:: POLYNOM2 semantic presentation

REAL is set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal Element of bool REAL
bool REAL is non empty set
NAT is non empty non trivial epsilon-transitive epsilon-connected ordinal non finite cardinal limit_cardinal set
bool NAT is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
K124(NAT) is V33() set
COMPLEX is set
RAT is set
INT is set
[:COMPLEX,COMPLEX:] is Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:REAL,REAL:] is Relation-like set
bool [:REAL,REAL:] is non empty set
[:[:REAL,REAL:],REAL:] is Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is Relation-like set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is Relation-like set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is Relation-like set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty non trivial Relation-like non finite set
[:[:NAT,NAT:],NAT:] is non empty non trivial Relation-like non finite set
bool [:[:NAT,NAT:],NAT:] is non empty non trivial non finite set
K266() is set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support set
{{},1} is non empty finite V38() set
K624() is non empty strict multMagma
the carrier of K624() is non empty set
K629() is non empty strict unital Group-like associative commutative V243() V244() V245() V246() V247() V248() multMagma
K630() is non empty strict associative commutative V246() V247() V248() M37(K629())
K631() is non empty strict unital associative commutative V246() V247() V248() V249() M40(K629(),K630())
K633() is non empty strict unital associative commutative multMagma
K634() is non empty strict unital associative commutative V249() M37(K633())
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
[:NAT,REAL:] is Relation-like set
bool [:NAT,REAL:] is non empty set
K662() is Relation-like NAT -defined Function-like total set
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg 1 is non empty trivial finite 1 -element Element of bool NAT
{1} is non empty trivial finite V38() 1 -element set
Seg 2 is non empty finite 2 -element Element of bool NAT
{1,2} is non empty finite V38() set
dom {} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support set
rng {} is empty trivial epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() V60() ext-real non positive non negative V211() V212() V213() V214() V215() V216() V217() V218() V221() V222() V223() V224() V226() FinSequence-yielding finite-support set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support Element of NAT
card {} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support set
Bags {} is non empty functional Element of bool (Bags {})
Bags {} is non empty set
bool (Bags {}) is non empty set
{{}} is non empty trivial functional finite V38() 1 -element set
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is Element of bool n
x is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
field x is set
f is set
p is set
[f,p] is set
[p,f] is set
f is set
p is set
q is set
[f,p] is set
[p,q] is set
[f,q] is set
f is set
[f,f] is set
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is Element of bool n
x is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
f is set
p is set
[f,p] is set
[p,f] is set
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is Element of bool n
x is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
field x is set
f is set
p is set
q is set
[f,p] is set
[p,q] is set
[f,q] is set
f is set
p is set
[f,p] is set
[p,f] is set
f is set
p is set
[f,p] is set
[p,f] is set
dom x is Element of bool n
rng x is Element of bool n
(dom x) \/ (rng x) is Element of bool n
[p,p] is set
[f,f] is set
f is set
[f,f] is set
n is non empty unital associative multMagma
the carrier of n is non empty set
power n is non empty Relation-like [: the carrier of n,NAT:] -defined the carrier of n -valued Function-like total V27([: the carrier of n,NAT:], the carrier of n) Element of bool [:[: the carrier of n,NAT:], the carrier of n:]
[: the carrier of n,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of n,NAT:], the carrier of n:] is non empty non trivial non finite set
L is Element of the carrier of n
x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x + f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(power n) . (L,(x + f)) is Element of the carrier of n
[L,(x + f)] is set
(power n) . [L,(x + f)] is set
(power n) . (L,x) is Element of the carrier of n
[L,x] is set
(power n) . [L,x] is set
(power n) . (L,f) is Element of the carrier of n
[L,f] is set
(power n) . [L,f] is set
((power n) . (L,x)) * ((power n) . (L,f)) is Element of the carrier of n
the multF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n . (((power n) . (L,x)),((power n) . (L,f))) is Element of the carrier of n
[((power n) . (L,x)),((power n) . (L,f))] is set
the multF of n . [((power n) . (L,x)),((power n) . (L,f))] is set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x + p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(power n) . (L,(x + p)) is Element of the carrier of n
[L,(x + p)] is set
(power n) . [L,(x + p)] is set
(power n) . (L,p) is Element of the carrier of n
[L,p] is set
(power n) . [L,p] is set
((power n) . (L,x)) * ((power n) . (L,p)) is Element of the carrier of n
the multF of n . (((power n) . (L,x)),((power n) . (L,p))) is Element of the carrier of n
[((power n) . (L,x)),((power n) . (L,p))] is set
the multF of n . [((power n) . (L,x)),((power n) . (L,p))] is set
p + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
x + (p + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(power n) . (L,(x + (p + 1))) is Element of the carrier of n
[L,(x + (p + 1))] is set
(power n) . [L,(x + (p + 1))] is set
(x + p) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(power n) . (L,((x + p) + 1)) is Element of the carrier of n
[L,((x + p) + 1)] is set
(power n) . [L,((x + p) + 1)] is set
(((power n) . (L,x)) * ((power n) . (L,p))) * L is Element of the carrier of n
the multF of n . ((((power n) . (L,x)) * ((power n) . (L,p))),L) is Element of the carrier of n
[(((power n) . (L,x)) * ((power n) . (L,p))),L] is set
the multF of n . [(((power n) . (L,x)) * ((power n) . (L,p))),L] is set
((power n) . (L,p)) * L is Element of the carrier of n
the multF of n . (((power n) . (L,p)),L) is Element of the carrier of n
[((power n) . (L,p)),L] is set
the multF of n . [((power n) . (L,p)),L] is set
((power n) . (L,x)) * (((power n) . (L,p)) * L) is Element of the carrier of n
the multF of n . (((power n) . (L,x)),(((power n) . (L,p)) * L)) is Element of the carrier of n
[((power n) . (L,x)),(((power n) . (L,p)) * L)] is set
the multF of n . [((power n) . (L,x)),(((power n) . (L,p)) * L)] is set
(power n) . (L,(p + 1)) is Element of the carrier of n
[L,(p + 1)] is set
(power n) . [L,(p + 1)] is set
((power n) . (L,x)) * ((power n) . (L,(p + 1))) is Element of the carrier of n
the multF of n . (((power n) . (L,x)),((power n) . (L,(p + 1)))) is Element of the carrier of n
[((power n) . (L,x)),((power n) . (L,(p + 1)))] is set
the multF of n . [((power n) . (L,x)),((power n) . (L,(p + 1)))] is set
x + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(power n) . (L,(x + 0)) is Element of the carrier of n
[L,(x + 0)] is set
(power n) . [L,(x + 0)] is set
1_ n is Element of the carrier of n
((power n) . (L,x)) * (1_ n) is Element of the carrier of n
the multF of n . (((power n) . (L,x)),(1_ n)) is Element of the carrier of n
[((power n) . (L,x)),(1_ n)] is set
the multF of n . [((power n) . (L,x)),(1_ n)] is set
(power n) . (L,0) is Element of the carrier of n
[L,0] is set
(power n) . [L,0] is set
((power n) . (L,x)) * ((power n) . (L,0)) is Element of the carrier of n
the multF of n . (((power n) . (L,x)),((power n) . (L,0))) is Element of the carrier of n
[((power n) . (L,x)),((power n) . (L,0))] is set
the multF of n . [((power n) . (L,x)),((power n) . (L,0))] is set
F_Real is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable almost_left_invertible strict unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
n is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
dom n is finite Element of bool NAT
L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg f is finite f -element Element of bool NAT
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is finite Element of bool n
x is Element of n
{x} is non empty trivial finite 1 -element set
{x} \/ L is non empty finite set
f is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
p is set
n is non empty right_zeroed left_zeroed addLoopStr
the carrier of n is non empty set
0. n is V80(n) Element of the carrier of n
L is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
dom L is finite Element of bool NAT
Sum L is Element of the carrier of n
x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
L /. x is Element of the carrier of n
[:NAT, the carrier of n:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of n:] is non empty non trivial non finite set
len L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
f . (len L) is Element of the carrier of n
f . 0 is Element of the carrier of n
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= q ) } is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f . p9 is Element of the carrier of n
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
f . (p9 + 1) is Element of the carrier of n
L . (p9 + 1) is set
L /. (p9 + 1) is Element of the carrier of n
(0. n) + (L /. (p9 + 1)) is Element of the carrier of n
the addF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the addF of n . ((0. n),(L /. (p9 + 1))) is Element of the carrier of n
[(0. n),(L /. (p9 + 1))] is set
the addF of n . [(0. n),(L /. (p9 + 1))] is set
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg q is finite q -element Element of bool NAT
L . (p9 + 1) is set
L /. (p9 + 1) is Element of the carrier of n
(0. n) + (L /. (p9 + 1)) is Element of the carrier of n
the addF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the addF of n . ((0. n),(L /. (p9 + 1))) is Element of the carrier of n
[(0. n),(L /. (p9 + 1))] is set
the addF of n . [(0. n),(L /. (p9 + 1))] is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
L . (p9 + 1) is set
L /. (p9 + 1) is Element of the carrier of n
(L /. x) + (L /. (p9 + 1)) is Element of the carrier of n
the addF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the addF of n . ((L /. x),(L /. (p9 + 1))) is Element of the carrier of n
[(L /. x),(L /. (p9 + 1))] is set
the addF of n . [(L /. x),(L /. (p9 + 1))] is set
(L /. x) + (0. n) is Element of the carrier of n
the addF of n . ((L /. x),(0. n)) is Element of the carrier of n
[(L /. x),(0. n)] is set
the addF of n . [(L /. x),(0. n)] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is non empty left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of n is non empty set
0. n is V80(n) Element of the carrier of n
L is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
dom L is finite Element of bool NAT
Product L is Element of the carrier of n
x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
L /. x is Element of the carrier of n
f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p is finite Element of bool NAT
Product p is Element of the carrier of n
[:NAT, the carrier of n:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of n:] is non empty non trivial non finite set
p . 1 is set
the multF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n "**" p is Element of the carrier of n
Seg (len p) is finite len p -element Element of bool NAT
p /. 1 is Element of the carrier of n
q is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
q . 1 is Element of the carrier of n
q . (len p) is Element of the carrier of n
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p /. q is Element of the carrier of n
Seg f is finite f -element Element of bool NAT
p | (Seg f) is Relation-like NAT -defined Seg f -defined NAT -defined the carrier of n -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of n:]
q is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
dom q is finite Element of bool NAT
p . (len p) is set
<*(p . (len p))*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support set
q ^ <*(p . (len p))*> is non empty Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
p /. (len p) is Element of the carrier of n
<*(p /. (len p))*> is non empty trivial Relation-like NAT -defined the carrier of n -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support M9( the carrier of n,K256( the carrier of n))
K256( the carrier of n) is non empty functional FinSequence-membered M8( the carrier of n)
q ^ <*(p /. (len p))*> is non empty Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product q is Element of the carrier of n
(Product q) * (0. n) is Element of the carrier of n
the multF of n . ((Product q),(0. n)) is Element of the carrier of n
[(Product q),(0. n)] is set
the multF of n . [(Product q),(0. n)] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p /. p9 is Element of the carrier of n
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p /. p9 is Element of the carrier of n
p | f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
(p | f) . p9 is set
p . p9 is set
q . p9 is set
q /. p9 is Element of the carrier of n
Product q is Element of the carrier of n
(Product q) * (p /. (len p)) is Element of the carrier of n
the multF of n . ((Product q),(p /. (len p))) is Element of the carrier of n
[(Product q),(p /. (len p))] is set
the multF of n . [(Product q),(p /. (len p))] is set
(0. n) * (p /. (len p)) is Element of the carrier of n
the multF of n . ((0. n),(p /. (len p))) is Element of the carrier of n
[(0. n),(p /. (len p))] is set
the multF of n . [(0. n),(p /. (len p))] is set
p is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p is finite Element of bool NAT
p /. q is Element of the carrier of n
Product p is Element of the carrier of n
len L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom f is finite Element of bool NAT
Product f is Element of the carrier of n
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f /. p is Element of the carrier of n
f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is non empty Abelian add-associative addLoopStr
the carrier of n is non empty set
L is Element of the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom x is finite Element of bool NAT
Sum f is Element of the carrier of n
Sum x is Element of the carrier of n
L + (Sum x) is Element of the carrier of n
the addF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the addF of n . (L,(Sum x)) is Element of the carrier of n
[L,(Sum x)] is set
the addF of n . [L,(Sum x)] is set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f /. p is Element of the carrier of n
x /. p is Element of the carrier of n
L + (x /. p) is Element of the carrier of n
the addF of n . (L,(x /. p)) is Element of the carrier of n
[L,(x /. p)] is set
the addF of n . [L,(x /. p)] is set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f /. p is Element of the carrier of n
x /. p is Element of the carrier of n
L + (x /. p) is Element of the carrier of n
the addF of n . (L,(x /. p)) is Element of the carrier of n
[L,(x /. p)] is set
the addF of n . [L,(x /. p)] is set
[:NAT, the carrier of n:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of n:] is non empty non trivial non finite set
0. n is V80(n) Element of the carrier of n
q is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
q . (len f) is Element of the carrier of n
q . 0 is Element of the carrier of n
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= p9 ) } is set
dom f is finite Element of bool NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg q9 is finite q9 -element Element of bool NAT
p9 is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
p9 . (len x) is Element of the carrier of n
p9 . 0 is Element of the carrier of n
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 . bg is Element of the carrier of n
q . bg is Element of the carrier of n
L + (p9 . bg) is Element of the carrier of n
the addF of n . (L,(p9 . bg)) is Element of the carrier of n
[L,(p9 . bg)] is set
the addF of n . [L,(p9 . bg)] is set
bg + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 . (bg + 1) is Element of the carrier of n
q . (bg + 1) is Element of the carrier of n
L + (p9 . (bg + 1)) is Element of the carrier of n
the addF of n . (L,(p9 . (bg + 1))) is Element of the carrier of n
[L,(p9 . (bg + 1))] is set
the addF of n . [L,(p9 . (bg + 1))] is set
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
L + (x /. (bg + 1)) is Element of the carrier of n
the addF of n . (L,(x /. (bg + 1))) is Element of the carrier of n
[L,(x /. (bg + 1))] is set
the addF of n . [L,(x /. (bg + 1))] is set
(q . bg) + (L + (x /. (bg + 1))) is Element of the carrier of n
the addF of n . ((q . bg),(L + (x /. (bg + 1)))) is Element of the carrier of n
[(q . bg),(L + (x /. (bg + 1)))] is set
the addF of n . [(q . bg),(L + (x /. (bg + 1)))] is set
(q . bg) + (x /. (bg + 1)) is Element of the carrier of n
the addF of n . ((q . bg),(x /. (bg + 1))) is Element of the carrier of n
[(q . bg),(x /. (bg + 1))] is set
the addF of n . [(q . bg),(x /. (bg + 1))] is set
L + ((q . bg) + (x /. (bg + 1))) is Element of the carrier of n
the addF of n . (L,((q . bg) + (x /. (bg + 1)))) is Element of the carrier of n
[L,((q . bg) + (x /. (bg + 1)))] is set
the addF of n . [L,((q . bg) + (x /. (bg + 1)))] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg p is finite p -element Element of bool NAT
