s1 -defined NAT -defined cL -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,cL:]
[:NAT,cL:] is Relation-like set
bool [:NAT,cL:] is non empty set
p is set
rng ((cL) | ((dom cs1a) \ s1)) is finite Element of bool cL
dom ((cL) | ((dom cs1a) \ s1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool ((dom cs1a) \ s1)
bool ((dom cs1a) \ s1) is non empty finite V41() set
a is set
((cL) | ((dom cs1a) \ s1)) . a is set
(cL) . a is set
findom cs1a is finite Element of K102(NAT)
(findom cs1a) \ s1 is finite Element of bool (findom cs1a)
bool (findom cs1a) is non empty finite V41() set
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . c is V33() real ext-real set
(cL) . c is set
s . ((cL) . c) is V33() real ext-real set
a is set
(cL) . a is set
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . c is V33() real ext-real set
(cL) . c is set
s . ((cL) . c) is V33() real ext-real set
((findom cs1a) \ s1) /\ (dom cs1a) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
card ((dom cs1a) \ s1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
card (dom cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card (dom cs1a)) - (card s1) is V33() real V49() ext-real set
(card (support s)) + (card s1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
proj1 (((support s)) ") is set
dom ((cL) * (((support s)) ")) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(dom ((cL) * (((support s)) "))) /\ (proj1 i) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
p is set
a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . a is V33() real ext-real set
(cL) . a is set
s . ((cL) . a) is V33() real ext-real set
len (t ^ (p1 |-> 0)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len (p1 |-> 0) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cs1) + (len (p1 |-> 0)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (t ^ (p1 |-> 0)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
p is set
(dom ((cL) * (((support s)) "))) \/ (proj1 i) is set
cs1a . p is V33() real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . p is V33() real ext-real set
a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . a is V33() real ext-real set
(cL) . a is set
s . ((cL) . a) is V33() real ext-real set
cs1a . p is V33() real ext-real set
proj1 (((cL) * (((support s)) ")) +* i) is set
len t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom t1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
rng i is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool REAL
proj2 i is set
p is set
a is set
i . a is set
i /. a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(i /. a) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p1 + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p is set
((cL) * (((support s)) ")) . p is set
proj2 ((cL) * (((support s)) ")) is finite set
proj2 (((support s)) ") is set
(((cL) * (((support s)) ")) +* i) . p is set
(((cL) * (((support s)) ")) +* i) . p is set
i . p is set
[:(dom (t ^ (p1 |-> 0))),(dom (t ^ (p1 |-> 0))):] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(dom (t ^ (p1 |-> 0))),(dom (t ^ (p1 |-> 0))):] is non empty finite V41() set
dom (p1 |-> 0) is finite p1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
rng t1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool REAL
a is set
p is Relation-like dom (t ^ (p1 |-> 0)) -defined dom (t ^ (p1 |-> 0)) -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(dom (t ^ (p1 |-> 0))),(dom (t ^ (p1 |-> 0))):]
rng p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool REAL
dom p is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool (dom (t ^ (p1 |-> 0)))
bool (dom (t ^ (p1 |-> 0))) is non empty finite V41() set
c is set
p . c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . c is set
(((support s)) ") . ((cL) . c) is set
proj2 (((support s)) ") is set
((cL) * (((support s)) ")) . c is set
i . c is set
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom t is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
((support s)) . c is set
(cL) " is Relation-like Function-like one-to-one set
((cL) ") . (((support s)) . c) is set
((cL) * s) . (((cL) ") . (((support s)) . c)) is V33() real ext-real set
(cL) . (((cL) ") . (((support s)) . c)) is set
s . ((cL) . (((cL) ") . (((support s)) . c))) is V33() real ext-real set
s . (((support s)) . c) is V33() real ext-real set
px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . px is V33() real ext-real set
p . (((cL) ") . (((support s)) . c)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((cL) * (((support s)) ")) . (((cL) ") . (((support s)) . c)) is set
(((support s)) ") . ((cL) . (((cL) ") . (((support s)) . c))) is set
(((support s)) ") . (((support s)) . c) is set
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(len t) + px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(len t) + px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t1 . px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p . (t1 . px) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . (t1 . px) is set
i /. (t1 . px) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(i /. (t1 . px)) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . (t1 . px) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(i . (t1 . px)) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
px + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom t is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(t ^ (p1 |-> 0)) * p is Relation-like dom (t ^ (p1 |-> 0)) -defined REAL -valued Function-like finite complex-valued ext-real-valued real-valued finite-support Element of bool [:(dom (t ^ (p1 |-> 0))),REAL:]
[:(dom (t ^ (p1 |-> 0))),REAL:] is Relation-like complex-valued ext-real-valued real-valued set
bool [:(dom (t ^ (p1 |-> 0))),REAL:] is non empty set
dom ((t ^ (p1 |-> 0)) * p) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool (dom (t ^ (p1 |-> 0)))
a is set
cs1a . a is V33() real ext-real set
i /. a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
j is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is set
(i /. a) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(p1 |-> 0) . c is Relation-like non-empty empty-yielding NAT -defined RAT -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() set
px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(t ^ (p1 |-> 0)) . px is V33() real ext-real set
((t ^ (p1 |-> 0)) * p) . a is V33() real ext-real set
cs1a . a is V33() real ext-real set
p . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . a is set
s . ((cL) . a) is V33() real ext-real set
(((support s)) ") . ((cL) . a) is set
px is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . px is V33() real ext-real set
((cL) * (((support s)) ")) . a is set
c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((support s)) . c is set
s . (((support s)) . c) is V33() real ext-real set
cs1 . c is V33() real ext-real set
(t ^ (p1 |-> 0)) . c is V33() real ext-real set
((t ^ (p1 |-> 0)) * p) . a is V33() real ext-real set
cs1a . a is V33() real ext-real set
a is set
proj1 p is finite set
c is set
p . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
p . c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
p . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
p . c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . a is set
(((support s)) ") . ((cL) . a) is set
((support s)) . ((((support s)) ") . ((cL) . a)) is set
(cL) " is Relation-like Function-like one-to-one set
((cL) ") . ((cL) . a) is set
(cL) . c is set
((cL) * (((support s)) ")) . c is set
(((support s)) ") . ((cL) . c) is set
((cL) * (((support s)) ")) . a is set
((cL) ") . ((cL) . c) is set
(cL) . a is set
(((support s)) ") . ((cL) . a) is set
proj2 (((support s)) ") is set
((cL) * (((support s)) ")) . a is set
i . c is set
i /. c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(i /. c) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL) . c is set
(((support s)) ") . ((cL) . c) is set
proj2 (((support s)) ") is set
((cL) * (((support s)) ")) . c is set
i . a is set
i /. a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(i /. a) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i /. a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i /. c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . c is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . c is set
(i /. c) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
i . a is set
(i /. a) + (card (support s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum (p1 |-> 0) is V33() real ext-real Element of REAL
K786() "**" (p1 |-> 0) is V33() real ext-real Element of REAL
Sum t is V33() real ext-real Element of REAL
K786() "**" t is V33() real ext-real Element of REAL
(Sum t) + (Sum (p1 |-> 0)) is V33() real ext-real Element of REAL
Sum (t ^ (p1 |-> 0)) is V33() real ext-real Element of REAL
K786() "**" (t ^ (p1 |-> 0)) is V33() real ext-real Element of REAL
L is set
cL is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
s is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
x is Relation-like L -defined Function-like total complex-valued ext-real-valued real-valued finite-support set
s + x is Relation-like L -defined Function-like total set
(L,cL) is V33() real ext-real set
(L,s) is V33() real ext-real set
(L,x) is V33() real ext-real set
(L,s) + (L,x) is V33() real ext-real set
support cL is finite set
support s is finite set
support x is finite set
(support s) \/ (support x) is finite set
dom x is Element of bool L
bool L is non empty set
dom s is Element of bool L
cs1 is finite Element of bool L
(cs1) is Relation-like NAT -defined cs1 -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cs1
the Relation-like NAT -defined cs1 -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cs1 is Relation-like NAT -defined cs1 -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of cs1
(cs1) * s is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
cs1a is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum cs1a is V33() real ext-real Element of REAL
K786() "**" cs1a is V33() real ext-real Element of REAL
rng (cs1) is finite Element of bool cs1
bool cs1 is non empty finite V41() set
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom (cs1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(cs1) * cL is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
cs1 is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum cs1 is V33() real ext-real Element of REAL
K786() "**" cs1 is V33() real ext-real Element of REAL
dom cL is Element of bool L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs1) * x is Relation-like NAT -defined Function-like finite complex-valued ext-real-valued real-valued finite-support set
t is Relation-like NAT -defined REAL -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued finite-support FinSequence of REAL
Sum t is V33() real ext-real Element of REAL
K786() "**" t is V33() real ext-real Element of REAL
dom t is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a /. s1 is V33() real ext-real Element of REAL
cs1a . s1 is V33() real ext-real set
(cs1) . s1 is set
s . ((cs1) . s1) is V33() real ext-real set
cs1 . s1 is V33() real ext-real set
cL . ((cs1) . s1) is V33() real ext-real set
t /. s1 is V33() real ext-real Element of REAL
t . s1 is V33() real ext-real set
x . ((cs1) . s1) is V33() real ext-real set
(cs1a /. s1) + (t /. s1) is V33() real ext-real Element of REAL
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty unital associative commutative multMagma
the carrier of L is non empty set
cL is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom cL is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
[:(dom cL),(dom cL):] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(dom cL),(dom cL):] is non empty finite V41() set
s is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
Product s is Element of the carrier of L
the multF of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like non empty total quasi_total V32( the carrier of L) associative Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is Relation-like non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is Relation-like non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L "**" s is Element of the carrier of L
Product cL is Element of the carrier of L
the multF of L "**" cL is Element of the carrier of L
x is Relation-like dom cL -defined dom cL -valued Function-like one-to-one total quasi_total onto bijective finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(dom cL),(dom cL):]
cL * x is Relation-like dom cL -defined the carrier of L -valued Function-like finite finite-support Element of bool [:(dom cL), the carrier of L:]
[:(dom cL), the carrier of L:] is Relation-like set
bool [:(dom cL), the carrier of L:] is non empty set
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
L is non empty non trivial ZeroStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
NonZero L is non empty Element of bool the carrier of L
bool the carrier of L is non empty set
[#] L is non empty non proper Element of bool the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
{(0. L)} is non empty trivial finite 1 -element set
([#] L) \ {(0. L)} is Element of bool the carrier of L
the Element of NonZero L is Element of NonZero L
s is Element of the carrier of L
<%s%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element Element of bool the carrier of L
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
(0_. L) . 0 is Element of the carrier of L
x . 0 is Element of the carrier of L
L is non empty non degenerated non trivial multLoopStr_0
the carrier of L is non empty non trivial set
cL is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
len <%cL,(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . ((len cL) -' 1) is Element of the carrier of L
((len cL) -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . 0 is Element of the carrier of L
<%(cL . 0)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' (0_. L) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cL *' (0_. L)) . s is Element of the carrier of L
x is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum x is Element of the carrier of L
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . f is set
f -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . (f -' 1) is Element of the carrier of L
(s + 1) -' f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(0_. L) . ((s + 1) -' f) is Element of the carrier of L
(cL . (f -' 1)) * ((0_. L) . ((s + 1) -' f)) is Element of the carrier of L
(cL . (f -' 1)) * (0. L) is Element of the carrier of L
(0_. L) . s is Element of the carrier of L
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
f + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . cs is Element of the carrier of L
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . cs1 is Element of the carrier of L
cs + cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs + cs1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cL *' s) . (cs + cs1) is Element of the carrier of L
cs1a is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum cs1a is Element of the carrier of L
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
((cs + cs1) + 1) -' (len cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 + (cs + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cs1 + (cs + 1)) -' (len cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs + 1) + cs1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . t is set
t -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . (t -' 1) is Element of the carrier of L
((cs + cs1) + 1) -' t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . (((cs + cs1) + 1) -' t) is Element of the carrier of L
(cL . (t -' 1)) * (s . (((cs + cs1) + 1) -' t)) is Element of the carrier of L
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t + s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 + s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((cs + cs1) + 1) -' cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs1 + s1) + cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((cs1 + s1) + cs1) -' cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . cs1 is set
cs1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . (cs1 -' 1) is Element of the carrier of L
(cL . (cs1 -' 1)) * (0. L) is Element of the carrier of L
(len cL) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len cL) + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . cs1 is set