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
(q . bg) + (f /. (bg + 1)) is Element of the carrier of n
the addF of n . ((q . bg),(f /. (bg + 1))) is Element of the carrier of n
[(q . bg),(f /. (bg + 1))] is set
the addF of n . [(q . bg),(f /. (bg + 1))] is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
(q . bg) + (f /. (bg + 1)) is Element of the carrier of n
the addF of n . ((q . bg),(f /. (bg + 1))) is Element of the carrier of n
[(q . bg),(f /. (bg + 1))] is set
the addF of n . [(q . bg),(f /. (bg + 1))] is set
(L + (p9 . bg)) + (x /. (bg + 1)) is Element of the carrier of n
the addF of n . ((L + (p9 . bg)),(x /. (bg + 1))) is Element of the carrier of n
[(L + (p9 . bg)),(x /. (bg + 1))] is set
the addF of n . [(L + (p9 . bg)),(x /. (bg + 1))] is set
(p9 . bg) + (x /. (bg + 1)) is Element of the carrier of n
the addF of n . ((p9 . bg),(x /. (bg + 1))) is Element of the carrier of n
[(p9 . bg),(x /. (bg + 1))] is set
the addF of n . [(p9 . bg),(x /. (bg + 1))] is set
L + ((p9 . bg) + (x /. (bg + 1))) is Element of the carrier of n
the addF of n . (L,((p9 . bg) + (x /. (bg + 1)))) is Element of the carrier of n
[L,((p9 . bg) + (x /. (bg + 1)))] is set
the addF of n . [L,((p9 . bg) + (x /. (bg + 1)))] is set
L + (p9 . 0) is Element of the carrier of n
the addF of n . (L,(p9 . 0)) is Element of the carrier of n
[L,(p9 . 0)] is set
the addF of n . [L,(p9 . 0)] is set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is non empty associative commutative doubleLoopStr
the carrier of n is non empty set
L is Element of the carrier of n
x is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f is Relation-like NAT -defined the carrier of n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of n
len f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom x is finite Element of bool NAT
Product f is Element of the carrier of n
Product x is Element of the carrier of n
L * (Product x) is Element of the carrier of n
the multF of n is non empty Relation-like [: the carrier of n, the carrier of n:] -defined the carrier of n -valued Function-like total V27([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is non empty Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is non empty set
the multF of n . (L,(Product x)) is Element of the carrier of n
[L,(Product x)] is set
the multF of n . [L,(Product x)] is set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f /. p is Element of the carrier of n
x /. p is Element of the carrier of n
L * (x /. p) is Element of the carrier of n
the multF of n . (L,(x /. p)) is Element of the carrier of n
[L,(x /. p)] is set
the multF of n . [L,(x /. p)] is set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
f /. p is Element of the carrier of n
x /. p is Element of the carrier of n
L * (x /. p) is Element of the carrier of n
the multF of n . (L,(x /. p)) is Element of the carrier of n
[L,(x /. p)] is set
the multF of n . [L,(x /. p)] is set
the multF of n "**" x is Element of the carrier of n
the multF of n "**" f is Element of the carrier of n
dom f is finite Element of bool NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg q is finite q -element Element of bool NAT
[:NAT, the carrier of n:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of n:] is non empty non trivial non finite set
x . 1 is set
p9 is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
p9 . 1 is Element of the carrier of n
p9 . (len x) is Element of the carrier of n
f . 1 is set
q9 is non empty Relation-like NAT -defined the carrier of n -valued Function-like total V27( NAT , the carrier of n) Element of bool [:NAT, the carrier of n:]
q9 . 1 is Element of the carrier of n
q9 . (len x) is Element of the carrier of n
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 . bg is Element of the carrier of n
q9 . bg is Element of the carrier of n
L * (p9 . bg) is Element of the carrier of n
the multF of n . (L,(p9 . bg)) is Element of the carrier of n
[L,(p9 . bg)] is set
the multF of n . [L,(p9 . bg)] is set
bg + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 . (bg + 1) is Element of the carrier of n
q9 . (bg + 1) is Element of the carrier of n
L * (p9 . (bg + 1)) is Element of the carrier of n
the multF of n . (L,(p9 . (bg + 1))) is Element of the carrier of n
[L,(p9 . (bg + 1))] is set
the multF of n . [L,(p9 . (bg + 1))] is set
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
L * (x /. (bg + 1)) is Element of the carrier of n
the multF of n . (L,(x /. (bg + 1))) is Element of the carrier of n
[L,(x /. (bg + 1))] is set
the multF of n . [L,(x /. (bg + 1))] is set
(p9 . bg) * (L * (x /. (bg + 1))) is Element of the carrier of n
the multF of n . ((p9 . bg),(L * (x /. (bg + 1)))) is Element of the carrier of n
[(p9 . bg),(L * (x /. (bg + 1)))] is set
the multF of n . [(p9 . bg),(L * (x /. (bg + 1)))] is set
(p9 . bg) * (x /. (bg + 1)) is Element of the carrier of n
the multF of n . ((p9 . bg),(x /. (bg + 1))) is Element of the carrier of n
[(p9 . bg),(x /. (bg + 1))] is set
the multF of n . [(p9 . bg),(x /. (bg + 1))] is set
((p9 . bg) * (x /. (bg + 1))) * L is Element of the carrier of n
the multF of n . (((p9 . bg) * (x /. (bg + 1))),L) is Element of the carrier of n
[((p9 . bg) * (x /. (bg + 1))),L] is set
the multF of n . [((p9 . bg) * (x /. (bg + 1))),L] is set
(p9 . (bg + 1)) * L is Element of the carrier of n
the multF of n . ((p9 . (bg + 1)),L) is Element of the carrier of n
[(p9 . (bg + 1)),L] is set
the multF of n . [(p9 . (bg + 1)),L] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg q is finite q -element Element of bool NAT
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
(q9 . bg) * (f /. (bg + 1)) is Element of the carrier of n
the multF of n . ((q9 . bg),(f /. (bg + 1))) is Element of the carrier of n
[(q9 . bg),(f /. (bg + 1))] is set
the multF of n . [(q9 . bg),(f /. (bg + 1))] is set
the multF of n . ((q9 . bg),(x . (bg + 1))) is set
[(q9 . bg),(x . (bg + 1))] is set
the multF of n . [(q9 . bg),(x . (bg + 1))] is set
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
x . (bg + 1) is set
x /. (bg + 1) is Element of the carrier of n
f . (bg + 1) is set
f /. (bg + 1) is Element of the carrier of n
(q9 . bg) * (f /. (bg + 1)) is Element of the carrier of n
the multF of n . ((q9 . bg),(f /. (bg + 1))) is Element of the carrier of n
[(q9 . bg),(f /. (bg + 1))] is set
the multF of n . [(q9 . bg),(f /. (bg + 1))] is set
(L * (p9 . bg)) * (x /. (bg + 1)) is Element of the carrier of n
the multF of n . ((L * (p9 . bg)),(x /. (bg + 1))) is Element of the carrier of n
[(L * (p9 . bg)),(x /. (bg + 1))] is set
the multF of n . [(L * (p9 . bg)),(x /. (bg + 1))] is set
(p9 . bg) * (x /. (bg + 1)) is Element of the carrier of n
the multF of n . ((p9 . bg),(x /. (bg + 1))) is Element of the carrier of n
[(p9 . bg),(x /. (bg + 1))] is set
the multF of n . [(p9 . bg),(x /. (bg + 1))] is set
L * ((p9 . bg) * (x /. (bg + 1))) is Element of the carrier of n
the multF of n . (L,((p9 . bg) * (x /. (bg + 1)))) is Element of the carrier of n
[L,((p9 . bg) * (x /. (bg + 1)))] is set
the multF of n . [L,((p9 . bg) * (x /. (bg + 1)))] is set
x /. 1 is Element of the carrier of n
f /. 1 is Element of the carrier of n
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x /. 1 is Element of the carrier of n
L * (x /. 1) is Element of the carrier of n
the multF of n . (L,(x /. 1)) is Element of the carrier of n
[L,(x /. 1)] is set
the multF of n . [L,(x /. 1)] is set
L * (p9 . 1) is Element of the carrier of n
the multF of n . (L,(p9 . 1)) is Element of the carrier of n
[L,(p9 . 1)] is set
the multF of n . [L,(p9 . 1)] is set
L * (p9 . 1) is Element of the carrier of n
the multF of n . (L,(p9 . 1)) is Element of the carrier of n
[L,(p9 . 1)] is set
the multF of n . [L,(p9 . 1)] is set
L * (p9 . 1) is Element of the carrier of n
the multF of n . (L,(p9 . 1)) is Element of the carrier of n
[L,(p9 . 1)] is set
the multF of n . [L,(p9 . 1)] is set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support Element of bool n
x is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
SgmX (x,L) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
rng (SgmX (x,L)) is finite set
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is finite Element of bool n
x is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
SgmX (x,L) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom (SgmX (x,L)) is finite Element of bool NAT
f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (x,L)) /. f is Element of n
(SgmX (x,L)) /. p is Element of n
n is set
L is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom L is finite Element of bool NAT
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is finite Element of bool n
x is Element of n
{x} is non empty trivial finite 1 -element set
{x} \/ L is non empty finite set
f is finite Element of bool n
p is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
SgmX (p,f) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom (SgmX (p,f)) is finite Element of bool NAT
SgmX (p,L) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
field p is set
dom (SgmX (p,L)) is finite Element of bool NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg q9 is finite q9 -element Element of bool NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
len (SgmX (p,f)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card L) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
len (SgmX (p,L)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(len (SgmX (p,L))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. bg is Element of n
bg - 1 is V11() V12() V45() ext-real Element of REAL
Seg (len (SgmX (p,f))) is finite len (SgmX (p,f)) -element Element of bool NAT
1 - 1 is V11() V12() V45() ext-real Element of REAL
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
bg + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (p9 + 1) is Element of n
(SgmX (p,L)) /. (p9 + 1) is Element of n
(SgmX (p,f)) . (p9 + 1) is set
rng (SgmX (p,f)) is finite set
rng (SgmX (p,L)) is finite set
p9 is set
(SgmX (p,L)) . p9 is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
(SgmX (p,L)) /. m is Element of n
(SgmX (p,L)) . (p9 + 1) is set
m is set
(SgmX (p,f)) . m is set
(SgmX (p,L)) /. m is Element of n
[((SgmX (p,L)) /. (p9 + 1)),((SgmX (p,L)) /. m)] is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
[((SgmX (p,f)) /. m),((SgmX (p,L)) /. m)] is set
[((SgmX (p,f)) /. m),((SgmX (p,f)) /. (p9 + 1))] is set
(SgmX (p,f)) . m is set
(SgmX (p,L)) /. m is Element of n
[((SgmX (p,f)) /. (p9 + 1)),((SgmX (p,f)) /. m)] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. p9 is Element of n
(SgmX (p,L)) /. p9 is Element of n
(SgmX (p,f)) /. 1 is Element of n
(SgmX (p,L)) /. 1 is Element of n
(SgmX (p,f)) . 1 is set
rng (SgmX (p,f)) is finite set
rng (SgmX (p,L)) is finite set
p9 is set
(SgmX (p,L)) . p9 is set
(SgmX (p,L)) . 1 is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,L)) /. m is Element of n
[((SgmX (p,L)) /. 1),((SgmX (p,L)) /. m)] is set
[((SgmX (p,L)) /. 1),((SgmX (p,f)) /. 1)] is set
m is set
(SgmX (p,f)) . m is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
[((SgmX (p,f)) /. 1),((SgmX (p,f)) /. m)] is set
[((SgmX (p,f)) /. 1),((SgmX (p,L)) /. 1)] is set
n is set
bool n is non empty set
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is finite Element of bool n
x is Element of n
{x} is non empty trivial finite 1 -element set
{x} \/ L is non empty finite set
f is finite Element of bool n
p is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
SgmX (p,f) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom (SgmX (p,f)) is finite Element of bool NAT
SgmX (p,L) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX (p,L)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
field p is set
dom (SgmX (p,L)) is finite Element of bool NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg q9 is finite q9 -element Element of bool NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. q is Element of n
len (SgmX (p,f)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len (SgmX (p,f))) is finite len (SgmX (p,f)) -element Element of bool NAT
1 - 1 is V11() V12() V45() ext-real Element of REAL
q - 1 is V11() V12() V45() ext-real Element of REAL
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
q + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card L) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(len (SgmX (p,L))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(p9 + 1) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. ((p9 + 1) + 1) is Element of n
(SgmX (p,L)) /. (p9 + 1) is Element of n
1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 + (1 + 1) is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 + 2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (p9 + 2) is Element of n
(SgmX (p,f)) . (p9 + 2) is set
rng (SgmX (p,f)) is finite set
rng (SgmX (p,L)) is finite set
p9 is set
(SgmX (p,L)) . p9 is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,L)) /. m is Element of n
(SgmX (p,L)) . m is set
(SgmX (p,L)) . (p9 + 1) is set
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (m + 1) is Element of n
(SgmX (p,f)) /. m is Element of n
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
(SgmX (p,L)) /. m is Element of n
m - 1 is V11() V12() V45() ext-real Element of REAL
(p9 + 1) - 1 is V11() V12() V45() ext-real Element of REAL
(m - 1) + 1 is V11() V12() V45() ext-real Element of REAL
m + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,L)) /. (m - 1) is Element of n
m is set
(SgmX (p,f)) . m is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
(SgmX (p,f)) . m is set
Seg (len (SgmX (p,L))) is finite len (SgmX (p,L)) -element Element of bool NAT
[((SgmX (p,L)) /. (p9 + 1)),((SgmX (p,L)) /. m)] is set
[((SgmX (p,L)) /. m),((SgmX (p,L)) /. (p9 + 1))] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (p9 + 1) is Element of n
(SgmX (p,L)) /. p9 is Element of n
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (q + 1) is Element of n
(SgmX (p,L)) /. q is Element of n
(SgmX (p,L)) . q is set
rng (SgmX (p,L)) is finite set
rng (SgmX (p,f)) is finite set
p9 is set
(SgmX (p,f)) . p9 is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. m is Element of n
(SgmX (p,f)) . m is set
(SgmX (p,L)) /. m is Element of n
[((SgmX (p,f)) /. (q + 1)),((SgmX (p,f)) /. m)] is set
(SgmX (p,f)) . (q + 1) is set
m is set
(SgmX (p,L)) . m is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,L)) /. m is Element of n
(SgmX (p,L)) . m is set
(SgmX (p,f)) /. m is Element of n
[((SgmX (p,f)) /. m),((SgmX (p,f)) /. (q + 1))] is set
n is non empty set
bool n is non empty set
[:n,n:] is non empty Relation-like set
bool [:n,n:] is non empty set
L is finite Element of bool n
x is Element of n
{x} is non empty trivial finite 1 -element set
{x} \/ L is non empty finite set
f is finite Element of bool n
p is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
SgmX (p,f) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom (SgmX (p,f)) is finite Element of bool NAT
SgmX (p,L) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(SgmX (p,f)) /. (q + 1) is Element of n
Ins ((SgmX (p,L)),q,x) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX (p,f)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len (SgmX (p,f))) is finite len (SgmX (p,f)) -element Element of bool NAT
(q + 1) - 1 is V11() V12() V45() ext-real Element of REAL
(len (SgmX (p,f))) - 1 is V11() V12() V45() ext-real Element of REAL
q + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
card L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card L) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
len (SgmX (p,L)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(len (SgmX (p,L))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
len (Ins ((SgmX (p,L)),q,x)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len (Ins ((SgmX (p,L)),q,x))) is finite len (Ins ((SgmX (p,L)),q,x)) -element Element of bool NAT
dom (Ins ((SgmX (p,L)),q,x)) is finite Element of bool NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
(SgmX (p,f)) . p is set
(Ins ((SgmX (p,L)),q,x)) . p is set
(SgmX (p,f)) /. p is Element of n
(Ins ((SgmX (p,L)),q,x)) /. (q + 1) is Element of n
Seg q is finite q -element Element of bool NAT
(SgmX (p,L)) | (Seg q) is Relation-like NAT -defined Seg q -defined NAT -defined n -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,n:]
[:NAT,n:] is non empty non trivial Relation-like non finite set
bool [:NAT,n:] is non empty non trivial non finite set
dom ((SgmX (p,L)) | (Seg q)) is finite Element of bool NAT
(SgmX (p,L)) | q is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom ((SgmX (p,L)) | q) is finite Element of bool NAT
(SgmX (p,f)) /. p is Element of n
(SgmX (p,L)) /. p is Element of n
(Ins ((SgmX (p,L)),q,x)) /. p is Element of n
1 - 1 is V11() V12() V45() ext-real Element of REAL
p - 1 is V11() V12() V45() ext-real Element of REAL
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX (p,f)) /. (q + 1) is Element of n
(SgmX (p,L)) /. q is Element of n
(Ins ((SgmX (p,L)),q,x)) /. (q + 1) is Element of n
n is set
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
support L is finite set
dom (EmptyBag n) is Element of bool n
bool n is non empty set
x is set
L . x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(EmptyBag n) . x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom L is Element of bool n
n is set
[:n,NAT:] is Relation-like set
bool [:n,NAT:] is non empty set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x is set
rng L is V221() V222() V223() V224() V226() set
f is set
p is set
[f,p] is set
dom L is Element of bool n
bool n is non empty set
n is set
n is non empty set
the Element of n is Element of n
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
(EmptyBag n) +* ( the Element of n,1) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (EmptyBag n) is Element of bool n
bool n is non empty set
((EmptyBag n) +* ( the Element of n,1)) . the Element of n is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
the Element of n .--> 1 is Relation-like n -defined { the Element of n} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite V211() V212() V213() V214() finite-support set
{ the Element of n} is non empty trivial finite 1 -element set
{ the Element of n} --> 1 is non empty Relation-like non-empty { the Element of n} -defined NAT -valued RAT -valued INT -valued {1} -valued Function-like constant total V27({ the Element of n},{1}) finite V211() V212() V213() V214() finite-support Element of bool [:{ the Element of n},{1}:]
[:{ the Element of n},{1}:] is non empty Relation-like finite set
bool [:{ the Element of n},{1}:] is non empty finite V38() set
(EmptyBag n) +* ( the Element of n .--> 1) is Relation-like RAT -valued Function-like V211() V212() V213() V214() set
((EmptyBag n) +* ( the Element of n .--> 1)) . the Element of n is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom ( the Element of n .--> 1) is finite Element of bool n
( the Element of n .--> 1) . the Element of n is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(EmptyBag n) . the Element of n is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
n is set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
support L is finite set
bool n is non empty set
dom L is Element of bool n
x is set
n is epsilon-transitive epsilon-connected ordinal set
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,L) is finite Element of bool n
bool n is non empty set
p is set
q is set
[p,q] is set
[q,p] is set
q9 is epsilon-transitive epsilon-connected ordinal set
p9 is epsilon-transitive epsilon-connected ordinal set
n is set
L is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
x is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
L * x is Relation-like NAT -defined RAT -valued Function-like finite V211() V212() V213() V214() finite-support set
bool [:NAT,NAT:] is non empty non trivial non finite set
[:n,NAT:] is Relation-like set
bool [:n,NAT:] is non empty set
f is Relation-like n -defined NAT -valued Function-like V211() V212() V213() V214() Element of bool [:n,NAT:]
f * L is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
n is epsilon-transitive epsilon-connected ordinal set
x is non empty non trivial unital right_unital well-unital left_unital doubleLoopStr
the carrier of x is non empty non trivial set
[:n, the carrier of x:] is Relation-like set
bool [:n, the carrier of x:] is non empty set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,L) is finite Element of bool n
bool n is non empty set
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),(n,L)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX ((RelIncl n),(n,L))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
power x is non empty Relation-like [: the carrier of x,NAT:] -defined the carrier of x -valued Function-like total V27([: the carrier of x,NAT:], the carrier of x) Element of bool [:[: the carrier of x,NAT:], the carrier of x:]
[: the carrier of x,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of x,NAT:], the carrier of x:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of x,NAT:], the carrier of x:] is non empty non trivial non finite set
f is Relation-like n -defined the carrier of x -valued Function-like total V27(n, the carrier of x) Element of bool [:n, the carrier of x:]
f * (SgmX ((RelIncl n),(n,L))) is Relation-like NAT -defined the carrier of x -valued Function-like finite finite-support Element of bool [:NAT, the carrier of x:]
[:NAT, the carrier of x:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of x:] is non empty non trivial non finite set
(n,(SgmX ((RelIncl n),(n,L))),L) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
Seg (len (SgmX ((RelIncl n),(n,L)))) is finite len (SgmX ((RelIncl n),(n,L))) -element Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(f * (SgmX ((RelIncl n),(n,L)))) /. p9 is Element of the carrier of x
(n,(SgmX ((RelIncl n),(n,L))),L) /. p9 is Element of RAT
(power x) . (((f * (SgmX ((RelIncl n),(n,L)))) /. p9),((n,(SgmX ((RelIncl n),(n,L))),L) /. p9)) is set
[((f * (SgmX ((RelIncl n),(n,L)))) /. p9),((n,(SgmX ((RelIncl n),(n,L))),L) /. p9)] is set
(power x) . [((f * (SgmX ((RelIncl n),(n,L)))) /. p9),((n,(SgmX ((RelIncl n),(n,L))),L) /. p9)] is set
dom (SgmX ((RelIncl n),(n,L))) is finite Element of bool NAT
dom L is Element of bool n
rng (SgmX ((RelIncl n),(n,L))) is finite set
dom (n,(SgmX ((RelIncl n),(n,L))),L) is finite Element of bool NAT
(n,(SgmX ((RelIncl n),(n,L))),L) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of x
dom p9 is finite Element of bool NAT
Product p9 is Element of the carrier of x
len p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 /. q9 is Element of the carrier of x
(f * (SgmX ((RelIncl n),(n,L)))) /. q9 is Element of the carrier of x
(n,(SgmX ((RelIncl n),(n,L))),L) /. q9 is Element of RAT
(power x) . (((f * (SgmX ((RelIncl n),(n,L)))) /. q9),((n,(SgmX ((RelIncl n),(n,L))),L) /. q9)) is set
[((f * (SgmX ((RelIncl n),(n,L)))) /. q9),((n,(SgmX ((RelIncl n),(n,L))),L) /. q9)] is set
(power x) . [((f * (SgmX ((RelIncl n),(n,L)))) /. q9),((n,(SgmX ((RelIncl n),(n,L))),L) /. q9)] is set
q is Element of the carrier of x
p9 is Element of the carrier of x
q9 is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of x
len q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product q9 is Element of the carrier of x
q9 is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of x
len q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product q9 is Element of the carrier of x
p is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of x
len p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product p is Element of the carrier of x
p is Relation-like NAT -defined the carrier of x -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of x
len p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product p is Element of the carrier of x
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg (len p) is finite len p -element Element of bool NAT
dom p is finite Element of bool NAT
Seg (len q9) is finite len q9 -element Element of bool NAT
dom q9 is finite Element of bool NAT
q9 . q is set
q9 /. q is Element of the carrier of x
(f * (SgmX ((RelIncl n),(n,L)))) /. q is Element of the carrier of x
(n,(SgmX ((RelIncl n),(n,L))),L) /. q is Element of RAT
(power x) . (((f * (SgmX ((RelIncl n),(n,L)))) /. q),((n,(SgmX ((RelIncl n),(n,L))),L) /. q)) is set
[((f * (SgmX ((RelIncl n),(n,L)))) /. q),((n,(SgmX ((RelIncl n),(n,L))),L) /. q)] is set
(power x) . [((f * (SgmX ((RelIncl n),(n,L)))) /. q),((n,(SgmX ((RelIncl n),(n,L))),L) /. q)] is set
p /. q is Element of the carrier of x
p . q is set
n is set
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
(n,(EmptyBag n)) is finite Element of bool n
bool n is non empty set
L is non empty set
the Element of L is Element of L
(EmptyBag n) . the Element of L is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
n is epsilon-transitive epsilon-connected ordinal set
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
L is non empty non trivial unital right_unital well-unital left_unital doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
1. L is Element of the carrier of L
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,(EmptyBag n),L,x) is Element of the carrier of L
bool n is non empty set
(n,(EmptyBag n)) is finite Element of bool n
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),(n,(EmptyBag n))) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX ((RelIncl n),(n,(EmptyBag n)))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
power L is non empty Relation-like [: the carrier of L,NAT:] -defined the carrier of L -valued Function-like total V27([: the carrier of L,NAT:], the carrier of L) Element of bool [:[: the carrier of L,NAT:], the carrier of L:]
[: the carrier of L,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial non finite set
x * (SgmX ((RelIncl n),(n,(EmptyBag n)))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
(n,(SgmX ((RelIncl n),(n,(EmptyBag n)))),(EmptyBag n)) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
q 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 q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product q is Element of the carrier of L
p is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support Element of bool n
SgmX ((RelIncl n),p) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
<*> the carrier of L is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of L -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
1_ L is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non trivial unital right_unital well-unital left_unital doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
power L is non empty Relation-like [: the carrier of L,NAT:] -defined the carrier of L -valued Function-like total V27([: the carrier of L,NAT:], the carrier of L) Element of bool [:[: the carrier of L,NAT:], the carrier of L:]
[: the carrier of L,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial non finite set
x is set
{x} is non empty trivial finite 1 -element set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,f) is finite Element of bool n
bool n is non empty set
f . x is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p is finite Element of bool n
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),p) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
p9 is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,f,L,p9) is Element of the carrier of L
p9 . x is set
(power L) . ((p9 . x),(f . x)) is set
[(p9 . x),(f . x)] is set
(power L) . [(p9 . x),(f . x)] is set
rng p9 is set
dom p9 is Element of bool n
rng (SgmX ((RelIncl n),p)) is finite set
dom (SgmX ((RelIncl n),p)) is finite Element of bool NAT
(SgmX ((RelIncl n),p)) . 1 is set
p9 * (SgmX ((RelIncl n),p)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
dom (p9 * (SgmX ((RelIncl n),p))) is finite Element of bool NAT
(p9 * (SgmX ((RelIncl n),p))) /. 1 is Element of the carrier of L
(p9 * (SgmX ((RelIncl n),p))) . 1 is set
p9 . ((SgmX ((RelIncl n),p)) . 1) is set
dom f is Element of bool n
(n,(SgmX ((RelIncl n),p)),f) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
dom (n,(SgmX ((RelIncl n),p)),f) is finite Element of bool NAT
(n,(SgmX ((RelIncl n),p)),f) /. 1 is Element of RAT
(n,(SgmX ((RelIncl n),p)),f) . 1 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
f . ((SgmX ((RelIncl n),p)) . 1) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
q9 is Element of the carrier of L
(power L) . (q9,(f . x)) is Element of the carrier of L
[q9,(f . x)] is set
(power L) . [q9,(f . x)] is set
p is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((RelIncl n),p)) /. q is epsilon-transitive epsilon-connected ordinal Element of n
(SgmX ((RelIncl n),p)) . q is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= p9 ) } is set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((RelIncl n),p)) /. 1 is epsilon-transitive epsilon-connected ordinal Element of n
SgmX ((RelIncl n),(n,f)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX ((RelIncl n),(n,f))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 * (SgmX ((RelIncl n),(n,f))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
(n,(SgmX ((RelIncl n),(n,f))),f) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
p 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 p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product p is Element of the carrier of L
q is set
len (SgmX ((RelIncl n),p)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p . 1 is set
p /. 1 is Element of the carrier of L
(power L) . (((p9 * (SgmX ((RelIncl n),p))) /. 1),((n,(SgmX ((RelIncl n),p)),f) /. 1)) is set
[((p9 * (SgmX ((RelIncl n),p))) /. 1),((n,(SgmX ((RelIncl n),p)),f) /. 1)] is set
(power L) . [((p9 * (SgmX ((RelIncl n),p))) /. 1),((n,(SgmX ((RelIncl n),p)),f) /. 1)] is set
<*((power L) . ((p9 . x),(f . x)))*> is non empty trivial Relation-like NAT -defined Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support set
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
power L is non empty Relation-like [: the carrier of L,NAT:] -defined the carrier of L -valued Function-like total V27([: the carrier of L,NAT:], the carrier of L) Element of bool [:[: the carrier of L,NAT:], the carrier of L:]
[: the carrier of L,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial non finite set
x is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,x) is finite Element of bool n
bool n is non empty set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,f) is finite Element of bool n
p is set
{p} is non empty trivial finite 1 -element set
(n,x) \/ {p} is non empty finite set
f . p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),(n,f)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
SgmX ((RelIncl n),(n,x)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
q9 is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
q9 . p is set
(power L) . ((q9 . p),(f . p)) is set
[(q9 . p),(f . p)] is set
(power L) . [(q9 . p),(f . p)] is set
(n,f,L,q9) is Element of the carrier of L
(n,x,L,q9) is Element of the carrier of L
dom q9 is Element of bool n
p is Element of the carrier of L
p * (n,x,L,q9) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . (p,(n,x,L,q9)) is Element of the carrier of L
[p,(n,x,L,q9)] is set
the multF of L . [p,(n,x,L,q9)] is set
rng (SgmX ((RelIncl n),(n,f))) is finite set
dom (SgmX ((RelIncl n),(n,f))) is finite Element of bool NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
(SgmX ((RelIncl n),(n,f))) . q is set
(SgmX ((RelIncl n),(n,f))) /. q is epsilon-transitive epsilon-connected ordinal Element of n
len (SgmX ((RelIncl n),(n,f))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len (SgmX ((RelIncl n),(n,f)))) is finite len (SgmX ((RelIncl n),(n,f))) -element Element of bool NAT
1 - 1 is V11() V12() V45() ext-real Element of REAL
q - 1 is V11() V12() V45() ext-real Element of REAL
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
q + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
p9 is epsilon-transitive epsilon-connected ordinal Element of n
{p9} is non empty trivial finite 1 -element set
f . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
{(EmptyBag n)} is non empty trivial functional finite 1 -element set
p9 is non empty epsilon-transitive epsilon-connected ordinal set
[:p9, the carrier of L:] is non empty Relation-like set
bool [:p9, the carrier of L:] is non empty set
p9 is epsilon-transitive epsilon-connected ordinal Element of n
m is Relation-like NAT -defined p9 -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of p9
len m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is Relation-like p9 -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 is non empty Relation-like p9 -defined the carrier of L -valued Function-like total V27(p9, the carrier of L) Element of bool [:p9, the carrier of L:]
(p9,m,L,p9) is Element of the carrier of L
q9 * m is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
(p9,m,m) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
u 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 u is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product u is Element of the carrier of L
m is Relation-like NAT -defined p9 -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of p9
len m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is Relation-like p9 -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(p9,p9,L,p9) is Element of the carrier of L
q9 * m is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
(p9,m,p9) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
u 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 u is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product u is Element of the carrier of L
Ins (u,bg,p) 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 m) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(p9,p9) is finite Element of bool p9
bool p9 is non empty set
card (p9,p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (p9,p9)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(p9,m) is finite Element of bool p9
card (p9,m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
len (Ins (u,bg,p)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is epsilon-transitive epsilon-connected ordinal Element of p9
Ins (m,bg,m) is Relation-like NAT -defined p9 -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of p9
i is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
u . i is set
(Ins (u,bg,p)) . i is set
Seg (len u) is finite len u -element Element of bool NAT
u /. i is Element of the carrier of L
q9 * (Ins (m,bg,m)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
(q9 * (Ins (m,bg,m))) /. i is Element of the carrier of L
(p9,(Ins (m,bg,m)),m) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