((cs + cs1) + 1) -' cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . (((cs + cs1) + 1) -' cs1) is Element of the carrier of L
(0. L) * (s . (((cs + cs1) + 1) -' cs1)) is Element of the carrier of L
cs1a . (len cL) is set
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . ((len cL) -' 1) is Element of the carrier of L
s . (((cs + cs1) + 1) -' (len cL)) is Element of the carrier of L
(cL . ((len cL) -' 1)) * (s . (((cs + cs1) + 1) -' (len cL))) is Element of the carrier of L
(cL . cs) * (s . cs1) is Element of the carrier of L
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
s is Element of the carrier of (Polynom-Ring L)
x is Element of the carrier of (Polynom-Ring L)
s * x is Element of the carrier of (Polynom-Ring L)
0. (Polynom-Ring L) is V107( Polynom-Ring L) Element of the carrier of (Polynom-Ring L)
the ZeroF of (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f *' cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots cL is Element of bool the carrier of L
bool the carrier of L is non empty set
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots s is Element of bool the carrier of L
(Roots cL) \/ (Roots s) is Element of bool the carrier of L
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots (cL *' s) is Element of bool the carrier of L
x is set
f is Element of the carrier of L
eval (cL,f) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
eval (s,f) is Element of the carrier of L
(eval (cL,f)) * (eval (s,f)) is Element of the carrier of L
eval ((cL *' s),f) is Element of the carrier of L
f is Element of the carrier of L
eval (s,f) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
eval (cL,f) is Element of the carrier of L
(eval (cL,f)) * (eval (s,f)) is Element of the carrier of L
eval ((cL *' s),f) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots (cL *' s) is Element of bool the carrier of L
bool the carrier of L is non empty set
Roots cL is Element of bool the carrier of L
Roots s is Element of bool the carrier of L
(Roots cL) \/ (Roots s) is Element of bool the carrier of L
x is set
f is Element of the carrier of L
eval ((cL *' s),f) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
eval (cL,f) is Element of the carrier of L
eval (s,f) is Element of the carrier of L
(eval (cL,f)) * (eval (s,f)) is Element of the carrier of L
f is Element of the carrier of L
eval (cL,f) is Element of the carrier of L
eval (s,f) is Element of the carrier of L
(eval (cL,f)) * (eval (s,f)) is Element of the carrier of L
eval ((cL *' s),f) is Element of the carrier of L
f is Element of the carrier of L
eval (s,f) is Element of the carrier of L
eval (cL,f) is Element of the carrier of L
(eval (cL,f)) * (eval (s,f)) is Element of the carrier of L
eval ((cL *' s),f) is Element of the carrier of L
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Element of the carrier of (Polynom-Ring L)
- s is Element of the carrier of (Polynom-Ring L)
cL + (- cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL - cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
f is Element of the carrier of (Polynom-Ring L)
s + f is Element of the carrier of (Polynom-Ring L)
0. (Polynom-Ring L) is V107( Polynom-Ring L) Element of the carrier of (Polynom-Ring L)
the ZeroF of (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL - s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL + (- s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Element of the carrier of (Polynom-Ring L)
f is Element of the carrier of (Polynom-Ring L)
x - f is Element of the carrier of (Polynom-Ring L)
- f is Element of the carrier of (Polynom-Ring L)
x + (- f) is Element of the carrier of (Polynom-Ring L)
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(cL *' s) - (cL *' x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- (cL *' x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(cL *' s) + (- (cL *' x)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s - x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s + (- x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' (s - x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict right-distributive left-distributive distributive Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
cs1 is Element of the carrier of (Polynom-Ring L)
g is Element of the carrier of (Polynom-Ring L)
cs1 - g is Element of the carrier of (Polynom-Ring L)
- g is Element of the carrier of (Polynom-Ring L)
cs1 + (- g) is Element of the carrier of (Polynom-Ring L)
cs is Element of the carrier of (Polynom-Ring L)
cs * cs1 is Element of the carrier of (Polynom-Ring L)
cs * g is Element of the carrier of (Polynom-Ring L)
(cs * cs1) - (cs * g) is Element of the carrier of (Polynom-Ring L)
- (cs * g) is Element of the carrier of (Polynom-Ring L)
(cs * cs1) + (- (cs * g)) is Element of the carrier of (Polynom-Ring L)
cs * (cs1 - g) is Element of the carrier of (Polynom-Ring L)
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL - s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL + (- s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
0. (Polynom-Ring L) is V107( Polynom-Ring L) Element of the carrier of (Polynom-Ring L)
the ZeroF of (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
f is Element of the carrier of (Polynom-Ring L)
cs is Element of the carrier of (Polynom-Ring L)
f - cs is Element of the carrier of (Polynom-Ring L)
- cs is Element of the carrier of (Polynom-Ring L)
f + (- cs) is Element of the carrier of (Polynom-Ring L)
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(cL *' s) - (cL *' x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- (cL *' x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(cL *' s) + (- (cL *' x)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s - x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
- x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s + (- x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' (s - x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s `^ cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (s,cL) is set
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s `^ x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (s,x) is set
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s `^ (x + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (s,(x + 1)) is set
(s `^ x) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(1_. L) . 0 is Element of the carrier of L
(0_. L) . 0 is Element of the carrier of L
s `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (s,0) is set
L is non empty left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x `^ cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (x,cL) is set
x `^ s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,s) is set
(x `^ cL) *' (x `^ s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x `^ (cL + s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,(cL + s)) is set
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x `^ f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,f) is set
(x `^ cL) *' (x `^ f) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL + f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x `^ (cL + f) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,(cL + f)) is set
f + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x `^ (f + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,(f + 1)) is set
(x `^ cL) *' (x `^ (f + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL + (f + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x `^ (cL + (f + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,(cL + (f + 1))) is set
(x `^ f) *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(x `^ cL) *' ((x `^ f) *' x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(x `^ (cL + f)) *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(cL + f) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x `^ ((cL + f) + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,((cL + f) + 1)) is set
x `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,0) is set
(x `^ cL) *' (x `^ 0) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(x `^ cL) *' (1_. L) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x `^ (cL + 0) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (x,(cL + 0)) is set
L is non empty multLoopStr_0
the carrier of L is non empty set
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
<%(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
dom (0_. L) is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V288() Element of bool NAT
cL is set
(1_. L) . cL is set
<%(1. L)%> . cL is set
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(1_. L) . cL is set
(0_. L) . s is Element of the carrier of L
<%(1. L)%> . cL is set
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty left_add-cancelable right_add-cancelable right_complementable unital right-distributive right_unital well-unital left_unital add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' <%(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
cL *' (1_. L) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (cL *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len (cL *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . ((len cL) -' 1) is Element of the carrier of L
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . ((len s) -' 1) is Element of the carrier of L
(cL . ((len cL) -' 1)) * (s . ((len s) -' 1)) is Element of the carrier of L
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (cL *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len cL) + (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(1 + 1) - 1 is V33() real V49() ext-real Element of REAL
((len cL) + (len s)) - 1 is V33() real V49() ext-real Element of REAL
L is non empty left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (cL *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . ((len cL) -' 1) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . ((len s) -' 1) is Element of the carrier of L
(cL . ((len cL) -' 1)) * (s . ((len s) -' 1)) is Element of the carrier of L
(len cL) + (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((len cL) + (len s)) - 1 is V33() real V49() ext-real Element of REAL
(len s) - 1 is V33() real V49() ext-real Element of REAL
1 - 1 is V33() real V49() ext-real Element of REAL
(len cL) + ((len s) - 1) is V33() real V49() ext-real Element of REAL
0 + (len cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty left_add-cancelable right_add-cancelable right_complementable left-distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Element of the carrier of L
s is Element of the carrier of L
<%cL,s%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%cL,s%> *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(<%cL,s%> *' x) . 0 is Element of the carrier of L
x . 0 is Element of the carrier of L
cL * (x . 0) is Element of the carrier of L
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum cs is Element of the carrier of L
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs . 1 is set
cs1 is Element of the carrier of L
<*cs1*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,s%> . (1 -' 1) is Element of the carrier of L
(0 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . ((0 + 1) -' 1) is Element of the carrier of L
(<%cL,s%> . (1 -' 1)) * (x . ((0 + 1) -' 1)) is Element of the carrier of L
<%cL,s%> . 0 is Element of the carrier of L
(<%cL,s%> . 0) * (x . ((0 + 1) -' 1)) is Element of the carrier of L
(<%cL,s%> . 0) * (x . 0) is Element of the carrier of L
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(<%cL,s%> *' x) . (g + 1) is Element of the carrier of L
x . (g + 1) is Element of the carrier of L
cL * (x . (g + 1)) is Element of the carrier of L
x . g is Element of the carrier of L
s * (x . g) is Element of the carrier of L
(cL * (x . (g + 1))) + (s * (x . g)) is Element of the carrier of L
(g + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Sum cs1a is Element of the carrier of L
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
g + 2 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 + 2 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a /. 2 is Element of the carrier of L
cs1a . 2 is set
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(1 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,s%> . ((1 + 1) -' 1) is Element of the carrier of L
((g + 1) + 1) -' 2 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . (((g + 1) + 1) -' 2) is Element of the carrier of L
(<%cL,s%> . ((1 + 1) -' 1)) * (x . (((g + 1) + 1) -' 2)) is Element of the carrier of L
<%cL,s%> . 1 is Element of the carrier of L
(<%cL,s%> . 1) * (x . (((g + 1) + 1) -' 2)) is Element of the carrier of L
g + (1 + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(g + (1 + 1)) -' 2 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . ((g + (1 + 1)) -' 2) is Element of the carrier of L
s * (x . ((g + (1 + 1)) -' 2)) is Element of the carrier of L
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a . cs1 is set
cs1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,s%> . (cs1 -' 1) is Element of the carrier of L
((g + 1) + 1) -' cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . (((g + 1) + 1) -' cs1) is Element of the carrier of L
(<%cL,s%> . (cs1 -' 1)) * (x . (((g + 1) + 1) -' cs1)) is Element of the carrier of L
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,s%> . s1 is Element of the carrier of L
(<%cL,s%> . s1) * (x . (((g + 1) + 1) -' cs1)) is Element of the carrier of L
(0. L) * (x . (((g + 1) + 1) -' cs1)) is Element of the carrier of L
cs1a /. 1 is Element of the carrier of L
cs1a . 1 is set
((g + 1) + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . (((g + 1) + 1) -' 1) is Element of the carrier of L
(<%cL,s%> . (1 -' 1)) * (x . (((g + 1) + 1) -' 1)) is Element of the carrier of L
(<%cL,s%> . 0) * (x . (((g + 1) + 1) -' 1)) is Element of the carrier of L
(<%cL,s%> . 0) * (x . (g + 1)) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
<%cL,(1. L)%> *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (<%cL,(1. L)%> *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len <%cL,(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len <%cL,(1. L)%>) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> . ((len <%cL,(1. L)%>) -' 1) is Element of the carrier of L
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . ((len s) -' 1) is Element of the carrier of L
(<%cL,(1. L)%> . ((len <%cL,(1. L)%>) -' 1)) * (s . ((len s) -' 1)) is Element of the carrier of L
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(1 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> . ((1 + 1) -' 1) is Element of the carrier of L
(<%cL,(1. L)%> . ((1 + 1) -' 1)) * (s . ((len s) -' 1)) is Element of the carrier of L
<%cL,(1. L)%> . 1 is Element of the carrier of L
(<%cL,(1. L)%> . 1) * (s . ((len s) -' 1)) is Element of the carrier of L
(1. L) * (s . ((len s) -' 1)) is Element of the carrier of L
(len s) + (1 + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len s) + (1 + 1)) - 1 is V33() real V49() ext-real Element of REAL
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,x) is set
len (<%cL,(1. L)%> `^ x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ (x + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,(x + 1)) is set
len (<%cL,(1. L)%> `^ (x + 1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(x + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ 1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,1) is set
f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(<%cL,(1. L)%> `^ 1) *' f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((<%cL,(1. L)%> `^ 1) *' f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> *' f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (<%cL,(1. L)%> *' f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,0) is set
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
len (<%cL,(1. L)%> `^ 0) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
cL is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,s) is set
len (<%cL,(1. L)%> `^ s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,f) is set
(<%cL,(1. L)%> `^ f) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((<%cL,(1. L)%> `^ f) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f + (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ (f + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,(f + 1)) is set
(<%cL,(1. L)%> `^ (f + 1)) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((<%cL,(1. L)%> `^ (f + 1)) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(f + 1) + (len s) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ 1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,1) is set
(<%cL,(1. L)%> `^ 1) *' (<%cL,(1. L)%> `^ f) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
((<%cL,(1. L)%> `^ 1) *' (<%cL,(1. L)%> `^ f)) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (((<%cL,(1. L)%> `^ 1) *' (<%cL,(1. L)%> `^ f)) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> *' (<%cL,(1. L)%> `^ f) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%cL,(1. L)%> *' (<%cL,(1. L)%> `^ f)) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((<%cL,(1. L)%> *' (<%cL,(1. L)%> `^ f)) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%cL,(1. L)%> *' ((<%cL,(1. L)%> `^ f) *' s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (<%cL,(1. L)%> *' ((<%cL,(1. L)%> `^ f) *' s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(f + (len s)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%cL,(1. L)%> `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%cL,(1. L)%>,0) is set
(<%cL,(1. L)%> `^ 0) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((<%cL,(1. L)%> `^ 0) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(1_. L) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len ((1_. L) *' s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + (len s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
<%cL,(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,cL),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%cL,(1. L)%> *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . ((len s) -' 1) is Element of the carrier of L
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len x) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . ((len x) -' 1) is Element of the carrier of L
(len x) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x . ((len s) -' 1) is Element of the carrier of L
(<%cL,(1. L)%> *' x) . ((len s) -' 1) is Element of the carrier of L
cL * (x . ((len s) -' 1)) is Element of the carrier of L
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . cs1 is Element of the carrier of L
(1. L) * (x . cs1) is Element of the carrier of L
(cL * (x . ((len s) -' 1))) + ((1. L) * (x . cs1)) is Element of the carrier of L
(0. L) + ((1. L) * (x . cs1)) is Element of the carrier of L
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . f is Element of the carrier of L
s + f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cL . (s + f) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs is set
x . cs is set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s + cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . (s + cs1) is Element of the carrier of L
f . cs is set
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,cL,0) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is set
(L,cL,0) . x is set
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . (0 + f) is Element of the carrier of L
cL . x is set
L is non empty ZeroStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f is set
dom (L,s,cL) is non empty complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V288() Element of bool NAT
(L,s,cL) . f is set
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . (cL + cs) is Element of the carrier of L
L is non empty non degenerated non trivial multLoopStr_0
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (L,s,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (L,s,cL)) + cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) - cL is V33() real V49() ext-real Element of REAL
cL - cL is V33() real V49() ext-real Element of REAL
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cs1 + 1) - 1 is V33() real V49() ext-real Element of REAL
cs - 1 is V33() real V49() ext-real Element of REAL
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL + g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL + g) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s . (cL + g) is Element of the carrier of L
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
(L,s,cL) . g is Element of the carrier of L
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1 + cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s,cL) . cs1 is Element of the carrier of L
cL + cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . (cL + cs1) is Element of the carrier of L
cs + cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Element of the carrier of L
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,x,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,x,s),cL) is Element of the carrier of L
(L,x,(s + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,x,(s + 1)),cL) is Element of the carrier of L
cL * (eval ((L,x,(s + 1)),cL)) is Element of the carrier of L
x . s is Element of the carrier of L
(cL * (eval ((L,x,(s + 1)),cL))) + (x . s) is Element of the carrier of L
len (L,x,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
power L is Relation-like [: the carrier of L,NAT:] -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of L,NAT:], the carrier of L:]
[: the carrier of L,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of L,NAT:], the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial non finite set
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
Sum cs1 is Element of the carrier of L
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len (L,x,(s + 1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
Sum g is Element of the carrier of L
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom g is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
rng g is finite Element of bool the carrier of L
bool the carrier of L is non empty set
cL multfield is Relation-like the carrier of L -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [: the carrier of L, the carrier of L:]
[: the carrier of L, the carrier of L:] is Relation-like non empty set
bool [: the carrier of L, the carrier of L:] is non empty set
dom (cL multfield) is non empty Element of bool the carrier of L
cL * g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
(cL multfield) * g is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
dom ((cL multfield) * g) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
1_ L is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
(len (L,x,(s + 1))) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len (L,x,(s + 1))) + 1) + s is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len (L,x,(s + 1))) + (s + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len (L,x,s)) + s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<*(x . s)*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
len <*(x . s)*> is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len (cL * g) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<*(x . s)*> ^ (cL * g) is Relation-like NAT -defined the carrier of L -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len (<*(x . s)*> ^ (cL * g)) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
dom <*(x . s)*> is non empty trivial finite 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 . cs1a is set
1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,x,s) . (1 -' 1) is Element of the carrier of L
(power L) . (cL,(1 -' 1)) is Element of the carrier of L
((L,x,s) . (1 -' 1)) * ((power L) . (cL,(1 -' 1))) is Element of the carrier of L
(L,x,s) . 0 is Element of the carrier of L
((L,x,s) . 0) * ((power L) . (cL,(1 -' 1))) is Element of the carrier of L
(power L) . (cL,0) is Element of the carrier of L
((L,x,s) . 0) * ((power L) . (cL,0)) is Element of the carrier of L
((L,x,s) . 0) * (1. L) is Element of the carrier of L
s + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . (s + 0) is Element of the carrier of L
<*(x . s)*> . 1 is set
(<*(x . s)*> ^ (cL * g)) . cs1a is set
1 - 1 is V33() real V49() ext-real Element of REAL
cs1a - 1 is V33() real V49() ext-real Element of REAL
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(1 + 1) - 1 is V33() real V49() ext-real Element of REAL
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs1 -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len g) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len g) + 1) - 1 is V33() real V49() ext-real Element of REAL
g . cs1 is set
cs1 . cs1a is set
(L,x,s) . (cs1a -' 1) is Element of the carrier of L
(power L) . (cL,(cs1a -' 1)) is Element of the carrier of L
((L,x,s) . (cs1a -' 1)) * ((power L) . (cL,(cs1a -' 1))) is Element of the carrier of L
s + cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . (s + cs1) is Element of the carrier of L
(power L) . (cL,cs1) is Element of the carrier of L
(x . (s + cs1)) * ((power L) . (cL,cs1)) is Element of the carrier of L
(s + 1) + (cs1 -' 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x . ((s + 1) + (cs1 -' 1)) is Element of the carrier of L
(power L) . (cL,(cs1 -' 1)) is Element of the carrier of L
((power L) . (cL,(cs1 -' 1))) * cL is Element of the carrier of L
(x . ((s + 1) + (cs1 -' 1))) * (((power L) . (cL,(cs1 -' 1))) * cL) is Element of the carrier of L
(x . ((s + 1) + (cs1 -' 1))) * ((power L) . (cL,(cs1 -' 1))) is Element of the carrier of L
cL * ((x . ((s + 1) + (cs1 -' 1))) * ((power L) . (cL,(cs1 -' 1)))) is Element of the carrier of L
(L,x,(s + 1)) . (cs1 -' 1) is Element of the carrier of L
((L,x,(s + 1)) . (cs1 -' 1)) * ((power L) . (cL,(cs1 -' 1))) is Element of the carrier of L
cL * (((L,x,(s + 1)) . (cs1 -' 1)) * ((power L) . (cL,(cs1 -' 1)))) is Element of the carrier of L
t is Element of the carrier of L
cL * t is Element of the carrier of L
(cL * g) . cs1 is set
(<*(x . s)*> ^ (cL * g)) . cs1a is set
cL * (Sum g) is Element of the carrier of L
Sum (cL * g) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Roots cL is Element of bool the carrier of L
bool the carrier of L is non empty set
cL . 0 is Element of the carrier of L
<%(cL . 0)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
s is set
x is Element of the carrier of L
eval (cL,x) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cL is Element of the carrier of L
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
Roots s is Element of bool the carrier of L
bool the carrier of L is non empty set
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1a + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
cs . cs1a is Element of the carrier of L
(L,s,(cs1a + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(cs1a + 1)),cL) is Element of the carrier of L
eval ((0_. L),cL) is Element of the carrier of L
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs . cs1a is Element of the carrier of L
cs1a + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(L,s,(cs1a + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(cs1a + 1)),cL) is Element of the carrier of L
(L,s,(len s)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(len s)),cL) is Element of the carrier of L
cL * (eval ((L,s,(len s)),cL)) is Element of the carrier of L
s . ((len s) -' 1) is Element of the carrier of L
(cL * (eval ((L,s,(len s)),cL))) + (s . ((len s) -' 1)) is Element of the carrier of L
cL * (eval ((0_. L),cL)) is Element of the carrier of L
(cL * (eval ((0_. L),cL))) + (s . ((len s) -' 1)) is Element of the carrier of L
cL * (0. L) is Element of the carrier of L
(cL * (0. L)) + (s . ((len s) -' 1)) is Element of the carrier of L
(0. L) + (s . ((len s) -' 1)) is Element of the carrier of L
((len s) -' 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
t is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len t) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cs1 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((cs1 + 1) -' 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t . s1 is Element of the carrier of L
s1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(L,s,(s1 + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(s1 + 1)),cL) is Element of the carrier of L
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len x) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
f is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len f) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . cs is Element of the carrier of L
cs + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(L,s,(cs + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(cs + 1)),cL) is Element of the carrier of L
f . cs is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Element of the carrier of L
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,cL,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len (L,cL,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (L,cL,s)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
Roots s is Element of bool the carrier of L
bool the carrier of L is non empty set
L is non empty left_add-cancelable right_add-cancelable right_complementable unital left-distributive right_unital well-unital left_unital add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
Roots <%(- cL),(1. L)%> is Element of bool the carrier of L
bool the carrier of L is non empty set
{cL} is non empty trivial finite 1 -element Element of bool the carrier of L
s is set
x is Element of the carrier of L
eval (<%(- cL),(1. L)%>,x) is Element of the carrier of L
(- cL) + x is Element of the carrier of L
- x is Element of the carrier of L
eval (<%(- cL),(1. L)%>,cL) is Element of the carrier of L
(- cL) + cL is Element of the carrier of L
L is non empty non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval (<%(- cL),(1. L)%>,cL) is Element of the carrier of L
(- cL) + cL is Element of the carrier of L
eval (s,cL) is Element of the carrier of L
eval (x,cL) is Element of the carrier of L
(eval (<%(- cL),(1. L)%>,cL)) * (eval (x,cL)) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,cL,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' (L,cL,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len <%(- cL),(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len <%(- cL),(1. L)%>) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> . ((len <%(- cL),(1. L)%>) -' 1) is Element of the carrier of L
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(1 + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> . ((1 + 1) -' 1) is Element of the carrier of L
<%(- cL),(1. L)%> . 1 is Element of the carrier of L
len (L,cL,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (L,cL,s)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) . ((len (L,cL,s)) -' 1) is Element of the carrier of L
(<%(- cL),(1. L)%> . ((len <%(- cL),(1. L)%>) -' 1)) * ((L,cL,s) . ((len (L,cL,s)) -' 1)) is Element of the carrier of L
len (<%(- cL),(1. L)%> *' (L,cL,s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len <%(- cL),(1. L)%>) + (len (L,cL,s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((len <%(- cL),(1. L)%>) + (len (L,cL,s))) - 1 is V33() real V49() ext-real Element of REAL
(len (L,cL,s)) + (1 + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len (L,cL,s)) + (1 + 1)) - 1 is V33() real V49() ext-real Element of REAL
(len (L,cL,s)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
eval (s,cL) is Element of the carrier of L
(L,s,0) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,0),cL) is Element of the carrier of L
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(L,s,(0 + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(0 + 1)),cL) is Element of the carrier of L
cL * (eval ((L,s,(0 + 1)),cL)) is Element of the carrier of L
s . 0 is Element of the carrier of L
(cL * (eval ((L,s,(0 + 1)),cL))) + (s . 0) is Element of the carrier of L
(- cL) * (eval ((L,s,(0 + 1)),cL)) is Element of the carrier of L
((- cL) * (eval ((L,s,(0 + 1)),cL))) + (0. L) is Element of the carrier of L
(L,s,1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,1),cL) is Element of the carrier of L
cL * (eval ((L,s,1),cL)) is Element of the carrier of L
((- cL) * (eval ((L,s,(0 + 1)),cL))) + (cL * (eval ((L,s,1),cL))) is Element of the carrier of L
(((- cL) * (eval ((L,s,(0 + 1)),cL))) + (cL * (eval ((L,s,1),cL)))) + (s . 0) is Element of the carrier of L
- (cL * (eval ((L,s,(0 + 1)),cL))) is Element of the carrier of L
(- (cL * (eval ((L,s,(0 + 1)),cL)))) + (cL * (eval ((L,s,1),cL))) is Element of the carrier of L
((- (cL * (eval ((L,s,(0 + 1)),cL)))) + (cL * (eval ((L,s,1),cL)))) + (s . 0) is Element of the carrier of L