(p9,(Ins (m,bg,m)),m) /. i is Element of RAT
(power L) . (((q9 * (Ins (m,bg,m))) /. i),((p9,(Ins (m,bg,m)),m) /. i)) is set
[((q9 * (Ins (m,bg,m))) /. i),((p9,(Ins (m,bg,m)),m) /. i)] is set
(power L) . [((q9 * (Ins (m,bg,m))) /. i),((p9,(Ins (m,bg,m)),m) /. i)] is set
dom u is finite Element of bool NAT
Seg (len (Ins (u,bg,p))) is finite len (Ins (u,bg,p)) -element Element of bool NAT
dom (Ins (u,bg,p)) is finite Element of bool NAT
i - 1 is V11() V12() V45() ext-real Element of REAL
m . q is set
{m} is non empty trivial finite 1 -element set
dom (q9 * m) is finite Element of bool NAT
(q9 * m) /. q is Element of the carrier of L
(q9 * m) . q is set
q9 . m is set
dom m is Element of bool p9
dom (p9,m,m) is finite Element of bool NAT
(p9,m,m) /. q is Element of RAT
(p9,m,m) . q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(power L) . ((q9 . m),(m . m)) is set
[(q9 . m),(m . m)] is set
(power L) . [(q9 . m),(m . m)] is set
(Ins (u,bg,p)) /. i is Element of the carrier of L
len (Ins (m,bg,m)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom (Ins (m,bg,m)) is finite Element of bool NAT
Seg ((len m) + 1) is non empty finite (len m) + 1 -element (len m) + 1 -element Element of bool NAT
(len m) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg bg is finite bg -element Element of bool NAT
u | (Seg bg) is Relation-like NAT -defined Seg bg -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
dom (u | (Seg bg)) is finite Element of bool NAT
u | bg 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 (u | bg) is finite Element of bool NAT
m | (Seg bg) is Relation-like NAT -defined Seg bg -defined NAT -defined p9 -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT,p9:]
[:NAT,p9:] is non empty non trivial Relation-like non finite set
bool [:NAT,p9:] is non empty non trivial non finite set
rng m is finite set
Seg (len m) is finite len m -element Element of bool NAT
dom m is finite Element of bool NAT
m . i is set
dom (q9 * m) is finite Element of bool NAT
m /. i is epsilon-transitive epsilon-connected ordinal Element of p9
((len m) + 1) - 1 is V11() V12() V45() ext-real Element of REAL
rng (Ins (m,bg,m)) is finite set
dom p9 is Element of bool p9
dom (p9,m,p9) is finite Element of bool NAT
dom (m | (Seg bg)) is finite Element of bool NAT
m | bg is Relation-like NAT -defined p9 -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of p9
dom (m | bg) is finite Element of bool NAT
(Ins (m,bg,m)) . i is set
dom (q9 * (Ins (m,bg,m))) is finite Element of bool NAT
(q9 * (Ins (m,bg,m))) . i is set
q9 . ((Ins (m,bg,m)) . i) is set
(Ins (m,bg,m)) /. i is epsilon-transitive epsilon-connected ordinal Element of p9
q9 . ((Ins (m,bg,m)) /. i) is set
q9 . (m /. i) is set
q9 . (m . i) is set
(q9 * m) . i is set
(q9 * m) /. i is Element of the carrier of L
dom m is Element of bool p9
dom (p9,(Ins (m,bg,m)),m) is finite Element of bool NAT
(p9,(Ins (m,bg,m)),m) . i is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . ((Ins (m,bg,m)) . i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . ((Ins (m,bg,m)) /. i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . (m /. i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . (m /. i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . (m . i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9,m,p9) . i is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9,m,p9) /. i is Element of RAT
u /. i is Element of the carrier of L
(Ins (u,bg,p)) /. i is Element of the carrier of L
(i - 1) + 1 is V11() V12() V45() ext-real Element of REAL
i + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
rng m is finite set
(len m) - 1 is V11() V12() V45() ext-real Element of REAL
i1 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len m) is finite len m -element Element of bool NAT
dom m is finite Element of bool NAT
m . i1 is set
dom (q9 * m) is finite Element of bool NAT
dom p9 is Element of bool p9
dom (p9,m,p9) is finite Element of bool NAT
m /. i1 is epsilon-transitive epsilon-connected ordinal Element of p9
i1 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
rng (Ins (m,bg,m)) is finite set
(Ins (m,bg,m)) . i is set
dom (q9 * (Ins (m,bg,m))) is finite Element of bool NAT
(q9 * (Ins (m,bg,m))) . i is set
q9 . ((Ins (m,bg,m)) . i) is set
(Ins (m,bg,m)) /. i is epsilon-transitive epsilon-connected ordinal Element of p9
q9 . ((Ins (m,bg,m)) /. i) is set
q9 . (m /. i1) is set
q9 . (m . i1) is set
(q9 * m) . i1 is set
(q9 * m) /. i1 is Element of the carrier of L
dom m is Element of bool p9
dom (p9,(Ins (m,bg,m)),m) is finite Element of bool NAT
(p9,(Ins (m,bg,m)),m) . i is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . ((Ins (m,bg,m)) . i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . ((Ins (m,bg,m)) /. i) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
m . (m /. i1) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . (m /. i1) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . (m . i1) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9,m,p9) . i1 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9,m,p9) /. i1 is Element of RAT
u /. i1 is Element of the carrier of L
(Ins (u,bg,p)) /. i is Element of the carrier of L
u | bg 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
<*p*> is non empty trivial Relation-like NAT -defined the carrier of L -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
(u | bg) ^ <*p*> is non empty Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
u /^ bg 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
((u | bg) ^ <*p*>) ^ (u /^ bg) is non empty 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 (((u | bg) ^ <*p*>) ^ (u /^ bg)) is Element of the carrier of L
Product ((u | bg) ^ <*p*>) is Element of the carrier of L
Product (u /^ bg) is Element of the carrier of L
(Product ((u | bg) ^ <*p*>)) * (Product (u /^ bg)) is Element of the carrier of L
the multF of L . ((Product ((u | bg) ^ <*p*>)),(Product (u /^ bg))) is Element of the carrier of L
[(Product ((u | bg) ^ <*p*>)),(Product (u /^ bg))] is set
the multF of L . [(Product ((u | bg) ^ <*p*>)),(Product (u /^ bg))] is set
Product (u | bg) is Element of the carrier of L
Product <*p*> is Element of the carrier of L
(Product (u | bg)) * (Product <*p*>) is Element of the carrier of L
the multF of L . ((Product (u | bg)),(Product <*p*>)) is Element of the carrier of L
[(Product (u | bg)),(Product <*p*>)] is set
the multF of L . [(Product (u | bg)),(Product <*p*>)] is set
((Product (u | bg)) * (Product <*p*>)) * (Product (u /^ bg)) is Element of the carrier of L
the multF of L . (((Product (u | bg)) * (Product <*p*>)),(Product (u /^ bg))) is Element of the carrier of L
[((Product (u | bg)) * (Product <*p*>)),(Product (u /^ bg))] is set
the multF of L . [((Product (u | bg)) * (Product <*p*>)),(Product (u /^ bg))] is set
(Product (u | bg)) * (Product (u /^ bg)) is Element of the carrier of L
the multF of L . ((Product (u | bg)),(Product (u /^ bg))) is Element of the carrier of L
[(Product (u | bg)),(Product (u /^ bg))] is set
the multF of L . [(Product (u | bg)),(Product (u /^ bg))] is set
((Product (u | bg)) * (Product (u /^ bg))) * (Product <*p*>) is Element of the carrier of L
the multF of L . (((Product (u | bg)) * (Product (u /^ bg))),(Product <*p*>)) is Element of the carrier of L
[((Product (u | bg)) * (Product (u /^ bg))),(Product <*p*>)] is set
the multF of L . [((Product (u | bg)) * (Product (u /^ bg))),(Product <*p*>)] is set
(u | bg) ^ (u /^ bg) 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 ((u | bg) ^ (u /^ bg)) is Element of the carrier of L
(Product ((u | bg) ^ (u /^ bg))) * (Product <*p*>) is Element of the carrier of L
the multF of L . ((Product ((u | bg) ^ (u /^ bg))),(Product <*p*>)) is Element of the carrier of L
[(Product ((u | bg) ^ (u /^ bg))),(Product <*p*>)] is set
the multF of L . [(Product ((u | bg) ^ (u /^ bg))),(Product <*p*>)] is set
(Product u) * (Product <*p*>) is Element of the carrier of L
the multF of L . ((Product u),(Product <*p*>)) is Element of the carrier of L
[(Product u),(Product <*p*>)] is set
the multF of L . [(Product u),(Product <*p*>)] is set
(p9,p9,L,p9) * p is Element of the carrier of L
the multF of L . ((p9,p9,L,p9),p) is Element of the carrier of L
[(p9,p9,L,p9),p] is set
the multF of L . [(p9,p9,L,p9),p] is set
Product (Ins (u,bg,p)) is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,x) is finite Element of bool n
bool n is non empty set
f is set
{f} is non empty trivial finite 1 -element set
f is set
{f} is non empty trivial finite 1 -element set
p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x + p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
q is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,(x + p),L,q) is Element of the carrier of L
(n,x,L,q) is Element of the carrier of L
(n,p,L,q) is Element of the carrier of L
(n,x,L,q) * (n,p,L,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((n,x,L,q),(n,p,L,q)) is Element of the carrier of L
[(n,x,L,q),(n,p,L,q)] is set
the multF of L . [(n,x,L,q),(n,p,L,q)] is set
(n,(x + p)) is finite Element of bool n
(n,p) is finite Element of bool n
(n,p) \/ {f} is non empty finite set
p9 is set
(x + p) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
x . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x . p9) + (p . p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
0 + (p . p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),(n,p)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
SgmX ((RelIncl n),(n,(x + p))) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
dom q is Element of bool n
len (SgmX ((RelIncl n),(n,(x + p)))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
power L is non empty Relation-like [: the carrier of L,NAT:] -defined the carrier of L -valued Function-like total V27([: the carrier of L,NAT:], the carrier of L) Element of bool [:[: the carrier of L,NAT:], the carrier of L:]
[: the carrier of L,NAT:] is non empty non trivial Relation-like non finite set
[:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:[: the carrier of L,NAT:], the carrier of L:] is non empty non trivial non finite set
q * (SgmX ((RelIncl n),(n,(x + p)))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
(n,(SgmX ((RelIncl n),(n,(x + p)))),(x + p)) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
p 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 p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product p is Element of the carrier of L
len (SgmX ((RelIncl n),(n,p))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q * (SgmX ((RelIncl n),(n,p))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
(n,(SgmX ((RelIncl n),(n,p))),p) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
q 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 q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Product q is Element of the carrier of L
dom (SgmX ((RelIncl n),(n,p))) is finite Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
bg is set
(x + p) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
bg is set
rng (SgmX ((RelIncl n),(n,p))) is finite set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
(SgmX ((RelIncl n),(n,p))) . bg is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p is Element of bool n
dom (n,(SgmX ((RelIncl n),(n,p))),p) is finite Element of bool NAT
(n,(SgmX ((RelIncl n),(n,p))),p) /. p9 is Element of RAT
(n,(SgmX ((RelIncl n),(n,p))),p) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
rng q is set
dom (SgmX ((RelIncl n),(n,(x + p)))) is finite Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
dom (x + p) is Element of bool n
(n,(SgmX ((RelIncl n),(n,p))),(x + p)) is Relation-like NAT -defined NAT -valued RAT -valued Function-like finite V211() V212() V213() V214() finite-support Element of bool [:NAT,NAT:]
dom (n,(SgmX ((RelIncl n),(n,p))),(x + p)) is finite Element of bool NAT
(n,(SgmX ((RelIncl n),(n,p))),(x + p)) /. p9 is Element of RAT
(n,(SgmX ((RelIncl n),(n,p))),(x + p)) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x + p) . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom q is finite Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is Relation-like NAT -defined NAT -valued Function-like V211() V212() V213() V214() Element of bool [:NAT,NAT:]
m /. p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p /. p9 is Element of the carrier of L
(q * (SgmX ((RelIncl n),(n,p)))) /. p9 is Element of the carrier of L
(power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((n,(SgmX ((RelIncl n),(n,p))),(x + p)) /. p9)) is set
[((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((n,(SgmX ((RelIncl n),(n,p))),(x + p)) /. p9)] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((n,(SgmX ((RelIncl n),(n,p))),(x + p)) /. p9)] is set
x . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x . f) + (p . f) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((x . f) + (p . f))) is Element of the carrier of L
[((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((x . f) + (p . f))] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,p)))) /. p9),((x . f) + (p . f))] is set
(power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f)) is Element of the carrier of L
[((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f)] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f)] is set
(power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)) is Element of the carrier of L
[((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)] is set
((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))) * ((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9))) is Element of the carrier of L
the multF of L . (((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)))) is Element of the carrier of L
[((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)))] is set
the multF of L . [((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(m /. p9)))] is set
q /. p9 is Element of the carrier of L
((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))) * (q /. p9) is Element of the carrier of L
the multF of L . (((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),(q /. p9)) is Element of the carrier of L
[((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),(q /. p9)] is set
the multF of L . [((power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. p9),(x . f))),(q /. p9)] is set
rng (SgmX ((RelIncl n),(n,(x + p)))) is finite set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p /. m is Element of the carrier of L
q /. m is Element of the carrier of L
(SgmX ((RelIncl n),(n,p))) . m is set
(n,(SgmX ((RelIncl n),(n,p))),p) /. m is Element of RAT
(n,(SgmX ((RelIncl n),(n,p))),p) . m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p . ((SgmX ((RelIncl n),(n,p))) . m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(SgmX ((RelIncl n),(n,p))) /. m is epsilon-transitive epsilon-connected ordinal Element of n
p . ((SgmX ((RelIncl n),(n,p))) /. m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
Seg m is finite m -element Element of bool NAT
(SgmX ((RelIncl n),(n,(x + p)))) /. m is epsilon-transitive epsilon-connected ordinal Element of n
(SgmX ((RelIncl n),(n,(x + p)))) . m is set
(SgmX ((RelIncl n),(n,(x + p)))) . p9 is set
(SgmX ((RelIncl n),(n,(x + p)))) . m is set
dom (n,(SgmX ((RelIncl n),(n,(x + p)))),(x + p)) is finite Element of bool NAT
(n,(SgmX ((RelIncl n),(n,(x + p)))),(x + p)) /. m is Element of RAT
(n,(SgmX ((RelIncl n),(n,(x + p)))),(x + p)) . m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x + p) . ((SgmX ((RelIncl n),(n,(x + p)))) . m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x + p) . ((SgmX ((RelIncl n),(n,(x + p)))) /. m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(q * (SgmX ((RelIncl n),(n,(x + p))))) /. m is Element of the carrier of L
(power L) . (((q * (SgmX ((RelIncl n),(n,(x + p))))) /. m),((x + p) . ((SgmX ((RelIncl n),(n,(x + p)))) /. m))) is Element of the carrier of L
[((q * (SgmX ((RelIncl n),(n,(x + p))))) /. m),((x + p) . ((SgmX ((RelIncl n),(n,(x + p)))) /. m))] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,(x + p))))) /. m),((x + p) . ((SgmX ((RelIncl n),(n,(x + p)))) /. m))] is set
(q * (SgmX ((RelIncl n),(n,p)))) /. m is Element of the carrier of L
(power L) . (((q * (SgmX ((RelIncl n),(n,p)))) /. m),(p . ((SgmX ((RelIncl n),(n,p))) /. m))) is Element of the carrier of L
[((q * (SgmX ((RelIncl n),(n,p)))) /. m),(p . ((SgmX ((RelIncl n),(n,p))) /. m))] is set
(power L) . [((q * (SgmX ((RelIncl n),(n,p)))) /. m),(p . ((SgmX ((RelIncl n),(n,p))) /. m))] is set
dom x is Element of bool n
q . f is set
m is Element of the carrier of L
(power L) . (m,(x . f)) is Element of the carrier of L
[m,(x . f)] is set
(power L) . [m,(x . f)] is set
u is Element of the carrier of L
dom (q * (SgmX ((RelIncl n),(n,p)))) is finite Element of bool NAT
(q * (SgmX ((RelIncl n),(n,p)))) . p9 is set
(SgmX ((RelIncl n),(n,p))) . p9 is set
q . ((SgmX ((RelIncl n),(n,p))) . p9) is set
u * (q /. p9) is Element of the carrier of L
the multF of L . (u,(q /. p9)) is Element of the carrier of L
[u,(q /. p9)] is set
the multF of L . [u,(q /. p9)] is set
u * (Product q) is Element of the carrier of L
the multF of L . (u,(Product q)) is Element of the carrier of L
[u,(Product q)] is set
the multF of L . [u,(Product q)] is set
dom x is Element of bool n
rng q is set
q . f is set
p9 is Element of the carrier of L
(x + p) . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(power L) . (p9,((x + p) . f)) is Element of the carrier of L