(0. L) + (s . 0) is Element of the carrier of L
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s . cs1 is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . cs1 is Element of the carrier of L
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s . (cs1 + 1) is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . (cs1 + 1) is Element of the carrier of L
(L,cL,s) . (cs1 + 1) is Element of the carrier of L
(cs1 + 1) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(L,s,((cs1 + 1) + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,((cs1 + 1) + 1)),cL) is Element of the carrier of L
(L,cL,s) . cs1 is Element of the carrier of L
(L,s,(cs1 + 1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,s,(cs1 + 1)),cL) is Element of the carrier of L
eval ((0_. L),cL) is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . (cs1 + 1) is Element of the carrier of L
(- cL) * ((L,cL,s) . (cs1 + 1)) is Element of the carrier of L
(1. L) * ((L,cL,s) . cs1) is Element of the carrier of L
((- cL) * ((L,cL,s) . (cs1 + 1))) + ((1. L) * ((L,cL,s) . cs1)) is Element of the carrier of L
(0. L) + ((1. L) * ((L,cL,s) . cs1)) is Element of the carrier of L
(0. L) + (0. L) is Element of the carrier of L
cL * (eval ((L,s,((cs1 + 1) + 1)),cL)) is Element of the carrier of L
s . (cs1 + 1) is Element of the carrier of L
(cL * (eval ((L,s,((cs1 + 1) + 1)),cL))) + (s . (cs1 + 1)) is Element of the carrier of L
cL * ((L,cL,s) . (cs1 + 1)) is Element of the carrier of L
- (cL * ((L,cL,s) . (cs1 + 1))) is Element of the carrier of L
(- (cL * ((L,cL,s) . (cs1 + 1)))) + ((L,cL,s) . cs1) is Element of the carrier of L
(- (cL * ((L,cL,s) . (cs1 + 1)))) + (cL * (eval ((L,s,((cs1 + 1) + 1)),cL))) is Element of the carrier of L
((- (cL * ((L,cL,s) . (cs1 + 1)))) + (cL * (eval ((L,s,((cs1 + 1) + 1)),cL)))) + (s . (cs1 + 1)) is Element of the carrier of L
(0. L) + (s . (cs1 + 1)) is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . (cs1 + 1) is Element of the carrier of L
(- cL) * ((L,cL,s) . (cs1 + 1)) is Element of the carrier of L
(1. L) * ((L,cL,s) . cs1) is Element of the carrier of L
((- cL) * ((L,cL,s) . (cs1 + 1))) + ((1. L) * ((L,cL,s) . cs1)) is Element of the carrier of L
((- cL) * ((L,cL,s) . (cs1 + 1))) + ((L,cL,s) . cs1) is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . 0 is Element of the carrier of L
(L,cL,s) . 0 is Element of the carrier of L
(- cL) * ((L,cL,s) . 0) is Element of the carrier of L
s . 0 is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . 0 is Element of the carrier of L
s . cs is Element of the carrier of L
(<%(- cL),(1. L)%> *' (L,cL,s)) . cs is Element of the carrier of L
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval (s,cL) is Element of the carrier of L
eval (<%(- cL),(1. L)%>,cL) is Element of the carrier of L
eval (x,cL) is Element of the carrier of L
(eval (<%(- cL),(1. L)%>,cL)) * (eval (x,cL)) is Element of the carrier of L
(- cL) + cL is Element of the carrier of L
((- cL) + cL) * (eval (x,cL)) is Element of the carrier of L
(0. L) * (eval (x,cL)) is Element of the carrier of L
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Roots x is Element of bool the carrier of L
bool the carrier of L is non empty set
card (Roots x) is epsilon-transitive epsilon-connected ordinal cardinal set
f is Element of the carrier of L
- f is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- f),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(- f)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- f)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(L,f,x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- f),(1. L)%> *' (L,f,x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots <%(- f),(1. L)%> is Element of bool the carrier of L
Roots (L,f,x) is Element of bool the carrier of L
(Roots <%(- f),(1. L)%>) \/ (Roots (L,f,x)) is Element of bool the carrier of L
{f} is non empty trivial finite 1 -element Element of bool the carrier of L
len (L,f,x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (L,f,x)) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
card (Roots (L,f,x)) is epsilon-transitive epsilon-connected ordinal cardinal set
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is finite set
g is finite set
cs1 \/ g is finite set
card g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
t is finite set
card t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
cs1a + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
f is set
cs is Element of the carrier of L
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Roots s is Element of bool the carrier of L
bool the carrier of L is non empty set
card (Roots s) is epsilon-transitive epsilon-connected ordinal cardinal set
Roots cL is Element of bool the carrier of L
bool the carrier of L is non empty set
card (Roots cL) is epsilon-transitive epsilon-connected ordinal cardinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Roots cL is Element of bool the carrier of L
bool the carrier of L is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
cL is Element of the carrier of L
- cL is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b₂ being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st s = (<%(- cL),(1. L)%> `^ b<sub>1</sub>) *' b<sub>2</sub> } is set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : S<sub>1</sub>[b<sub>1</sub>] } is set
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(1_. L) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,0) is set
(<%(- cL),(1. L)%> `^ 0) *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,cs1) is set
g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ cs1) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ cs1) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 + (len cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : not len s <= b₁ } is set
cs is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
max cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
max cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
cs1 is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
max cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
s is Element of the carrier of L
(L,s,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
- s is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- s),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- s)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- s)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b₂ being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st cL = (<%(- s),(1. L)%> `^ b<sub>1</sub>) *' b<sub>2</sub> } is set
cs is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- s),(1. L)%> `^ cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- s),(1. L)%>,cs1) is set
(L,s,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- s),(1. L)%> *' (L,s,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- s),(1. L)%> `^ 1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- s),(1. L)%>,1) is set
g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- s),(1. L)%> `^ cs1) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- s),(1. L)%> `^ cs1) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1a + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- s),(1. L)%> `^ cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- s),(1. L)%>,cs1) is set
<%(- s),(1. L)%> *' (<%(- s),(1. L)%> `^ cs1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- s),(1. L)%> *' (<%(- s),(1. L)%> `^ cs1)) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- s),(1. L)%> `^ cs1) *' g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- s),(1. L)%> *' ((<%(- s),(1. L)%> `^ cs1) *' g) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(L,cL,<%(- cL),(1. L)%>) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b₂ being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st <%(- cL),(1. L)%> = (<%(- cL),(1. L)%> `^ b<sub>1</sub>) *' b<sub>2</sub> } is set
f is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,cs) is set
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ cs) *' cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ cs) *' cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len <%(- cL),(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs + (len cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
2 + 0 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ 1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,1) is set
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ 1) *' (1_. L) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Roots cL is finite Element of bool the carrier of L
bool the carrier of L is non empty set
[: the carrier of L,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [: the carrier of L,NAT:] is non empty non trivial non finite set
s is Relation-like the carrier of L -defined NAT -valued Function-like non empty total quasi_total complex-valued ext-real-valued real-valued natural-valued Element of bool [: the carrier of L,NAT:]
dom s is non empty Element of bool the carrier of L
support s is set
x is set
s . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Element of the carrier of L
s . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,f,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Element of the carrier of L
(L,f,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support x is finite set
f is Element of the carrier of L
x . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,f,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support s is finite set
x is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support x is finite set
f is set
s . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is Element of the carrier of L
(L,cs,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(L,<%(- cL),(1. L)%>) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
{cL} is non empty trivial finite 1 -element Element of bool the carrier of L
bool the carrier of L is non empty set
( the carrier of L,{cL},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{cL} --> 1 is Relation-like {cL} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{cL},NAT:]
[:{cL},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{cL},NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:{cL},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({cL} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
Roots <%(- cL),(1. L)%> is finite Element of bool the carrier of L
support (L,<%(- cL),(1. L)%>) is finite set
x is set
(L,<%(- cL),(1. L)%>) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Element of the carrier of L
(L,f,<%(- cL),(1. L)%>) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,{cL},1) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,<%(- cL),(1. L)%>) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,{cL},1) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Element of the carrier of L
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
s *' x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,cL,(s *' x)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL,s) + (L,cL,x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
- cL is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b₂ being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st s *' x = (<%(- cL),(1. L)%> `^ b<sub>1</sub>) *' b<sub>2</sub> } is set
cs is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b<sub>2</sub> being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st s = (<%(- cL),(1. L)%> `^ b₁) *' b₂ } is set
cs1 is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,g) is set
cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ g) *' cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ g) *' cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
{ b<sub>1</sub> where b<sub>1</sub> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b<sub>2</sub> being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st x = (<%(- cL),(1. L)%> `^ b₁) *' b₂ } is set
cs1 is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ t is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,t) is set
s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ t) *' s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ t) *' s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ p1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,p1) is set
f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ p1) *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ p1) *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ g) *' cs1a) *' (<%(- cL),(1. L)%> `^ p1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(((<%(- cL),(1. L)%> `^ g) *' cs1a) *' (<%(- cL),(1. L)%> `^ p1)) *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ g) *' (<%(- cL),(1. L)%> `^ p1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ g) *' (<%(- cL),(1. L)%> `^ p1)) *' cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(((<%(- cL),(1. L)%> `^ g) *' (<%(- cL),(1. L)%> `^ p1)) *' cs1a) *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
g + p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ (g + p1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,(g + p1)) is set
(<%(- cL),(1. L)%> `^ (g + p1)) *' cs1a is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ (g + p1)) *' cs1a) *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs1a *' f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ (g + p1)) *' (cs1a *' f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0 + (g + p1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t - (g + p1) is V33() real V49() ext-real Element of REAL
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ t1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,t1) is set
<%(- cL),(1. L)%> `^ 1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,1) is set
pj1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ pj1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,pj1) is set
(<%(- cL),(1. L)%> `^ 1) *' (<%(- cL),(1. L)%> `^ pj1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' (<%(- cL),(1. L)%> `^ pj1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ t1) *' s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ pj1) *' s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' ((<%(- cL),(1. L)%> `^ pj1) *' s1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(0_. L) . 1 is Element of the carrier of L
<%(- cL),(1. L)%> . 1 is Element of the carrier of L
t1 + (g + p1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(<%(- cL),(1. L)%> `^ (g + p1)) *' (<%(- cL),(1. L)%> `^ t1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ (g + p1)) *' (<%(- cL),(1. L)%> `^ t1)) *' s1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ (g + p1)) *' ((<%(- cL),(1. L)%> `^ t1) *' s1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Roots (cs1a *' f1) is Element of bool the carrier of L
bool the carrier of L is non empty set
Roots cs1a is Element of bool the carrier of L
Roots f1 is Element of bool the carrier of L
(Roots cs1a) \/ (Roots f1) is Element of bool the carrier of L
(L,cL,cs1a) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' (L,cL,cs1a) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ g) *' <%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ g) *' <%(- cL),(1. L)%>) *' (L,cL,cs1a) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ g) *' (<%(- cL),(1. L)%> `^ 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ g) *' (<%(- cL),(1. L)%> `^ 1)) *' (L,cL,cs1a) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ (g + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,(g + 1)) is set
(<%(- cL),(1. L)%> `^ (g + 1)) *' (L,cL,cs1a) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,cL,f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- cL),(1. L)%> *' (L,cL,f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ p1) *' <%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ p1) *' <%(- cL),(1. L)%>) *' (L,cL,f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- cL),(1. L)%> `^ p1) *' (<%(- cL),(1. L)%> `^ 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
((<%(- cL),(1. L)%> `^ p1) *' (<%(- cL),(1. L)%> `^ 1)) *' (L,cL,f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
p1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ (p1 + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,(p1 + 1)) is set
(<%(- cL),(1. L)%> `^ (p1 + 1)) *' (L,cL,f1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,(cL *' s)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,s) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) + (L,s) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
x is set
(L,(cL *' s)) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Element of the carrier of L
(L,f,(cL *' s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,f,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,f,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,f,cL) + (L,f,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL) . x) + (L,f,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,s) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL) . x) + ((L,s) . x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((L,cL) + (L,s)) . x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
cL *' s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,(cL *' s)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,s) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,cL) + (L,s) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(cL *' s))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,cL)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,cL)) + ( the carrier of L,(L,s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,cL) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,cL)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Roots cL is finite Element of bool the carrier of L
bool the carrier of L is non empty set
support (L,cL) is finite set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,x) is set
(L,(<%(- cL),(1. L)%> `^ x)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(<%(- cL),(1. L)%> `^ x))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ (x + 1) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,(x + 1)) is set
(L,(<%(- cL),(1. L)%> `^ (x + 1))) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(<%(- cL),(1. L)%> `^ (x + 1)))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(<%(- cL),(1. L)%> `^ x) *' <%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,<%(- cL),(1. L)%>) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,<%(- cL),(1. L)%>)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,(<%(- cL),(1. L)%> `^ x))) + ( the carrier of L,(L,<%(- cL),(1. L)%>)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
{cL} is non empty trivial finite 1 -element Element of bool the carrier of L
bool the carrier of L is non empty set
( the carrier of L,{cL},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{cL} --> 1 is Relation-like {cL} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{cL},NAT:]
[:{cL},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{cL},NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:{cL},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({cL} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
( the carrier of L,( the carrier of L,{cL},1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x + ( the carrier of L,( the carrier of L,{cL},1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
card {cL} is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
len (1_. L) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- cL),(1. L)%> `^ 0 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- cL),(1. L)%>,0) is set
(L,(<%(- cL),(1. L)%> `^ 0)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(<%(- cL),(1. L)%> `^ 0))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() algebraic-closed doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
(L,cL) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,cL)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,x) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,x)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len x) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 - 1 is V33() real V49() ext-real Element of REAL
f is Element of the carrier of L
(L,f,x) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
- f is Element of the carrier of L
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- f),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- f)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- f)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : ex b₂ being Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:] st x = (<%(- f),(1. L)%> `^ b<sub>1</sub>) *' b<sub>2</sub> } is set
cs1 is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
max cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() set
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- f),(1. L)%> `^ g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() doubleLoopStr
power (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L),NAT:] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):]
the carrier of (Polynom-Ring L) is non empty set
[: the carrier of (Polynom-Ring L),NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
[:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:[: the carrier of (Polynom-Ring L),NAT:], the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(power (Polynom-Ring L)) . (<%(- f),(1. L)%>,g) is set
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- f),(1. L)%> `^ g) *' cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(<%(- f),(1. L)%> `^ g) *' cs1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
t is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len t) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t . ((len t) -' 1) is Element of the carrier of L
len (<%(- f),(1. L)%> `^ g) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (<%(- f),(1. L)%> `^ g)) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(<%(- f),(1. L)%> `^ g) . ((len (<%(- f),(1. L)%> `^ g)) -' 1) is Element of the carrier of L
((<%(- f),(1. L)%> `^ g) . ((len (<%(- f),(1. L)%> `^ g)) -' 1)) * (t . ((len t) -' 1)) is Element of the carrier of L
g * 2 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(g * 2) - g is V33() real V49() ext-real Element of REAL
((g * 2) - g) + 1 is V33() real V49() ext-real Element of REAL
g + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len <%(- f),(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (<%(- f),(1. L)%> `^ g)) + (len t) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((len (<%(- f),(1. L)%> `^ g)) + (len t)) - 1 is V33() real V49() ext-real Element of REAL
(L,(<%(- f),(1. L)%> `^ g)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(<%(- f),(1. L)%> `^ g))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,t) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,t)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,(<%(- f),(1. L)%> `^ g))) + ( the carrier of L,(L,t)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,(<%(- f),(1. L)%> `^ g))) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
2 * g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(2 * g) - g is V33() real V49() ext-real Element of REAL
((2 * g) - g) + 1 is V33() real V49() ext-real Element of REAL
(((2 * g) - g) + 1) - 1 is V33() real V49() ext-real Element of REAL
(L,(<%(- f),(1. L)%> `^ g)) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,(<%(- f),(1. L)%> `^ g))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(g + 1) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,t) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,(L,t)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(L,(<%(- f),(1. L)%> `^ g))) + ( the carrier of L,(L,t)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((len (<%(- f),(1. L)%> `^ g)) -' 1) + ((len t) -' 1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (<%(- f),(1. L)%> `^ g)) - 1 is V33() real V49() ext-real Element of REAL
((len (<%(- f),(1. L)%> `^ g)) - 1) + ((len t) -' 1) is V33() real V49() ext-real Element of REAL
(len t) - 1 is V33() real V49() ext-real Element of REAL
((len (<%(- f),(1. L)%> `^ g)) - 1) + ((len t) - 1) is V33() real V49() ext-real Element of REAL
(len (<%(- f),(1. L)%> `^ g)) + (len t) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
((len (<%(- f),(1. L)%> `^ g)) + (len t)) - 1 is V33() real V49() ext-real Element of REAL
(((len (<%(- f),(1. L)%> `^ g)) + (len t)) - 1) - 1 is V33() real V49() ext-real Element of REAL
(len x) - 1 is V33() real V49() ext-real Element of REAL
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL is Element of the carrier of L
- cL is Element of the carrier of L
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s |-> <%(- cL),(1. L)%> is Relation-like NAT -defined bool [:NAT, the carrier of L:] -valued Function-like finite s -element FinSequence-like FinSubsequence-like finite-support Element of s -tuples_on (bool [:NAT, the carrier of L:])
s -tuples_on (bool [:NAT, the carrier of L:]) is functional non empty FinSequence-membered FinSequenceSet of bool [:NAT, the carrier of L:]
(bool [:NAT, the carrier of L:]) * is functional non empty FinSequence-membered FinSequenceSet of bool [:NAT, the carrier of L:]
{ b₁ where b₁ is Relation-like NAT -defined bool [:NAT, the carrier of L:] -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of (bool [:NAT, the carrier of L:]) * : len b₁ = s } is set
Seg s is finite s -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(Seg s) --> <%(- cL),(1. L)%> is Relation-like Seg s -defined {<%(- cL),(1. L)%>} -valued Function-like total quasi_total finite FinSequence-like FinSubsequence-like finite-support Element of bool [:(Seg s),{<%(- cL),(1. L)%>}:]
{<%(- cL),(1. L)%>} is functional non empty trivial finite 1 -element set
[:(Seg s),{<%(- cL),(1. L)%>}:] is Relation-like finite set
bool [:(Seg s),{<%(- cL),(1. L)%>}:] is non empty finite V41() set
x is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . f is set
x is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . cs is set
f . cs is set
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
cL is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support cL is finite set
card (support cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
dom cL is Element of bool the carrier of L
bool the carrier of L is non empty set
rng ((support cL)) is finite Element of bool (support cL)
bool (support cL) is non empty finite V41() set
s is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
x is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
proj2 x is finite set
f is set
cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . cs is set
s /. cs is Element of the carrier of L
cL . (s /. cs) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(s /. cs),(cL . (s /. cs))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
f is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
FlattenSeq f is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
cs is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f . cs1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
s /. cs1 is Element of the carrier of L
cL . (s /. cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(s /. cs1),(cL . (s /. cs1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq f is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
cs1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq cs1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq cs1) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" (FlattenSeq cs1) is Element of the carrier of (Polynom-Ring L)
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
f . cs1a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
cs /. cs1a is Element of the carrier of L
cL . (cs /. cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cs /. cs1a),(cL . (cs /. cs1a))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
cs1 . cs1a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
L is non empty left_add-cancelable right_add-cancelable right_complementable unital commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
(L,(EmptyBag the carrier of L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
support (EmptyBag the carrier of L) is finite set
card (support (EmptyBag the carrier of L)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support (EmptyBag the carrier of L))) is Relation-like NAT -defined support (EmptyBag the carrier of L) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (EmptyBag the carrier of L)
the Relation-like NAT -defined support (EmptyBag the carrier of L) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (EmptyBag the carrier of L) is Relation-like NAT -defined support (EmptyBag the carrier of L) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (EmptyBag the carrier of L)
s is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq s is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq s) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total V32( the carrier of (Polynom-Ring L)) Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" (FlattenSeq s) is Element of the carrier of (Polynom-Ring L)
<*> ( the carrier of (Polynom-Ring L) *) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring L) * -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() Element of ( the carrier of (Polynom-Ring L) *) *
( the carrier of (Polynom-Ring L) *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L) *
1_ (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
1. (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
the OneF of (Polynom-Ring L) is Element of the carrier of (Polynom-Ring L)
<*> the carrier of (Polynom-Ring L) is Relation-like non-empty empty-yielding NAT -defined the carrier of (Polynom-Ring L) -valued epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural Function-like one-to-one constant functional empty V33() real Function-yielding finite finite-yielding V41() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V49() ext-real non positive non negative complex-valued ext-real-valued real-valued natural-valued complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V83() FinSequence-yielding finite-support V288() V289() V290() V291() Element of the carrier of (Polynom-Ring L) *
1_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Element of the carrier of L
{cL} is non empty trivial finite 1 -element Element of bool the carrier of L
bool the carrier of L is non empty set
( the carrier of L,{cL},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{cL} --> 1 is Relation-like {cL} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{cL},NAT:]
[:{cL},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{cL},NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:{cL},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({cL} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,{cL},1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
- cL is Element of the carrier of L
<%(- cL),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cL)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
support ( the carrier of L,{cL},1) is finite set
card (support ( the carrier of L,{cL},1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support ( the carrier of L,{cL},1))) is Relation-like NAT -defined support ( the carrier of L,{cL},1) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support ( the carrier of L,{cL},1)
the Relation-like NAT -defined support ( the carrier of L,{cL},1) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support ( the carrier of L,{cL},1) is Relation-like NAT -defined support ( the carrier of L,{cL},1) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support ( the carrier of L,{cL},1)
x is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq x is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq x) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" (FlattenSeq x) is Element of the carrier of (Polynom-Ring L)
<*cL*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
x . 1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
x /. 1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f /. 1 is Element of the carrier of L
f . 1 is set
( the carrier of L,{cL},1) . (f /. 1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len (L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
<*(x /. 1)*> is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like constant non empty trivial Function-yielding finite 1 -element FinSequence-like FinSubsequence-like FinSequence-yielding finite-support Element of ( the carrier of (Polynom-Ring L) *) *
( the carrier of (Polynom-Ring L) *) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L) *
(L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) /. 1 is Element of the carrier of (Polynom-Ring L)