[p9,((x + p) . f)] is set
(power L) . [p9,((x + p) . f)] is set
bg is Element of the carrier of L
x . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p . f is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(x . f) + (p . f) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(x . f) + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
bg * (n,p,L,q) is Element of the carrier of L
the multF of L . (bg,(n,p,L,q)) is Element of the carrier of L
[bg,(n,p,L,q)] is set
the multF of L . [bg,(n,p,L,q)] is set
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x + f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,(x + f),L,p) is Element of the carrier of L
(n,x,L,p) is Element of the carrier of L
(n,f,L,p) is Element of the carrier of L
(n,x,L,p) * (n,f,L,p) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((n,x,L,p),(n,f,L,p)) is Element of the carrier of L
[(n,x,L,p),(n,f,L,p)] is set
the multF of L . [(n,x,L,p),(n,f,L,p)] is set
(n,x) is finite Element of bool n
bool n is non empty set
card (n,x) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,p9) is finite Element of bool n
card (n,p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(p9 + f),L,p) is Element of the carrier of L
(n,p9,L,p) is Element of the carrier of L
(n,p9,L,p) * (n,f,L,p) is Element of the carrier of L
the multF of L . ((n,p9,L,p),(n,f,L,p)) is Element of the carrier of L
[(n,p9,L,p),(n,f,L,p)] is set
the multF of L . [(n,p9,L,p),(n,f,L,p)] is set
RelIncl n is Relation-like n -defined n -valued total V27(n,n) reflexive antisymmetric transitive Element of bool [:n,n:]
[:n,n:] is Relation-like set
bool [:n,n:] is non empty set
SgmX ((RelIncl n),(n,p9)) is Relation-like NAT -defined n -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of n
len (SgmX ((RelIncl n),(n,p9))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))) is epsilon-transitive epsilon-connected ordinal Element of n
Seg (len (SgmX ((RelIncl n),(n,p9)))) is finite len (SgmX ((RelIncl n),(n,p9))) -element Element of bool NAT
dom (SgmX ((RelIncl n),(n,p9))) is finite Element of bool NAT
(SgmX ((RelIncl n),(n,p9))) . (len (SgmX ((RelIncl n),(n,p9)))) is set
rng (SgmX ((RelIncl n),(n,p9))) is finite set
p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom p9 is Element of bool n
((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> 0 is Relation-like {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite Function-yielding V48() V211() V212() V213() V214() finite-support set
{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} is non empty trivial finite 1 -element set
{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} --> 0 is non empty Relation-like {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} -defined NAT -valued RAT -valued INT -valued {0} -valued Function-like constant total V27({((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{0}) finite Function-yielding V48() V211() V212() V213() V214() finite-support Element of bool [:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{0}:]
{0} is non empty trivial functional finite V38() 1 -element set
[:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{0}:] is non empty Relation-like finite set
bool [:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{0}:] is non empty finite V38() set
p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> 0) is Relation-like RAT -valued Function-like V211() V212() V213() V214() set
(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0))) is finite Element of bool n
(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0))) \/ {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} is non empty finite set
p9 is set
p9 . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> 0) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} is non empty finite V38() set
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> 0) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} is non empty finite V38() set
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> 0) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() Relation-like Function-like finite cardinal V45() V46() ext-real non negative finite-support Element of NAT
p9 is set
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
card (n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (EmptyBag n) is Element of bool n
((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))) is Relation-like {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} -defined NAT -valued RAT -valued INT -valued Function-like one-to-one finite V211() V212() V213() V214() finite-support set
{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} --> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))) is non empty Relation-like {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))} -defined NAT -valued RAT -valued INT -valued {(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))} -valued Function-like constant total V27({((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))}) finite V211() V212() V213() V214() finite-support Element of bool [:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))}:]
{(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))} is non empty trivial finite V38() 1 -element set
[:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))}:] is non empty Relation-like finite set
bool [:{((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))},{(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))}:] is non empty finite V38() set
(EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is Relation-like RAT -valued Function-like V211() V212() V213() V214() set
(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))))) is finite Element of bool n
bg is set
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
(EmptyBag n) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
bg is set
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg is set
((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))))) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
p9 . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . bg) + (((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . bg) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
0 + (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))) .--> (p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))) is trivial finite Element of bool {((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))}
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(EmptyBag n) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() V46() ext-real non negative Element of NAT
((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) . bg) + (((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) . bg) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom ((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))))) is Element of bool n
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) + (p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) + (p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0))),L,p) is Element of the carrier of L
(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) is Element of the carrier of L
(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p) is Element of the carrier of L
(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p) is Element of the carrier of L
the multF of L . ((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)) is Element of the carrier of L
[(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
the multF of L . [(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,f,L,p) is Element of the carrier of L
the multF of L . ((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,f,L,p)) is Element of the carrier of L
[(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,f,L,p)] is set
the multF of L . [(n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p),(n,f,L,p)] is set
((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,f,L,p)) * (n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p) is Element of the carrier of L
the multF of L . (((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,f,L,p)),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)) is Element of the carrier of L
[((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,f,L,p)),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
the multF of L . [((n,(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)),L,p) * (n,f,L,p)),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
(p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f),L,p) is Element of the carrier of L
(n,((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f),L,p) * (n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p) is Element of the carrier of L
the multF of L . ((n,((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)) is Element of the carrier of L
[(n,((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
the multF of L . [(n,((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f),L,p),(n,((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))),L,p)] is set
((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) + ((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9)))))))) + ((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + f)),L,p) is Element of the carrier of L
((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))))) + f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(((p9 +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),0)) + ((EmptyBag n) +* (((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))),(p9 . ((SgmX ((RelIncl n),(n,p9))) /. (len (SgmX ((RelIncl n),(n,p9))))))))) + f),L,p) is Element of the carrier of L
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,q) is finite Element of bool n
card (n,q) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(q + f),L,p) is Element of the carrier of L
(n,q,L,p) is Element of the carrier of L
(n,q,L,p) * (n,f,L,p) is Element of the carrier of L
the multF of L . ((n,q,L,p),(n,f,L,p)) is Element of the carrier of L
[(n,q,L,p),(n,f,L,p)] is set
the multF of L . [(n,q,L,p),(n,f,L,p)] is set
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
1. L is V80(L) Element of the carrier of L
(1. L) * (n,f,L,p) is Element of the carrier of L
the multF of L . ((1. L),(n,f,L,p)) is Element of the carrier of L
[(1. L),(n,f,L,p)] is set
the multF of L . [(1. L),(n,f,L,p)] is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty left_add-cancelable right_add-cancelable right_complementable add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like addLoopStr
the carrier of L is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x - f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
- f is non empty Relation-like Bags n -defined Bags n -defined the carrier of L -valued Function-like total total V27( Bags n, the carrier of L) V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x + (- f) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
n is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of n is non empty non trivial set
L is epsilon-transitive epsilon-connected ordinal set
Bags L is non empty functional Element of bool (Bags L)
Bags L is non empty set
bool (Bags L) is non empty set
[:(Bags L), the carrier of n:] is non empty Relation-like set
bool [:(Bags L), the carrier of n:] is non empty set
0_ (L,n) is non empty Relation-like Bags L -defined the carrier of n -valued Function-like total V27( Bags L, the carrier of n) finite-Support Element of bool [:(Bags L), the carrier of n:]
x is non empty Relation-like Bags L -defined the carrier of n -valued Function-like total V27( Bags L, the carrier of n) finite-Support Element of bool [:(Bags L), the carrier of n:]
Support x is functional Element of bool (Bags L)
bool (Bags L) is non empty set
f is set
x . f is set
(0_ (L,n)) . f is set
p is Relation-like L -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x . p is Element of the carrier of n
0. n is V80(n) Element of the carrier of n
dom (0_ (L,n)) is non empty functional Element of bool (Bags L)
dom x is non empty functional Element of bool (Bags L)
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional Element of bool (Bags n)
bool (Bags n) is non empty set
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
field (BagOrder n) is set
p is set
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
[(EmptyBag n),q] is set
rng (BagOrder n) is functional Element of bool (Bags n)
dom (BagOrder n) is functional Element of bool (Bags n)
(dom (BagOrder n)) \/ (rng (BagOrder n)) is functional Element of bool (Bags n)
[:(field (BagOrder n)),(field (BagOrder n)):] is Relation-like set
bool [:(field (BagOrder n)),(field (BagOrder n)):] is non empty set
p is Relation-like field (BagOrder n) -defined field (BagOrder n) -valued Element of bool [:(field (BagOrder n)),(field (BagOrder n)):]
dom p is Element of bool (field (BagOrder n))
bool (field (BagOrder n)) is non empty set
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty set
Bags n is non empty functional Element of bool (Bags n)
bool (Bags n) is non empty set
L is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support x)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support x))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x * (SgmX ((BagOrder n),(Support x))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
f is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
Seg (len (SgmX ((BagOrder n),(Support x)))) is finite len (SgmX ((BagOrder n),(Support x))) -element Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(x * (SgmX ((BagOrder n),(Support x)))) /. p9 is Element of the carrier of L
(SgmX ((BagOrder n),(Support x))) /. p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support x))) /. p9)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. p9)),L,f) is Element of the carrier of L
((x * (SgmX ((BagOrder n),(Support x)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. p9)),L,f) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p9)),L,f)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p9)),L,f)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p9)),L,f)] is set
p9 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 p9 is finite Element of bool NAT
Sum p9 is Element of the carrier of L
len p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len p9) is finite len p9 -element Element of bool NAT
p9 /. q9 is Element of the carrier of L
(x * (SgmX ((BagOrder n),(Support x)))) /. q9 is Element of the carrier of L
(SgmX ((BagOrder n),(Support x))) /. q9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support x))) /. q9)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. q9)),L,f) is Element of the carrier of L
((x * (SgmX ((BagOrder n),(Support x)))) /. q9) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q9)),L,f) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. q9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q9)),L,f)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. q9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q9)),L,f)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. q9),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q9)),L,f)] is set
p is Element of the carrier of L
q is Element of the carrier of L
p9 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 p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p9 is Element of the carrier of L
p9 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 p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p9 is Element of the carrier of L
q9 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 q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum q9 is Element of the carrier of L
q9 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 q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum q9 is Element of the carrier of L
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
p9 . p is set
q9 . p is set
Seg (len q9) is finite len q9 -element Element of bool NAT
dom q9 is finite Element of bool NAT
Seg (len p9) is finite len p9 -element Element of bool NAT
dom p9 is finite Element of bool NAT
p9 /. p is Element of the carrier of L
(x * (SgmX ((BagOrder n),(Support x)))) /. p is Element of the carrier of L
(SgmX ((BagOrder n),(Support x))) /. p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support x))) /. p)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. p)),L,f) is Element of the carrier of L
((x * (SgmX ((BagOrder n),(Support x)))) /. p) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. p)),L,f) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p)),L,f)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p)),L,f)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support x))) /. p)),L,f)] is set
q9 /. p is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{f} is non empty trivial functional finite 1 -element set
x . f is Element of the carrier of L
p is functional finite Element of bool (Bags n)
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),p) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
dom x is non empty functional Element of bool (Bags n)
rng (SgmX ((BagOrder n),p)) is finite set
dom (SgmX ((BagOrder n),p)) is finite Element of bool NAT
(SgmX ((BagOrder n),p)) /. 1 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),p)) . 1 is Relation-like Function-like set
p9 is set
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((BagOrder n),p)) /. q9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),p)) . q9 is Relation-like Function-like set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is set