<*((L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) /. 1)*> is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
(L,(f /. 1),(( the carrier of L,{cL},1) . (f /. 1))) . 1 is set
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
cL is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support cL is finite set
card (support cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
( the carrier of L,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq s is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
len (FlattenSeq s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
Card s is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
((support cL)) * cL is Relation-like NAT -defined RAT -valued Function-like finite complex-valued ext-real-valued real-valued natural-valued finite-support set
cs is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
rng x is finite Element of bool the carrier of L
bool the carrier of L is non empty set
dom cL is Element of bool the carrier of L
cs1 is set
dom (Card s) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . cs1 is set
cL . (x . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . cs1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
x /. cs1 is Element of the carrier of L
cL . (x /. cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(x /. cs1),(cL . (x /. cs1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len (s . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
f . cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
card (s . cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
Sum f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
L is non empty left_add-cancelable right_add-cancelable right_complementable right-distributive left-distributive distributive add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
support cL is finite set
card (support cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
s is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom s is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq s is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
x is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
f is Element of the carrier of L
- f is Element of the carrier of L
<%(- f),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- f)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- f)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{<%(- f),(1. L)%>} is functional non empty trivial finite 1 -element Element of bool (bool [:NAT, the carrier of L:])
bool (bool [:NAT, the carrier of L:]) is non empty non trivial non finite set
(FlattenSeq s) " {<%(- f),(1. L)%>} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
card ((FlattenSeq s) " {<%(- f),(1. L)%>}) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
cL . f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom x is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT : not cL . f <= b₁ } is set
rng x is finite Element of bool the carrier of L
bool the carrier of L is non empty set
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
x . cs1a is set
cs1a -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s | (cs1a -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (s | (cs1a -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (s | (cs1a -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is set
t is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + t is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(Sum (Card (s | (cs1a -' 1)))) + (1 + t) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs1 is Relation-like Function-like set
proj1 cs1 is set
x /. cs1a is Element of the carrier of L
<%(- f),(1. L)%> . 0 is Element of the carrier of L
t is set
proj2 cs1 is set
s1 is set
cs1 . s1 is set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + p1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(Sum (Card (s | (cs1a -' 1)))) + (1 + p1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . cs1a is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
cL . (x /. cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(x /. cs1a),(cL . (x /. cs1a))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len (s . cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (s . cs1a) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(s . cs1a) . (1 + p1) is set
dom (FlattenSeq s) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(FlattenSeq s) . ((Sum (Card (s | (cs1a -' 1)))) + (1 + p1)) is set
(FlattenSeq s) . t is set
s1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(FlattenSeq s) . s1 is set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . p1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (s . p1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
p1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s | (p1 -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (s | (p1 -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (s | (p1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (s | (p1 -' 1)))) + f1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(s . p1) . f1 is set
x /. p1 is Element of the carrier of L
cL . (x /. p1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(x /. p1),(cL . (x /. p1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
- (x /. p1) is Element of the carrier of L
<%(- (x /. p1)),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(- (x /. p1))) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- (x /. p1))),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
x . p1 is set
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
t1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
len (s . p1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 . t1 is set
pj1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + pj1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(Sum (Card (s | (cs1a -' 1)))) + (1 + pj1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
t is set
s1 is set
cs1 . t is set
cs1 . s1 is set
p1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + p1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(Sum (Card (s | (cs1a -' 1)))) + (1 + p1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
f1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
1 + f1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(Sum (Card (s | (cs1a -' 1)))) + (1 + f1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
cs is set
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(FlattenSeq s) . cs1 is set
g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . g is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (s . g) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
g -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s | (g -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (s | (g -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (s | (g -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (s | (g -' 1)))) + cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(s . g) . cs1a is set
x /. g is Element of the carrier of L
x . g is set
cL . (x /. g) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(x /. g),(cL . (x /. g))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
- (x /. g) is Element of the carrier of L
<%(- (x /. g)),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,(0_. L),0,(- (x /. g))) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- (x /. g))),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
L is non empty left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
cL is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
s is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
cL + s is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,(cL + s)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
(L,cL) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,cL) *' (L,s) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
support (cL + s) is finite set
support cL is finite set
support s is finite set
(support cL) \/ (support s) is finite set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
card (support s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support s)) is Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s
the Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s is Relation-like NAT -defined support s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support s
f is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq f is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total V32( the carrier of (Polynom-Ring L)) associative Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" (FlattenSeq f) is Element of the carrier of (Polynom-Ring L)
card (support cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support cL)) is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
the Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL is Relation-like NAT -defined support cL -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support cL
cs1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq cs1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq cs1) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" (FlattenSeq cs1) is Element of the carrier of (Polynom-Ring L)
card (support (cL + s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
((support (cL + s))) is Relation-like NAT -defined support (cL + s) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (cL + s)
the Relation-like NAT -defined support (cL + s) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (cL + s) is Relation-like NAT -defined support (cL + s) -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of support (cL + s)
cs1a is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
len cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
dom cs1a is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
FlattenSeq cs1a is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
Product (FlattenSeq cs1a) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" (FlattenSeq cs1a) is Element of the carrier of (Polynom-Ring L)
len (FlattenSeq cs1a) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,(cL + s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len (FlattenSeq f) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(FlattenSeq cs1) ^ (FlattenSeq f) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
len ((FlattenSeq cs1) ^ (FlattenSeq f)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
len (FlattenSeq cs1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len (FlattenSeq cs1)) + (len (FlattenSeq f)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,cL) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
( the carrier of L,cL) + ( the carrier of L,s) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (FlattenSeq cs1a) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
dom ((FlattenSeq cs1) ^ (FlattenSeq f)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
len cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
t1 is set
rng (FlattenSeq cs1a) is finite Element of bool the carrier of (Polynom-Ring L)
bool the carrier of (Polynom-Ring L) is non empty set
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(FlattenSeq cs1a) . t1 is set
pj1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a . pj1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (cs1a . pj1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
pj1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a | (pj1 -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (cs1a | (pj1 -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (cs1a | (pj1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (cs1a | (pj1 -' 1)))) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs1a . pj1) . x is set
cs1 /. pj1 is Element of the carrier of L
(cL + s) . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cs1 /. pj1),((cL + s) . (cs1 /. pj1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
- (cs1 /. pj1) is Element of the carrier of L
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- (cs1 /. pj1)),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- (cs1 /. pj1))) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- (cs1 /. pj1))),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs1 . pj1 is set
rng cs1 is finite Element of bool the carrier of L
bool the carrier of L is non empty set
{t1} is non empty trivial finite 1 -element set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
card (((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1}) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(FlattenSeq cs1) " {t1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
card ((FlattenSeq cs1) " {t1}) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(FlattenSeq f) " {t1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
card ((FlattenSeq f) " {t1}) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of omega
(card ((FlattenSeq cs1) " {t1})) + (card ((FlattenSeq f) " {t1})) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim ((FlattenSeq cs1a),t1) is set
(FlattenSeq cs1a) " {t1} is finite set
card (Coim ((FlattenSeq cs1a),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
cL . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL . (cs1 /. pj1)) + (s . (cs1 /. pj1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL . (cs1 /. pj1)) + 0 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1) is set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite set
card (Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
s . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim ((FlattenSeq cs1a),t1) is set
(FlattenSeq cs1a) " {t1} is finite set
card (Coim ((FlattenSeq cs1a),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
cL . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL . (cs1 /. pj1)) + (s . (cs1 /. pj1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
0 + (s . (cs1 /. pj1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1) is set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite set
card (Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
s . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim ((FlattenSeq cs1a),t1) is set
(FlattenSeq cs1a) " {t1} is finite set
card (Coim ((FlattenSeq cs1a),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
cL . (cs1 /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cL . (cs1 /. pj1)) + (s . (cs1 /. pj1)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1) is set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite set
card (Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
t1 is set
rng (FlattenSeq cs1a) is finite Element of bool the carrier of (Polynom-Ring L)
bool the carrier of (Polynom-Ring L) is non empty set
rng ((FlattenSeq cs1) ^ (FlattenSeq f)) is finite Element of bool the carrier of (Polynom-Ring L)
rng (FlattenSeq cs1) is finite Element of bool the carrier of (Polynom-Ring L)
rng (FlattenSeq f) is finite Element of bool the carrier of (Polynom-Ring L)
(rng (FlattenSeq cs1)) \/ (rng (FlattenSeq f)) is finite Element of bool the carrier of (Polynom-Ring L)
dom (FlattenSeq cs1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(FlattenSeq cs1) . t1 is set
pj1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 . pj1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (cs1 . pj1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
pj1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1 | (pj1 -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (cs1 | (pj1 -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (cs1 | (pj1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (cs1 | (pj1 -' 1)))) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(cs1 . pj1) . x is set
g /. pj1 is Element of the carrier of L
cL . (g /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(g /. pj1),(cL . (g /. pj1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
- (g /. pj1) is Element of the carrier of L
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- (g /. pj1)),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- (g /. pj1))) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- (g /. pj1))),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom g is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
g . pj1 is set
rng g is finite Element of bool the carrier of L
bool the carrier of L is non empty set
(cL + s) . (g /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
rng cs1 is finite Element of bool the carrier of L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1 . i is set