len (SgmX ((BagOrder n),p)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x * (SgmX ((BagOrder n),p)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
dom (x * (SgmX ((BagOrder n),p))) is finite Element of bool NAT
(x * (SgmX ((BagOrder n),p))) /. 1 is Element of the carrier of L
(x * (SgmX ((BagOrder n),p))) . 1 is set
x . ((SgmX ((BagOrder n),p)) . 1) is set
p9 is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,x,p9) is Element of the carrier of L
(n,f,L,p9) is Element of the carrier of L
(x . f) * (n,f,L,p9) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((x . f),(n,f,L,p9)) is Element of the carrier of L
[(x . f),(n,f,L,p9)] is set
the multF of L . [(x . f),(n,f,L,p9)] is set
SgmX ((BagOrder n),(Support x)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support x))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x * (SgmX ((BagOrder n),(Support x))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
q9 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 q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum q9 is Element of the carrier of L
q9 . 1 is set
q9 /. 1 is Element of the carrier of L
(n,((SgmX ((BagOrder n),p)) /. 1)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,p9) is Element of the carrier of L
((x * (SgmX ((BagOrder n),p))) /. 1) * (n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,p9) is Element of the carrier of L
the multF of L . (((x * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,p9)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,p9)] is set
the multF of L . [((x * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,p9)] is set
((x * (SgmX ((BagOrder n),p))) /. 1) * (n,f,L,p9) is Element of the carrier of L
the multF of L . (((x * (SgmX ((BagOrder n),p))) /. 1),(n,f,L,p9)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),p))) /. 1),(n,f,L,p9)] is set
the multF of L . [((x * (SgmX ((BagOrder n),p))) /. 1),(n,f,L,p9)] is set
<*((x . f) * (n,f,L,p9))*> is non empty trivial Relation-like NAT -defined the carrier of L -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
0. L is V80(L) Element of the carrier of L
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(0_ (n,L)),x) is Element of the carrier of L
Support (0_ (n,L)) is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support (0_ (n,L)))) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support (0_ (n,L))))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(0_ (n,L)) * (SgmX ((BagOrder n),(Support (0_ (n,L))))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
p 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 p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p is Element of the carrier of L
the Relation-like n -defined Function-like Element of Support (0_ (n,L)) is Relation-like n -defined Function-like Element of Support (0_ (n,L))
(0_ (n,L)) . the Relation-like n -defined Function-like Element of Support (0_ (n,L)) is set
<*> the carrier of L is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() V12() Relation-like non-empty empty-yielding NAT -defined RAT -valued the carrier of L -valued Function-like one-to-one constant functional finite finite-yielding V38() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V45() Function-yielding V48() ext-real non positive non negative V211() V212() V213() V214() FinSequence-yielding finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
1_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
1. L is V80(L) Element of the carrier of L
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(1_ (n,L)),x) is Element of the carrier of L
EmptyBag n is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
{(EmptyBag n)} is non empty trivial functional finite 1 -element set
Support (1_ (n,L)) is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
p is set
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(1_ (n,L)) . q is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
p is functional finite Element of bool (Bags n)
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),p) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
p9 is set
q9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(1_ (n,L)) . q9 is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
rng (SgmX ((BagOrder n),p)) is finite set
dom (SgmX ((BagOrder n),p)) is finite Element of bool NAT
(SgmX ((BagOrder n),p)) /. 1 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),p)) . 1 is Relation-like Function-like set
p9 is set
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((BagOrder n),p)) /. q9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),p)) . q9 is Relation-like Function-like set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p is finite p -element Element of bool NAT
{ b₁ where b₁ is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT : ( 1 <= b₁ & b₁ <= p ) } is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom (1_ (n,L)) is non empty functional Element of bool (Bags n)
(1_ (n,L)) * (SgmX ((BagOrder n),p)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
dom ((1_ (n,L)) * (SgmX ((BagOrder n),p))) is finite Element of bool NAT
((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1 is Element of the carrier of L
((1_ (n,L)) * (SgmX ((BagOrder n),p))) . 1 is set
(1_ (n,L)) . ((SgmX ((BagOrder n),p)) . 1) is set
(1_ (n,L)) . (EmptyBag n) is Element of the carrier of L
len (SgmX ((BagOrder n),p)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 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 p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p9 is Element of the carrier of L
q9 is set
p9 . 1 is set
p9 /. 1 is Element of the carrier of L
(n,((SgmX ((BagOrder n),p)) /. 1)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,x) is Element of the carrier of L
(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1) * (n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,x) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,x)) is Element of the carrier of L
[(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,x)] is set
the multF of L . [(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(n,(n,((SgmX ((BagOrder n),p)) /. 1)),L,x)] is set
(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1) * (1. L) is Element of the carrier of L
the multF of L . ((((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(1. L)) is Element of the carrier of L
[(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(1. L)] is set
the multF of L . [(((1_ (n,L)) * (SgmX ((BagOrder n),p))) /. 1),(1. L)] is set
<*(1. L)*> is non empty trivial Relation-like NAT -defined the carrier of L -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
- x is non empty Relation-like Bags n -defined Bags n -defined the carrier of L -valued Function-like total total V27( Bags n, the carrier of L) V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
f is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(- x),f) is Element of the carrier of L
(n,L,x,f) is Element of the carrier of L
- (n,L,x,f) is Element of the carrier of L
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
Support (- x) is functional finite Element of bool (Bags n)
q is set
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
q9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x . q9 is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
(- x) . q9 is Element of the carrier of L
- (x . q9) is Element of the carrier of L
- (- (x . q9)) is Element of the carrier of L
- (0. L) is Element of the carrier of L
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support (- x))) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support (- x)))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(- x) * (SgmX ((BagOrder n),(Support (- x)))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
q 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 q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum q is Element of the carrier of L
SgmX ((BagOrder n),(Support x)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support x))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x * (SgmX ((BagOrder n),(Support x))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
p9 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 p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p9 is Element of the carrier of L
q9 is set
p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(- x) . q is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
x . q is Element of the carrier of L
- (x . q) is Element of the carrier of L
1. L is V80(L) Element of the carrier of L
- (1. L) is Element of the carrier of L
(- (1. L)) * p9 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 ((- (1. L)) * p9) is finite Element of bool NAT
dom p9 is finite Element of bool NAT
len ((- (1. L)) * p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg q9 is finite q9 -element Element of bool NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
q . q is set
((- (1. L)) * p9) . q is set
(- x) * (SgmX ((BagOrder n),(Support x))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
((- x) * (SgmX ((BagOrder n),(Support x)))) /. q is Element of the carrier of L
(x * (SgmX ((BagOrder n),(Support x)))) /. q is Element of the carrier of L
(- (1. L)) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((- (1. L)),((x * (SgmX ((BagOrder n),(Support x)))) /. q)) is Element of the carrier of L
[(- (1. L)),((x * (SgmX ((BagOrder n),(Support x)))) /. q)] is set
the multF of L . [(- (1. L)),((x * (SgmX ((BagOrder n),(Support x)))) /. q)] is set
(SgmX ((BagOrder n),(Support x))) /. q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Seg (len (SgmX ((BagOrder n),(Support x)))) is finite len (SgmX ((BagOrder n),(Support x))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support x))) is finite Element of bool NAT
dom x is non empty functional Element of bool (Bags n)
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (x * (SgmX ((BagOrder n),(Support x)))) is finite Element of bool NAT
dom (- x) is non empty functional Element of bool (Bags n)
dom ((- x) * (SgmX ((BagOrder n),(Support x)))) is finite Element of bool NAT
(- x) /. p9 is Element of the carrier of L
(- x) . p9 is Element of the carrier of L
x . p9 is Element of the carrier of L
- (x . p9) is Element of the carrier of L
x /. p9 is Element of the carrier of L
- (x /. p9) is Element of the carrier of L
(1. L) * (x /. p9) is Element of the carrier of L
the multF of L . ((1. L),(x /. p9)) is Element of the carrier of L
[(1. L),(x /. p9)] is set
the multF of L . [(1. L),(x /. p9)] is set
- ((1. L) * (x /. p9)) is Element of the carrier of L
(- (1. L)) * (x /. p9) is Element of the carrier of L
the multF of L . ((- (1. L)),(x /. p9)) is Element of the carrier of L
[(- (1. L)),(x /. p9)] is set
the multF of L . [(- (1. L)),(x /. p9)] is set
Seg p is finite p -element Element of bool NAT
q /. q is Element of the carrier of L
(SgmX ((BagOrder n),(Support x))) /. q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support x))) /. q)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f) is Element of the carrier of L
((- (1. L)) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q)) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f) is Element of the carrier of L
the multF of L . (((- (1. L)) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)) is Element of the carrier of L
[((- (1. L)) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
the multF of L . [((- (1. L)) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
(1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q) is Element of the carrier of L
the multF of L . ((1. L),((x * (SgmX ((BagOrder n),(Support x)))) /. q)) is Element of the carrier of L
[(1. L),((x * (SgmX ((BagOrder n),(Support x)))) /. q)] is set
the multF of L . [(1. L),((x * (SgmX ((BagOrder n),(Support x)))) /. q)] is set
- ((1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q)) is Element of the carrier of L
(- ((1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q))) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f) is Element of the carrier of L
the multF of L . ((- ((1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q))),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)) is Element of the carrier of L
[(- ((1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q))),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
the multF of L . [(- ((1. L) * ((x * (SgmX ((BagOrder n),(Support x)))) /. q))),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
- ((x * (SgmX ((BagOrder n),(Support x)))) /. q) is Element of the carrier of L
(- ((x * (SgmX ((BagOrder n),(Support x)))) /. q)) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f) is Element of the carrier of L
the multF of L . ((- ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)) is Element of the carrier of L
[(- ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
the multF of L . [(- ((x * (SgmX ((BagOrder n),(Support x)))) /. q)),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
((x * (SgmX ((BagOrder n),(Support x)))) /. q) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f) is Element of the carrier of L
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. q),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. q),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. q),(n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)] is set
- (((x * (SgmX ((BagOrder n),(Support x)))) /. q) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. q)),L,f)) is Element of the carrier of L
p9 /. q is Element of the carrier of L
- (p9 /. q) is Element of the carrier of L
(1. L) * (p9 /. q) is Element of the carrier of L
the multF of L . ((1. L),(p9 /. q)) is Element of the carrier of L
[(1. L),(p9 /. q)] is set
the multF of L . [(1. L),(p9 /. q)] is set
- ((1. L) * (p9 /. q)) is Element of the carrier of L
(- (1. L)) * (p9 /. q) is Element of the carrier of L
the multF of L . ((- (1. L)),(p9 /. q)) is Element of the carrier of L
[(- (1. L)),(p9 /. q)] is set
the multF of L . [(- (1. L)),(p9 /. q)] is set
((- (1. L)) * p9) /. q is Element of the carrier of L
Seg p is finite p -element Element of bool NAT
Sum ((- (1. L)) * p9) is Element of the carrier of L
(- (1. L)) * (Sum p9) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((- (1. L)),(Sum p9)) is Element of the carrier of L
[(- (1. L)),(Sum p9)] is set
the multF of L . [(- (1. L)),(Sum p9)] is set
(- (1. L)) * (n,L,x,f) is Element of the carrier of L
the multF of L . ((- (1. L)),(n,L,x,f)) is Element of the carrier of L
[(- (1. L)),(n,L,x,f)] is set
the multF of L . [(- (1. L)),(n,L,x,f)] is set
(1. L) * (n,L,x,f) is Element of the carrier of L
the multF of L . ((1. L),(n,L,x,f)) is Element of the carrier of L
[(1. L),(n,L,x,f)] is set
the multF of L . [(1. L),(n,L,x,f)] is set
- ((1. L) * (n,L,x,f)) is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support f is functional finite Element of bool (Bags n)
p is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,f,p) is Element of the carrier of L
(n,L,x,p) is Element of the carrier of L
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{q} is non empty trivial functional finite 1 -element set
(Support x) \/ {q} is non empty finite set
f . q is Element of the carrier of L
(n,q,L,p) is Element of the carrier of L
(f . q) * (n,q,L,p) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((f . q),(n,q,L,p)) is Element of the carrier of L
[(f . q),(n,q,L,p)] is set
the multF of L . [(f . q),(n,q,L,p)] is set
(n,L,x,p) + ((f . q) * (n,q,L,p)) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((n,L,x,p),((f . q) * (n,q,L,p))) is Element of the carrier of L
[(n,L,x,p),((f . q) * (n,q,L,p))] is set
the addF of L . [(n,L,x,p),((f . q) * (n,q,L,p))] is set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support f)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
SgmX ((BagOrder n),(Support x)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
rng (SgmX ((BagOrder n),(Support f))) is finite set
dom (SgmX ((BagOrder n),(Support f))) is finite Element of bool NAT
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
(SgmX ((BagOrder n),(Support f))) . p is Relation-like Function-like set
(SgmX ((BagOrder n),(Support f))) /. p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
len (SgmX ((BagOrder n),(Support f))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Seg (len (SgmX ((BagOrder n),(Support f)))) is finite len (SgmX ((BagOrder n),(Support f))) -element Element of bool NAT
1 - 1 is V11() V12() V45() ext-real Element of REAL
p - 1 is V11() V12() V45() ext-real Element of REAL
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Ins ((SgmX ((BagOrder n),(Support x))),p9,q) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support x))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
x * (SgmX ((BagOrder n),(Support x))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
bg 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 bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum bg is Element of the carrier of L
f * (SgmX ((BagOrder n),(Support f))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
p9 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 p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p9 is Element of the carrier of L
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