cs1 /. i is Element of the carrier of L
cs1a . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
(cL + s) . (cs1 /. i) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cs1 /. i),((cL + s) . (cs1 /. i))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len (cs1a . i) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (cs1a . i) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a | (i -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (cs1a | (i -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (cs1a | (i -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (cs1a | (i -' 1)))) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cs1a . i) . 1 is set
(FlattenSeq cs1a) . ((Sum (Card (cs1a | (i -' 1)))) + 1) is set
dom (FlattenSeq f) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
t1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(FlattenSeq f) . t1 is set
pj1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f . pj1 is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom (f . pj1) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
pj1 -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
f | (pj1 -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (f | (pj1 -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (f | (pj1 -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (f | (pj1 -' 1)))) + x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(f . pj1) . x is set
cs /. pj1 is Element of the carrier of L
s . (cs /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cs /. pj1),(s . (cs /. pj1))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
- (cs /. pj1) is Element of the carrier of L
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
<%(- (cs /. pj1)),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- (cs /. pj1))) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- (cs /. pj1))),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
len cs is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom cs is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
cs . pj1 is set
rng cs is finite Element of bool the carrier of L
bool the carrier of L is non empty set
(cL + s) . (cs /. pj1) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
rng cs1 is finite Element of bool the carrier of L
dom cs1 is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
cs1 . i is set
cs1 /. i is Element of the carrier of L
cs1a . i is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
(cL + s) . (cs1 /. i) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(L,(cs1 /. i),((cL + s) . (cs1 /. i))) is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len (cs1a . i) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom (cs1a . i) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a | (i -' 1) is Relation-like NAT -defined the carrier of (Polynom-Ring L) * -valued Function-like Function-yielding finite FinSequence-like FinSubsequence-like FinSequence-yielding finite-support FinSequence of the carrier of (Polynom-Ring L) *
Card (cs1a | (i -' 1)) is Relation-like NAT -defined NAT -valued Function-like finite FinSequence-like FinSubsequence-like complex-valued ext-real-valued real-valued natural-valued V263() finite-support FinSequence of NAT
Sum (Card (cs1a | (i -' 1))) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(Sum (Card (cs1a | (i -' 1)))) + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(cs1a . i) . 1 is set
(FlattenSeq cs1a) . ((Sum (Card (cs1a | (i -' 1)))) + 1) is set
{t1} is non empty trivial finite 1 -element set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Coim ((FlattenSeq cs1a),t1) is set
(FlattenSeq cs1a) " {t1} is finite set
card (Coim ((FlattenSeq cs1a),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1) is set
((FlattenSeq cs1) ^ (FlattenSeq f)) " {t1} is finite set
card (Coim (((FlattenSeq cs1) ^ (FlattenSeq f)),t1)) is epsilon-transitive epsilon-connected ordinal cardinal set
t1 is set
rng (FlattenSeq cs1a) is finite Element of bool the carrier of (Polynom-Ring L)
bool the carrier of (Polynom-Ring L) is non empty set
[:(dom (FlattenSeq cs1a)),(dom (FlattenSeq cs1a)):] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
bool [:(dom (FlattenSeq cs1a)),(dom (FlattenSeq cs1a)):] is non empty finite V41() set
Product ((FlattenSeq cs1) ^ (FlattenSeq f)) is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" ((FlattenSeq cs1) ^ (FlattenSeq f)) is Element of the carrier of (Polynom-Ring L)
(Product (FlattenSeq cs1)) * (Product (FlattenSeq f)) is Element of the carrier of (Polynom-Ring L)
t1 is Relation-like dom (FlattenSeq cs1a) -defined dom (FlattenSeq cs1a) -valued Function-like one-to-one total quasi_total onto bijective finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:(dom (FlattenSeq cs1a)),(dom (FlattenSeq cs1a)):]
((FlattenSeq cs1) ^ (FlattenSeq f)) * t1 is Relation-like dom (FlattenSeq cs1a) -defined the carrier of (Polynom-Ring L) -valued Function-like finite finite-support Element of bool [:(dom (FlattenSeq cs1a)), the carrier of (Polynom-Ring L):]
[:(dom (FlattenSeq cs1a)), the carrier of (Polynom-Ring L):] is Relation-like set
bool [:(dom (FlattenSeq cs1a)), the carrier of (Polynom-Ring L):] is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed domRing-like V184() V185() V186() V187() algebraic-closed doubleLoopStr
the carrier of L is non empty non trivial set
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
1. L is V107(L) Element of the carrier of L
the OneF of L is Element of the carrier of L
cL is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len cL is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len cL) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cL . ((len cL) -' 1) is Element of the carrier of L
(L,cL) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,(L,cL)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
s + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
x is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len x is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len x) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
x . ((len x) -' 1) is Element of the carrier of L
(L,x) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,(L,x)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f is Element of the carrier of L
- f is Element of the carrier of L
<%(- f),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- f)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- f)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
(L,f,x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
<%(- f),(1. L)%> *' (L,f,x) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
len <%(- f),(1. L)%> is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len <%(- f),(1. L)%>) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
<%(- f),(1. L)%> . ((len <%(- f),(1. L)%>) -' 1) is Element of the carrier of L
g is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len g) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
g . ((len g) -' 1) is Element of the carrier of L
(<%(- f),(1. L)%> . ((len <%(- f),(1. L)%>) -' 1)) * (g . ((len g) -' 1)) is Element of the carrier of L
1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(len g) + (1 + 1) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
((len g) + (1 + 1)) - 1 is V33() real V49() ext-real Element of REAL
(L,g) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,(L,g)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
{f} is non empty trivial finite 1 -element Element of bool the carrier of L
bool the carrier of L is non empty set
( the carrier of L,{f},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{f} --> 1 is Relation-like {f} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{f},NAT:]
[:{f},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{f},NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:{f},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({f} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,{f},1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(L,<%(- f),(1. L)%>) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
( the carrier of L,{f},1) + (L,g) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
s is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support (L) Element of bool [:NAT, the carrier of L:]
len s is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(len s) -' 1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
s . ((len s) -' 1) is Element of the carrier of L
(L,s) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
(L,(L,s)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
( the carrier of L,(L,s)) is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
<%(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
L is non empty left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty set
bool the carrier of L is non empty set
Polynom-Ring L is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of (Polynom-Ring L) is non empty set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
x is non empty finite Element of bool the carrier of L
card x is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
(x) is Relation-like NAT -defined x -valued Function-like one-to-one non empty onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of x
the Relation-like NAT -defined x -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of x is Relation-like NAT -defined x -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of x
( the carrier of L,x,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
x --> 1 is Relation-like x -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:x,NAT:]
[:x,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:x,NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:x,{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (x --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,x,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
f is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Product f is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) is Relation-like [: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] -defined the carrier of (Polynom-Ring L) -valued Function-like non empty total quasi_total V32( the carrier of (Polynom-Ring L)) associative Element of bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):]
[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):] is Relation-like non empty set
[:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is Relation-like non empty set
bool [:[: the carrier of (Polynom-Ring L), the carrier of (Polynom-Ring L):], the carrier of (Polynom-Ring L):] is non empty set
the multF of (Polynom-Ring L) "**" f is Element of the carrier of (Polynom-Ring L)
rng (x) is non empty finite Element of bool x
bool x is non empty finite V41() set
len (x) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
dom (x) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Seg cs1 is finite cs1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(x) | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined x -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,x:]
[:NAT,x:] is Relation-like non empty non trivial non finite set
bool [:NAT,x:] is non empty non trivial non finite set
rng ((x) | (Seg cs1)) is finite Element of bool x
f | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring L):]
[:NAT, the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
Seg (cs1 + 1) is non empty finite cs1 + 1 -element cs1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(x) | (Seg (cs1 + 1)) is Relation-like NAT -defined Seg (cs1 + 1) -defined NAT -defined x -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,x:]
rng ((x) | (Seg (cs1 + 1))) is finite Element of bool x
f | (Seg (cs1 + 1)) is Relation-like NAT -defined Seg (cs1 + 1) -defined NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring L):]
g is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(cs1 + 1) + cs1a is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(x) . (cs1 + 1) is set
g . (cs1 + 1) is set
f . (cs1 + 1) is set
cs1a is Element of the carrier of L
- cs1a is Element of the carrier of L
<%(- cs1a),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cs1a)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cs1a)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
s1 is finite Element of bool the carrier of L
p1 is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
( the carrier of L,s1,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
s1 --> 1 is Relation-like s1 -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s1,NAT:]
[:s1,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:s1,NAT:] is non empty set
[:s1,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (s1 --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,s1,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Product p1 is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" p1 is Element of the carrier of (Polynom-Ring L)
g | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring L):]
<*<%(- cs1a),(1. L)%>*> is Relation-like NAT -defined bool [:NAT, the carrier of L:] -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of (bool [:NAT, the carrier of L:]) *
(bool [:NAT, the carrier of L:]) * is functional non empty FinSequence-membered FinSequenceSet of bool [:NAT, the carrier of L:]
p1 ^ <*<%(- cs1a),(1. L)%>*> is Relation-like NAT -defined Function-like non empty finite FinSequence-like FinSubsequence-like finite-support set
cs1 is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of x
rng cs1 is finite Element of bool x
{(cs1 + 1)} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
(Seg cs1) \/ {(cs1 + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
t is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of x
(x) | {(cs1 + 1)} is Relation-like NAT -defined {(cs1 + 1)} -defined NAT -defined x -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,x:]
t \/ ((x) | {(cs1 + 1)}) is Relation-like NAT -defined x -valued finite Element of bool [:NAT,x:]
(cs1 + 1) .--> cs1a is finite set
{(cs1 + 1)} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
{(cs1 + 1)} --> cs1a is Relation-like {(cs1 + 1)} -defined {cs1a} -valued Function-like non empty total quasi_total finite finite-support Element of bool [:{(cs1 + 1)},{cs1a}:]
{cs1a} is non empty trivial finite 1 -element set
[:{(cs1 + 1)},{cs1a}:] is Relation-like non empty finite set
bool [:{(cs1 + 1)},{cs1a}:] is non empty finite V41() set
rng ((x) | {(cs1 + 1)}) is finite Element of bool x
{cs1a} is non empty trivial finite 1 -element Element of bool the carrier of L
s1 \/ {cs1a} is non empty finite Element of bool the carrier of L
( the carrier of L,{cs1a},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{cs1a} --> 1 is Relation-like {cs1a} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{cs1a},NAT:]
[:{cs1a},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{cs1a},NAT:] is non empty non trivial non finite set
[:{cs1a},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({cs1a} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,{cs1a},1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
pj1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
t1 is Element of the carrier of (Polynom-Ring L)
x is finite Element of bool the carrier of L
( the carrier of L,x,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
x --> 1 is Relation-like x -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:x,NAT:]
[:x,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:x,NAT:] is non empty set
[:x,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (x --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,x,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Product g is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" g is Element of the carrier of (Polynom-Ring L)
s1 /\ {cs1a} is finite Element of bool the carrier of L
i is set
dom ((x) | (Seg cs1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i is set
((x) | (Seg cs1)) . i is set
i is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(x) . i is set
( the carrier of L,s1,1) + ( the carrier of L,{cs1a},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f1 *' pj1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
(Product p1) * t1 is Element of the carrier of (Polynom-Ring L)
Seg (len f) is finite len f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring L):]
[:NAT, the carrier of (Polynom-Ring L):] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of (Polynom-Ring L):] is non empty non trivial non finite set
(x) | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined x -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,x:]
[:NAT,x:] is Relation-like non empty non trivial non finite set
bool [:NAT,x:] is non empty non trivial non finite set
(x) | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined x -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,x:]
rng ((x) | (Seg 1)) is finite Element of bool x
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of (Polynom-Ring L):]
(x) . 1 is set
cs1 is Element of x
cs1a is Relation-like NAT -defined x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of x
cs1a . 1 is set
cs1 is Element of the carrier of L
{cs1} is non empty trivial finite 1 -element Element of bool the carrier of L
t is finite Element of bool the carrier of L
( the carrier of L,t,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
t --> 1 is Relation-like t -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:t,NAT:]
[:t,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:t,NAT:] is non empty set
[:t,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (t --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,t,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
g is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
Product g is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" g is Element of the carrier of (Polynom-Ring L)
s1 is Element of x
<*s1*> is Relation-like NAT -defined x -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of x *
x * is functional non empty FinSequence-membered FinSequenceSet of x
p1 is Element of the carrier of (Polynom-Ring L)
<*p1*> is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of (Polynom-Ring L) *
the carrier of (Polynom-Ring L) * is functional non empty FinSequence-membered FinSequenceSet of the carrier of (Polynom-Ring L)
g . 1 is set
f . 1 is set
f1 is Element of the carrier of (Polynom-Ring L)
- cs1 is Element of the carrier of L
<%(- cs1),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cs1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cs1)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
rng ((x) | (Seg (len f))) is finite Element of bool x
cs1 is finite Element of bool the carrier of L
g is Relation-like NAT -defined the carrier of (Polynom-Ring L) -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of (Polynom-Ring L)
( the carrier of L,cs1,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
cs1 --> 1 is Relation-like cs1 -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cs1,NAT:]
[:cs1,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cs1,NAT:] is non empty set
[:cs1,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (cs1 --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,cs1,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
Product g is Element of the carrier of (Polynom-Ring L)
the multF of (Polynom-Ring L) "**" g is Element of the carrier of (Polynom-Ring L)
L is non empty non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital Abelian add-associative right_zeroed V184() V185() V186() V187() doubleLoopStr
the carrier of L is non empty non trivial set
bool the carrier of L is non empty set
1. L is Element of the carrier of L
the OneF of L is Element of the carrier of L
s is non empty finite Element of bool the carrier of L
card s is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of omega
(s) is Relation-like NAT -defined s -valued Function-like one-to-one non empty onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s
the Relation-like NAT -defined s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s is Relation-like NAT -defined s -valued Function-like one-to-one onto bijective finite FinSequence-like FinSubsequence-like finite-support FinSequence of s
( the carrier of L,s,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
Bags the carrier of L is non empty set
Bags the carrier of L is functional non empty Element of bool (Bags the carrier of L)
bool (Bags the carrier of L) is non empty set
EmptyBag the carrier of L is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
s --> 1 is Relation-like s -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s,NAT:]
[:s,NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:s,NAT:] is non empty non trivial non finite set
{1} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
[:s,{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (s --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,s,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is Relation-like non empty non trivial non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
x is Element of the carrier of L
eval ((L,( the carrier of L,s,1)),x) is Element of the carrier of L
f is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len f is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
dom f is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
Product f is Element of the carrier of L
the multF of L is Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like non empty total quasi_total V32( the carrier of L) associative Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is Relation-like non empty set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is Relation-like non empty set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L "**" f is Element of the carrier of L
rng (s) is non empty finite Element of bool s
bool s is non empty finite V41() set
len (s) is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
dom (s) is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
Seg cs1 is finite cs1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
(s) | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined s -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,s:]
[:NAT,s:] is Relation-like non empty non trivial non finite set
bool [:NAT,s:] is non empty non trivial non finite set
rng ((s) | (Seg cs1)) is finite Element of bool s
f | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
Seg (cs1 + 1) is non empty finite cs1 + 1 -element cs1 + 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
cs1 + 1 is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(s) | (Seg (cs1 + 1)) is Relation-like NAT -defined Seg (cs1 + 1) -defined NAT -defined s -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,s:]
rng ((s) | (Seg (cs1 + 1))) is finite Element of bool s
f | (Seg (cs1 + 1)) is Relation-like NAT -defined Seg (cs1 + 1) -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
len g is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
cs1a is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative set
(cs1 + 1) + cs1a is epsilon-transitive epsilon-connected ordinal natural non empty V33() real finite cardinal V49() ext-real positive non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of NAT
(s) . (cs1 + 1) is set
g . (cs1 + 1) is set
f . (cs1 + 1) is set
cs1a is Element of the carrier of L
- cs1a is Element of the carrier of L
<%(- cs1a),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cs1a)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cs1a)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
eval (<%(- cs1a),(1. L)%>,x) is Element of the carrier of L
s1 is finite Element of bool the carrier of L
p1 is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
( the carrier of L,s1,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
s1 --> 1 is Relation-like s1 -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:s1,NAT:]
[:s1,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:s1,NAT:] is non empty set
[:s1,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (s1 --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,s1,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,( the carrier of L,s1,1)),x) is Element of the carrier of L
Product p1 is Element of the carrier of L
the multF of L "**" p1 is Element of the carrier of L
g | (Seg cs1) is Relation-like NAT -defined Seg cs1 -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
<*(eval (<%(- cs1a),(1. L)%>,x))*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
p1 ^ <*(eval (<%(- cs1a),(1. L)%>,x))*> is Relation-like NAT -defined the carrier of L -valued Function-like non empty finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
cs1 is Relation-like NAT -defined s -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of s
rng cs1 is finite Element of bool s
{(cs1 + 1)} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
(Seg cs1) \/ {(cs1 + 1)} is non empty finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() Element of bool NAT
t is Relation-like NAT -defined s -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of s
(s) | {(cs1 + 1)} is Relation-like NAT -defined {(cs1 + 1)} -defined NAT -defined s -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,s:]
t \/ ((s) | {(cs1 + 1)}) is Relation-like NAT -defined s -valued finite Element of bool [:NAT,s:]
(cs1 + 1) .--> cs1a is finite set
{(cs1 + 1)} is non empty trivial finite V41() 1 -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V286() V287() V288() V289() V290() set
{(cs1 + 1)} --> cs1a is Relation-like {(cs1 + 1)} -defined {cs1a} -valued Function-like non empty total quasi_total finite finite-support Element of bool [:{(cs1 + 1)},{cs1a}:]
{cs1a} is non empty trivial finite 1 -element set
[:{(cs1 + 1)},{cs1a}:] is Relation-like non empty finite set
bool [:{(cs1 + 1)},{cs1a}:] is non empty finite V41() set
rng ((s) | {(cs1 + 1)}) is finite Element of bool s
{cs1a} is non empty trivial finite 1 -element Element of bool the carrier of L
s1 \/ {cs1a} is non empty finite Element of bool the carrier of L
t1 is finite Element of bool the carrier of L
( the carrier of L,t1,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
t1 --> 1 is Relation-like t1 -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:t1,NAT:]
[:t1,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:t1,NAT:] is non empty set
[:t1,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (t1 --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,t1,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,( the carrier of L,t1,1)),x) is Element of the carrier of L
Product g is Element of the carrier of L
the multF of L "**" g is Element of the carrier of L
( the carrier of L,{cs1a},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
{cs1a} --> 1 is Relation-like {cs1a} -defined NAT -valued RAT -valued INT -valued Function-like non empty total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:{cs1a},NAT:]
[:{cs1a},NAT:] is Relation-like RAT -valued INT -valued non empty non trivial non finite complex-valued ext-real-valued real-valued natural-valued set
bool [:{cs1a},NAT:] is non empty non trivial non finite set
[:{cs1a},{1}:] is Relation-like RAT -valued INT -valued non empty finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* ({cs1a} --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,{cs1a},1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
pj1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
s1 /\ {cs1a} is finite Element of bool the carrier of L
x is set
dom ((s) | (Seg cs1)) is finite complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
i is set
((s) | (Seg cs1)) . i is set
i is epsilon-transitive epsilon-connected ordinal natural V33() real finite cardinal V49() ext-real non negative V54() complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of NAT
(s) . i is set
( the carrier of L,s1,1) + ( the carrier of L,{cs1a},1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support set
f1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
f1 *' pj1 is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval (pj1,x) is Element of the carrier of L
(Product p1) * (eval (pj1,x)) is Element of the carrier of L
Seg (len f) is finite len f -element complex-membered ext-real-membered real-membered rational-membered integer-membered natural-membered V288() V289() V290() Element of bool NAT
f | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
(s) | (Seg (len f)) is Relation-like NAT -defined Seg (len f) -defined NAT -defined s -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,s:]
[:NAT,s:] is Relation-like non empty non trivial non finite set
bool [:NAT,s:] is non empty non trivial non finite set
(s) | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined s -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,s:]
rng ((s) | (Seg 1)) is finite Element of bool s
f | (Seg 1) is Relation-like NAT -defined Seg 1 -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
(s) . 1 is set
cs1 is Element of s
cs1a is Relation-like NAT -defined s -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of s
cs1a . 1 is set
cs1 is Element of the carrier of L
{cs1} is non empty trivial finite 1 -element Element of bool the carrier of L
t is finite Element of bool the carrier of L
( the carrier of L,t,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
t --> 1 is Relation-like t -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:t,NAT:]
[:t,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:t,NAT:] is non empty set
[:t,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (t --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,t,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,( the carrier of L,t,1)),x) is Element of the carrier of L
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
Product g is Element of the carrier of L
the multF of L "**" g is Element of the carrier of L
s1 is Element of s
<*s1*> is Relation-like NAT -defined s -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of s *
s * is functional non empty FinSequence-membered FinSequenceSet of s
p1 is Element of the carrier of L
<*p1*> is Relation-like NAT -defined the carrier of L -valued Function-like constant non empty trivial finite 1 -element FinSequence-like FinSubsequence-like finite-support Element of the carrier of L *
the carrier of L * is functional non empty FinSequence-membered FinSequenceSet of the carrier of L
g . 1 is set
f . 1 is set
f1 is Element of the carrier of L
- cs1 is Element of the carrier of L
<%(- cs1),(1. L)%> is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
0_. L is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support with_roots Element of bool [:NAT, the carrier of L:]
0. L is V107(L) Element of the carrier of L
the ZeroF of L is Element of the carrier of L
NAT --> (0. L) is Relation-like NAT -defined the carrier of L -valued T-Sequence-like Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
{(0. L)} is non empty trivial finite 1 -element set
[:NAT,{(0. L)}:] is Relation-like non empty non trivial non finite set
K832(NAT, the carrier of L,(0_. L),0,(- cs1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
K832(NAT, the carrier of L,K832(NAT, the carrier of L,(0_. L),0,(- cs1)),1,(1. L)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total Element of bool [:NAT, the carrier of L:]
eval (<%(- cs1),(1. L)%>,x) is Element of the carrier of L
rng ((s) | (Seg (len f))) is finite Element of bool s
cs1 is finite Element of bool the carrier of L
g is Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
( the carrier of L,cs1,1) is Relation-like the carrier of L -defined RAT -valued Function-like total complex-valued ext-real-valued real-valued natural-valued finite-support Element of Bags the carrier of L
cs1 --> 1 is Relation-like cs1 -defined NAT -valued RAT -valued INT -valued Function-like total quasi_total finite complex-valued ext-real-valued real-valued natural-valued finite-support Element of bool [:cs1,NAT:]
[:cs1,NAT:] is Relation-like RAT -valued INT -valued complex-valued ext-real-valued real-valued natural-valued set
bool [:cs1,NAT:] is non empty set
[:cs1,{1}:] is Relation-like RAT -valued INT -valued finite complex-valued ext-real-valued real-valued natural-valued set
(EmptyBag the carrier of L) +* (cs1 --> 1) is Relation-like RAT -valued Function-like complex-valued ext-real-valued real-valued natural-valued set
(L,( the carrier of L,cs1,1)) is Relation-like NAT -defined the carrier of L -valued Function-like non empty total quasi_total finite-Support Element of bool [:NAT, the carrier of L:]
eval ((L,( the carrier of L,cs1,1)),x) is Element of the carrier of L
Product g is Element of the carrier of L
the multF of L "**" g is Element of the carrier of L