f . p9 is Element of the carrier of L
(n,p9,L,p) is Element of the carrier of L
(f . p9) * (n,p9,L,p) is Element of the carrier of L
the multF of L . ((f . p9),(n,p9,L,p)) is Element of the carrier of L
[(f . p9),(n,p9,L,p)] is set
the multF of L . [(f . p9),(n,p9,L,p)] is set
Ins (bg,p9,((f . p9) * (n,p9,L,p))) 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 (SgmX ((BagOrder n),(Support x)))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
card (Support x) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (Support x)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
card (Support f) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
len (Ins (bg,p9,((f . p9) * (n,p9,L,p)))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
p9 . m is set
(Ins (bg,p9,((f . p9) * (n,p9,L,p)))) . m is set
Seg (len p9) is finite len p9 -element Element of bool NAT
dom p9 is finite Element of bool NAT
Seg (len (Ins (bg,p9,((f . p9) * (n,p9,L,p))))) is finite len (Ins (bg,p9,((f . p9) * (n,p9,L,p)))) -element Element of bool NAT
dom (Ins (bg,p9,((f . p9) * (n,p9,L,p)))) is finite Element of bool NAT
dom f is non empty functional Element of bool (Bags n)
dom (f * (SgmX ((BagOrder n),(Support f)))) is finite Element of bool NAT
(f * (SgmX ((BagOrder n),(Support f)))) /. p is Element of the carrier of L
(f * (SgmX ((BagOrder n),(Support f)))) . p is set
p9 /. m is Element of the carrier of L
(n,((SgmX ((BagOrder n),(Support f))) /. p)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support f))) /. p)),L,p) is Element of the carrier of L
((f * (SgmX ((BagOrder n),(Support f)))) /. p) * (n,(n,((SgmX ((BagOrder n),(Support f))) /. p)),L,p) is Element of the carrier of L
the multF of L . (((f * (SgmX ((BagOrder n),(Support f)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support f))) /. p)),L,p)) is Element of the carrier of L
[((f * (SgmX ((BagOrder n),(Support f)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support f))) /. p)),L,p)] is set
the multF of L . [((f * (SgmX ((BagOrder n),(Support f)))) /. p),(n,(n,((SgmX ((BagOrder n),(Support f))) /. p)),L,p)] is set
(Ins (bg,p9,((f . p9) * (n,p9,L,p)))) /. m is Element of the carrier of L
Ins ((SgmX ((BagOrder n),(Support x))),p9,p9) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is finite Element of bool NAT
Seg ((len (SgmX ((BagOrder n),(Support x)))) + 1) is non empty finite (len (SgmX ((BagOrder n),(Support x)))) + 1 -element (len (SgmX ((BagOrder n),(Support x)))) + 1 -element Element of bool NAT
(len (SgmX ((BagOrder n),(Support x)))) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg p9 is finite p9 -element Element of bool NAT
bg | (Seg p9) is Relation-like NAT -defined Seg p9 -defined NAT -defined the carrier of L -valued Function-like finite FinSubsequence-like finite-support Element of bool [:NAT, the carrier of L:]
dom (bg | (Seg p9)) is finite Element of bool NAT
bg | p9 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 (bg | p9) is finite Element of bool NAT
((len (SgmX ((BagOrder n),(Support x)))) + 1) - 1 is V11() V12() V45() ext-real Element of REAL
Seg (len (SgmX ((BagOrder n),(Support x)))) is finite len (SgmX ((BagOrder n),(Support x))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support x))) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support x))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),(Support x))) . m is Relation-like Function-like set
rng (SgmX ((BagOrder n),(Support x))) is finite set
(SgmX ((BagOrder n),(Support x))) | (Seg p9) is Relation-like NAT -defined Seg p9 -defined NAT -defined Bags n -valued Function-like finite FinSubsequence-like Function-yielding V48() finite-support Element of bool [:NAT,(Bags n):]
[:NAT,(Bags n):] is non empty non trivial Relation-like non finite set
bool [:NAT,(Bags n):] is non empty non trivial non finite set
rng (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is finite set
dom f is non empty functional Element of bool (Bags n)
dom ((SgmX ((BagOrder n),(Support x))) | (Seg p9)) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support x))) | p9 is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
dom ((SgmX ((BagOrder n),(Support x))) | p9) is finite Element of bool NAT
dom x is non empty functional Element of bool (Bags n)
dom (x * (SgmX ((BagOrder n),(Support x)))) is finite Element of bool NAT
(Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) . m is Relation-like Function-like set
f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
dom (f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) is finite Element of bool NAT
(f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) /. m is Element of the carrier of L
(f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) . m is set
f . ((Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) . m) is set
(Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
f . ((Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) /. m) is Element of the carrier of L
f . ((SgmX ((BagOrder n),(Support x))) /. m) is Element of the carrier of L
x . ((SgmX ((BagOrder n),(Support x))) /. m) is Element of the carrier of L
x . ((SgmX ((BagOrder n),(Support x))) . m) is set
(x * (SgmX ((BagOrder n),(Support x)))) . m is set
(x * (SgmX ((BagOrder n),(Support x)))) /. m is Element of the carrier of L
p9 /. m is Element of the carrier of L
(f * (SgmX ((BagOrder n),(Support f)))) /. m is Element of the carrier of L
(SgmX ((BagOrder n),(Support f))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support f))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p) is Element of the carrier of L
((f * (SgmX ((BagOrder n),(Support f)))) /. m) * (n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p) is Element of the carrier of L
the multF of L . (((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)) is Element of the carrier of L
[((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)] is set
the multF of L . [((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)] is set
(n,((SgmX ((BagOrder n),(Support x))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p) is Element of the carrier of L
((x * (SgmX ((BagOrder n),(Support x)))) /. m) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p) is Element of the carrier of L
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)] is set
bg /. m is Element of the carrier of L
(Ins (bg,p9,((f . p9) * (n,p9,L,p)))) /. m is Element of the carrier of L
m - 1 is V11() V12() V45() ext-real Element of REAL
(m - 1) + 1 is V11() V12() V45() ext-real Element of REAL
m + 0 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom f is non empty functional Element of bool (Bags n)
rng (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is finite set
dom x is non empty functional Element of bool (Bags n)
rng (SgmX ((BagOrder n),(Support x))) is finite set
(len bg) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
((len bg) + 1) - 1 is V11() V12() V45() ext-real Element of REAL
0 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
Seg (len (SgmX ((BagOrder n),(Support x)))) is finite len (SgmX ((BagOrder n),(Support x))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support x))) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support x))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),(Support x))) . m is Relation-like Function-like set
dom (x * (SgmX ((BagOrder n),(Support x)))) is finite Element of bool NAT
m + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
(Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) . m is Relation-like Function-like set
f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
dom (f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) is finite Element of bool NAT
(f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) /. m is Element of the carrier of L
(f * (Ins ((SgmX ((BagOrder n),(Support x))),p9,p9))) . m is set
f . ((Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) . m) is set
(Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
f . ((Ins ((SgmX ((BagOrder n),(Support x))),p9,p9)) /. m) is Element of the carrier of L
f . ((SgmX ((BagOrder n),(Support x))) /. m) is Element of the carrier of L
x . ((SgmX ((BagOrder n),(Support x))) /. m) is Element of the carrier of L
x . ((SgmX ((BagOrder n),(Support x))) . m) is set
(x * (SgmX ((BagOrder n),(Support x)))) . m is set
(x * (SgmX ((BagOrder n),(Support x)))) /. m is Element of the carrier of L
p9 /. m is Element of the carrier of L
(f * (SgmX ((BagOrder n),(Support f)))) /. m is Element of the carrier of L
(SgmX ((BagOrder n),(Support f))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support f))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p) is Element of the carrier of L
((f * (SgmX ((BagOrder n),(Support f)))) /. m) * (n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p) is Element of the carrier of L
the multF of L . (((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)) is Element of the carrier of L
[((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)] is set
the multF of L . [((f * (SgmX ((BagOrder n),(Support f)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support f))) /. m)),L,p)] is set
(n,((SgmX ((BagOrder n),(Support x))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p) is Element of the carrier of L
((x * (SgmX ((BagOrder n),(Support x)))) /. m) * (n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p) is Element of the carrier of L
the multF of L . (((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)) is Element of the carrier of L
[((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)] is set
the multF of L . [((x * (SgmX ((BagOrder n),(Support x)))) /. m),(n,(n,((SgmX ((BagOrder n),(Support x))) /. m)),L,p)] is set
bg /. m is Element of the carrier of L
(Ins (bg,p9,((f . p9) * (n,p9,L,p)))) /. m is Element of the carrier of L
bg | p9 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 . p9) * (n,p9,L,p))*> is non empty trivial Relation-like NAT -defined the carrier of L -valued Function-like constant finite 1 -element FinSequence-like FinSubsequence-like finite-support M9( the carrier of L,K256( the carrier of L))
K256( the carrier of L) is non empty functional FinSequence-membered M8( the carrier of L)
(bg | p9) ^ <*((f . p9) * (n,p9,L,p))*> is non empty Relation-like NAT -defined the carrier of L -valued Function-like finite FinSequence-like FinSubsequence-like finite-support FinSequence of the carrier of L
bg /^ p9 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
((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>) ^ (bg /^ p9) is non empty 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 (((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>) ^ (bg /^ p9)) is Element of the carrier of L
Sum ((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>) is Element of the carrier of L
Sum (bg /^ p9) is Element of the carrier of L
(Sum ((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>)) + (Sum (bg /^ p9)) is Element of the carrier of L
the addF of L . ((Sum ((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))) is Element of the carrier of L
[(Sum ((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))] is set
the addF of L . [(Sum ((bg | p9) ^ <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))] is set
Sum (bg | p9) is Element of the carrier of L
Sum <*((f . p9) * (n,p9,L,p))*> is Element of the carrier of L
(Sum (bg | p9)) + (Sum <*((f . p9) * (n,p9,L,p))*>) is Element of the carrier of L
the addF of L . ((Sum (bg | p9)),(Sum <*((f . p9) * (n,p9,L,p))*>)) is Element of the carrier of L
[(Sum (bg | p9)),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
the addF of L . [(Sum (bg | p9)),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
((Sum (bg | p9)) + (Sum <*((f . p9) * (n,p9,L,p))*>)) + (Sum (bg /^ p9)) is Element of the carrier of L
the addF of L . (((Sum (bg | p9)) + (Sum <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))) is Element of the carrier of L
[((Sum (bg | p9)) + (Sum <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))] is set
the addF of L . [((Sum (bg | p9)) + (Sum <*((f . p9) * (n,p9,L,p))*>)),(Sum (bg /^ p9))] is set
(Sum (bg | p9)) + (Sum (bg /^ p9)) is Element of the carrier of L
the addF of L . ((Sum (bg | p9)),(Sum (bg /^ p9))) is Element of the carrier of L
[(Sum (bg | p9)),(Sum (bg /^ p9))] is set
the addF of L . [(Sum (bg | p9)),(Sum (bg /^ p9))] is set
((Sum (bg | p9)) + (Sum (bg /^ p9))) + (Sum <*((f . p9) * (n,p9,L,p))*>) is Element of the carrier of L
the addF of L . (((Sum (bg | p9)) + (Sum (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)) is Element of the carrier of L
[((Sum (bg | p9)) + (Sum (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
the addF of L . [((Sum (bg | p9)) + (Sum (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
(bg | p9) ^ (bg /^ p9) 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 ((bg | p9) ^ (bg /^ p9)) is Element of the carrier of L
(Sum ((bg | p9) ^ (bg /^ p9))) + (Sum <*((f . p9) * (n,p9,L,p))*>) is Element of the carrier of L
the addF of L . ((Sum ((bg | p9) ^ (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)) is Element of the carrier of L
[(Sum ((bg | p9) ^ (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
the addF of L . [(Sum ((bg | p9) ^ (bg /^ p9))),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
(Sum bg) + (Sum <*((f . p9) * (n,p9,L,p))*>) is Element of the carrier of L
the addF of L . ((Sum bg),(Sum <*((f . p9) * (n,p9,L,p))*>)) is Element of the carrier of L
[(Sum bg),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
the addF of L . [(Sum bg),(Sum <*((f . p9) * (n,p9,L,p))*>)] is set
(n,L,x,p) + ((f . p9) * (n,p9,L,p)) is Element of the carrier of L
the addF of L . ((n,L,x,p),((f . p9) * (n,p9,L,p))) is Element of the carrier of L
[(n,L,x,p),((f . p9) * (n,p9,L,p))] is set
the addF of L . [(n,L,x,p),((f . p9) * (n,p9,L,p))] is set
Sum (Ins (bg,p9,((f . p9) * (n,p9,L,p)))) is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{f} is non empty trivial functional finite 1 -element set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{f} is non empty trivial functional finite 1 -element set
x . f is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
p is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x + p is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
q is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(x + p),q) is Element of the carrier of L
(n,L,x,q) is Element of the carrier of L
(n,L,p,q) is Element of the carrier of L
(n,L,x,q) + (n,L,p,q) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the addF of L . ((n,L,x,q),(n,L,p,q)) is Element of the carrier of L
[(n,L,x,q),(n,L,p,q)] is set
the addF of L . [(n,L,x,q),(n,L,p,q)] is set
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x + p) . p9 is Element of the carrier of L
p . p9 is Element of the carrier of L
x . p9 is Element of the carrier of L
(x . p9) + (p . p9) is Element of the carrier of L
the addF of L . ((x . p9),(p . p9)) is Element of the carrier of L
[(x . p9),(p . p9)] is set
the addF of L . [(x . p9),(p . p9)] is set
(0. L) + (p . p9) is Element of the carrier of L
the addF of L . ((0. L),(p . p9)) is Element of the carrier of L
[(0. L),(p . p9)] is set
the addF of L . [(0. L),(p . p9)] is set
Support p is functional finite Element of bool (Bags n)
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support p)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
Support (x + p) is functional finite Element of bool (Bags n)
SgmX ((BagOrder n),(Support (x + p))) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
(Support x) \/ (Support p) is functional finite Element of bool (Bags n)
len (SgmX ((BagOrder n),(Support p))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p * (SgmX ((BagOrder n),(Support p))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
[:NAT, the carrier of L:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of L:] is non empty non trivial non finite set
p 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 p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum p is Element of the carrier of L
len (SgmX ((BagOrder n),(Support (x + p)))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(x + p) * (SgmX ((BagOrder n),(Support (x + p)))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
q 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 q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
Sum q is Element of the carrier of L
p . f is Element of the carrier of L
- (p . f) is Element of the carrier of L
(Support (x + p)) \/ {f} is non empty finite set
p9 is set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x + p) . bg is Element of the carrier of L
p . bg is Element of the carrier of L
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x + p) . f is Element of the carrier of L
(x . f) + (p . f) is Element of the carrier of L
the addF of L . ((x . f),(p . f)) is Element of the carrier of L
[(x . f),(p . f)] is set
the addF of L . [(x . f),(p . f)] is set
p9 is set
(n,L,(x + p),q) + (0. L) is Element of the carrier of L
the addF of L . ((n,L,(x + p),q),(0. L)) is Element of the carrier of L
[(n,L,(x + p),q),(0. L)] is set
the addF of L . [(n,L,(x + p),q),(0. L)] is set
(n,f,L,q) is Element of the carrier of L
(p . f) * (n,f,L,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the multF of L . ((p . f),(n,f,L,q)) is Element of the carrier of L
[(p . f),(n,f,L,q)] is set
the multF of L . [(p . f),(n,f,L,q)] is set
- ((p . f) * (n,f,L,q)) is Element of the carrier of L
((p . f) * (n,f,L,q)) + (- ((p . f) * (n,f,L,q))) is Element of the carrier of L
the addF of L . (((p . f) * (n,f,L,q)),(- ((p . f) * (n,f,L,q)))) is Element of the carrier of L
[((p . f) * (n,f,L,q)),(- ((p . f) * (n,f,L,q)))] is set
the addF of L . [((p . f) * (n,f,L,q)),(- ((p . f) * (n,f,L,q)))] is set
(n,L,(x + p),q) + (((p . f) * (n,f,L,q)) + (- ((p . f) * (n,f,L,q)))) is Element of the carrier of L
the addF of L . ((n,L,(x + p),q),(((p . f) * (n,f,L,q)) + (- ((p . f) * (n,f,L,q))))) is Element of the carrier of L
[(n,L,(x + p),q),(((p . f) * (n,f,L,q)) + (- ((p . f) * (n,f,L,q))))] is set
the addF of L . [(n,L,(x + p),q),(((p . f) * (n,f,L,q)) + (- ((p . f) * (n,f,L,q))))] is set
(n,L,(x + p),q) + ((p . f) * (n,f,L,q)) is Element of the carrier of L
the addF of L . ((n,L,(x + p),q),((p . f) * (n,f,L,q))) is Element of the carrier of L
[(n,L,(x + p),q),((p . f) * (n,f,L,q))] is set
the addF of L . [(n,L,(x + p),q),((p . f) * (n,f,L,q))] is set
((n,L,(x + p),q) + ((p . f) * (n,f,L,q))) + (- ((p . f) * (n,f,L,q))) is Element of the carrier of L
the addF of L . (((n,L,(x + p),q) + ((p . f) * (n,f,L,q))),(- ((p . f) * (n,f,L,q)))) is Element of the carrier of L
[((n,L,(x + p),q) + ((p . f) * (n,f,L,q))),(- ((p . f) * (n,f,L,q)))] is set
the addF of L . [((n,L,(x + p),q) + ((p . f) * (n,f,L,q))),(- ((p . f) * (n,f,L,q)))] is set
(n,L,p,q) + (- ((p . f) * (n,f,L,q))) is Element of the carrier of L
the addF of L . ((n,L,p,q),(- ((p . f) * (n,f,L,q)))) is Element of the carrier of L
[(n,L,p,q),(- ((p . f) * (n,f,L,q)))] is set
the addF of L . [(n,L,p,q),(- ((p . f) * (n,f,L,q)))] is set
(x . f) * (n,f,L,q) is Element of the carrier of L
the multF of L . ((x . f),(n,f,L,q)) is Element of the carrier of L
[(x . f),(n,f,L,q)] is set
the multF of L . [(x . f),(n,f,L,q)] is set
(n,L,p,q) + ((x . f) * (n,f,L,q)) is Element of the carrier of L
the addF of L . ((n,L,p,q),((x . f) * (n,f,L,q))) is Element of the carrier of L
[(n,L,p,q),((x . f) * (n,f,L,q))] is set
the addF of L . [(n,L,p,q),((x . f) * (n,f,L,q))] is set
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
p . f is Element of the carrier of L
- (p . f) is Element of the carrier of L
(x . f) + (p . f) is Element of the carrier of L
the addF of L . ((x . f),(p . f)) is Element of the carrier of L
[(x . f),(p . f)] is set
the addF of L . [(x . f),(p . f)] is set
(- (p . f)) + (p . f) is Element of the carrier of L
the addF of L . ((- (p . f)),(p . f)) is Element of the carrier of L
[(- (p . f)),(p . f)] is set
the addF of L . [(- (p . f)),(p . f)] is set
(x . f) + (0. L) is Element of the carrier of L
the addF of L . ((x . f),(0. L)) is Element of the carrier of L
[(x . f),(0. L)] is set
the addF of L . [(x . f),(0. L)] is set
(p . f) + (- (p . f)) is Element of the carrier of L
the addF of L . ((p . f),(- (p . f))) is Element of the carrier of L
[(p . f),(- (p . f))] is set
the addF of L . [(p . f),(- (p . f))] is set
(x . f) + ((p . f) + (- (p . f))) is Element of the carrier of L
the addF of L . ((x . f),((p . f) + (- (p . f)))) is Element of the carrier of L
[(x . f),((p . f) + (- (p . f)))] is set
the addF of L . [(x . f),((p . f) + (- (p . f)))] is set
((- (p . f)) + (p . f)) + (- (p . f)) is Element of the carrier of L
the addF of L . (((- (p . f)) + (p . f)),(- (p . f))) is Element of the carrier of L
[((- (p . f)) + (p . f)),(- (p . f))] is set
the addF of L . [((- (p . f)) + (p . f)),(- (p . f))] is set
(0. L) + (- (p . f)) is Element of the carrier of L
the addF of L . ((0. L),(- (p . f))) is Element of the carrier of L
[(0. L),(- (p . f))] is set
the addF of L . [(0. L),(- (p . f))] is set
(x + p) . f is Element of the carrier of L
p9 is set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x + p) . bg is Element of the carrier of L
p . bg is Element of the carrier of L
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 is set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (SgmX ((BagOrder n),(Support (x + p)))) is finite Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
rng (SgmX ((BagOrder n),(Support p))) is finite set
dom (SgmX ((BagOrder n),(Support p))) is finite Element of bool NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
(SgmX ((BagOrder n),(Support p))) . bg is Relation-like Function-like set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg p9 is finite p9 -element Element of bool NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom (x + p) is non empty functional Element of bool (Bags n)
(SgmX ((BagOrder n),(Support p))) . p9 is Relation-like Function-like set
(x + p) * (SgmX ((BagOrder n),(Support p))) is Relation-like NAT -defined the carrier of L -valued Function-like finite finite-support Element of bool [:NAT, the carrier of L:]
dom ((x + p) * (SgmX ((BagOrder n),(Support p)))) is finite Element of bool NAT
((x + p) * (SgmX ((BagOrder n),(Support p)))) /. p9 is Element of the carrier of L
((x + p) * (SgmX ((BagOrder n),(Support p)))) . p9 is set
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p is non empty functional Element of bool (Bags n)
dom (p * (SgmX ((BagOrder n),(Support p)))) is finite Element of bool NAT
(p * (SgmX ((BagOrder n),(Support p)))) /. p9 is Element of the carrier of L
(p * (SgmX ((BagOrder n),(Support p)))) . p9 is set
dom p is finite Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative set
Seg m is finite m -element Element of bool NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
m is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q /. m is Element of the carrier of L
p /. m is Element of the carrier of L
Seg (len (SgmX ((BagOrder n),(Support (x + p))))) is finite len (SgmX ((BagOrder n),(Support (x + p)))) -element Element of bool NAT
(SgmX ((BagOrder n),(Support (x + p)))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),(Support (x + p)))) . m is Relation-like Function-like set
dom ((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) is finite Element of bool NAT
((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. m is Element of the carrier of L
((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) . m is set
(x + p) . ((SgmX ((BagOrder n),(Support (x + p)))) . m) is set
(x + p) . ((SgmX ((BagOrder n),(Support (x + p)))) /. m) is Element of the carrier of L
(SgmX ((BagOrder n),(Support (x + p)))) . p9 is Relation-like Function-like set
(SgmX ((BagOrder n),(Support p))) /. m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(SgmX ((BagOrder n),(Support p))) . m is Relation-like Function-like set
(p * (SgmX ((BagOrder n),(Support p)))) /. m is Element of the carrier of L
(p * (SgmX ((BagOrder n),(Support p)))) . m is set
p . ((SgmX ((BagOrder n),(Support p))) . m) is set
p . ((SgmX ((BagOrder n),(Support p))) /. m) is Element of the carrier of L
(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)),L,q) is Element of the carrier of L
(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. m) * (n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)),L,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the multF of L . ((((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. m),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)),L,q)) is Element of the carrier of L
[(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. m),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)),L,q)] is set
the multF of L . [(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. m),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. m)),L,q)] is set
(n,((SgmX ((BagOrder n),(Support p))) /. m)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support p))) /. m)),L,q) is Element of the carrier of L
(p . ((SgmX ((BagOrder n),(Support p))) /. m)) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. m)),L,q) is Element of the carrier of L
the multF of L . ((p . ((SgmX ((BagOrder n),(Support p))) /. m)),(n,(n,((SgmX ((BagOrder n),(Support p))) /. m)),L,q)) is Element of the carrier of L
[(p . ((SgmX ((BagOrder n),(Support p))) /. m)),(n,(n,((SgmX ((BagOrder n),(Support p))) /. m)),L,q)] is set
the multF of L . [(p . ((SgmX ((BagOrder n),(Support p))) /. m)),(n,(n,((SgmX ((BagOrder n),(Support p))) /. m)),L,q)] is set
(SgmX ((BagOrder n),(Support p))) /. p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
q /. p9 is Element of the carrier of L
((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. p9 is Element of the carrier of L
(SgmX ((BagOrder n),(Support (x + p)))) /. p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)),L,q) is Element of the carrier of L
(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)),L,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the multF of L . ((((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)),L,q)) is Element of the carrier of L
[(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)),L,q)] is set
the multF of L . [(((x + p) * (SgmX ((BagOrder n),(Support (x + p))))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support (x + p)))) /. p9)),L,q)] is set
(n,f,L,q) is Element of the carrier of L
((x . f) + (p . f)) * (n,f,L,q) is Element of the carrier of L
the multF of L . (((x . f) + (p . f)),(n,f,L,q)) is Element of the carrier of L
[((x . f) + (p . f)),(n,f,L,q)] is set
the multF of L . [((x . f) + (p . f)),(n,f,L,q)] is set
(x . f) * (n,f,L,q) is Element of the carrier of L
the multF of L . ((x . f),(n,f,L,q)) is Element of the carrier of L
[(x . f),(n,f,L,q)] is set
the multF of L . [(x . f),(n,f,L,q)] is set
(n,((SgmX ((BagOrder n),(Support p))) /. p9)) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q) is Element of the carrier of L
((p * (SgmX ((BagOrder n),(Support p)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q) is Element of the carrier of L
the multF of L . (((p * (SgmX ((BagOrder n),(Support p)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q)) is Element of the carrier of L
[((p * (SgmX ((BagOrder n),(Support p)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q)] is set
the multF of L . [((p * (SgmX ((BagOrder n),(Support p)))) /. p9),(n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q)] is set
((x . f) * (n,f,L,q)) + (((p * (SgmX ((BagOrder n),(Support p)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q)) is Element of the carrier of L
the addF of L . (((x . f) * (n,f,L,q)),(((p * (SgmX ((BagOrder n),(Support p)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q))) is Element of the carrier of L
[((x . f) * (n,f,L,q)),(((p * (SgmX ((BagOrder n),(Support p)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q))] is set
the addF of L . [((x . f) * (n,f,L,q)),(((p * (SgmX ((BagOrder n),(Support p)))) /. p9) * (n,(n,((SgmX ((BagOrder n),(Support p))) /. p9)),L,q))] is set
p /. p9 is Element of the carrier of L
((x . f) * (n,f,L,q)) + (p /. p9) is Element of the carrier of L
the addF of L . (((x . f) * (n,f,L,q)),(p /. p9)) is Element of the carrier of L
[((x . f) * (n,f,L,q)),(p /. p9)] is set
the addF of L . [((x . f) * (n,f,L,q)),(p /. p9)] is set
((x . f) * (n,f,L,q)) + (Sum p) is Element of the carrier of L
the addF of L . (((x . f) * (n,f,L,q)),(Sum p)) is Element of the carrier of L
[((x . f) * (n,f,L,q)),(Sum p)] is set
the addF of L . [((x . f) * (n,f,L,q)),(Sum p)] is set
p . f is Element of the carrier of L
- (p . f) is Element of the carrier of L
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
p . f is Element of the carrier of L
- (p . f) is Element of the carrier of L
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
(x + p) . f is Element of the carrier of L
p . f is Element of the carrier of L
(x . f) + (p . f) is Element of the carrier of L
the addF of L . ((x . f),(p . f)) is Element of the carrier of L
[(x . f),(p . f)] is set
the addF of L . [(x . f),(p . f)] is set
(x . f) + (0. L) is Element of the carrier of L
the addF of L . ((x . f),(0. L)) is Element of the carrier of L
[(x . f),(0. L)] is set
the addF of L . [(x . f),(0. L)] is set
p9 is set
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p . bg is Element of the carrier of L
(x + p) . bg is Element of the carrier of L
p9 is set
{f} \/ (Support p) is non empty finite set
(n,f,L,q) is Element of the carrier of L
((x + p) . f) * (n,f,L,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the multF of L . (((x + p) . f),(n,f,L,q)) is Element of the carrier of L
[((x + p) . f),(n,f,L,q)] is set
the multF of L . [((x + p) . f),(n,f,L,q)] is set
(n,L,p,q) + (((x + p) . f) * (n,f,L,q)) is Element of the carrier of L
the addF of L . ((n,L,p,q),(((x + p) . f) * (n,f,L,q))) is Element of the carrier of L
[(n,L,p,q),(((x + p) . f) * (n,f,L,q))] is set
the addF of L . [(n,L,p,q),(((x + p) . f) * (n,f,L,q))] is set
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
(n,L,p,q) + (n,L,x,q) is Element of the carrier of L
the addF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the addF of L . [(n,L,p,q),(n,L,x,q)] is set
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x + f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(x + f),p) is Element of the carrier of L
(n,L,x,p) is Element of the carrier of L
(n,L,f,p) is Element of the carrier of L
(n,L,x,p) + (n,L,f,p) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the addF of L . ((n,L,x,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,x,p),(n,L,f,p)] is set
the addF of L . [(n,L,x,p),(n,L,f,p)] is set
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
card (Support x) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional finite Element of bool (Bags n)
card (Support p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(p9 + f),p) is Element of the carrier of L
(n,L,p9,p) is Element of the carrier of L
(n,L,p9,p) + (n,L,f,p) is Element of the carrier of L
the addF of L . ((n,L,p9,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,f,p)] is set
the addF of L . [(n,L,p9,p),(n,L,f,p)] is set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support p9)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support p9))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9)))) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Seg (len (SgmX ((BagOrder n),(Support p9)))) is finite len (SgmX ((BagOrder n),(Support p9))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support p9))) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support p9))) . (len (SgmX ((BagOrder n),(Support p9)))) is Relation-like Function-like set
rng (SgmX ((BagOrder n),(Support p9))) is finite set
p9 . ((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))) is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))),(p9 . ((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))))) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
p9 +* (((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))),(0. L)) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
dom p9 is non empty functional Element of bool (Bags n)
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg .--> (0. L) is Relation-like {bg} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{bg} is non empty trivial functional finite 1 -element set
{bg} --> (0. L) is non empty Relation-like {bg} -defined the carrier of L -valued {(0. L)} -valued Function-like constant total V27({bg},{(0. L)}) finite finite-support Element of bool [:{bg},{(0. L)}:]
{(0. L)} is non empty trivial finite 1 -element set
[:{bg},{(0. L)}:] is non empty Relation-like finite set
bool [:{bg},{(0. L)}:] is non empty finite V38() set
p9 +* (bg .--> (0. L)) is Relation-like Function-like set
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional Element of bool (Bags n)
p9 is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . m is Element of the carrier of L
(bg .--> (0. L)) . bg is set
p9 . m is Element of the carrier of L
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional finite Element of bool (Bags n)
(Support p9) \/ {bg} is non empty finite set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
p9 . m is Element of the carrier of L
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . m is Element of the carrier of L
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . bg is Element of the carrier of L
(bg .--> (0. L)) . bg is set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
p9 . m is Element of the carrier of L
p9 . m is Element of the carrier of L
card (Support p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (Support p9)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
dom (0_ (n,L)) is non empty functional Element of bool (Bags n)
p9 . bg is Element of the carrier of L
bg .--> (p9 . bg) is Relation-like {bg} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{bg} --> (p9 . bg) is non empty Relation-like {bg} -defined the carrier of L -valued {(p9 . bg)} -valued Function-like constant total V27({bg},{(p9 . bg)}) finite finite-support Element of bool [:{bg},{(p9 . bg)}:]
{(p9 . bg)} is non empty trivial finite 1 -element set
[:{bg},{(p9 . bg)}:] is non empty Relation-like finite set
bool [:{bg},{(p9 . bg)}:] is non empty finite V38() set
(0_ (n,L)) +* (bg .--> (p9 . bg)) is Relation-like Function-like set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support m is functional Element of bool (Bags n)
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m . m is Element of the carrier of L
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
(0_ (n,L)) . m is Element of the carrier of L
m is set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . bg is Element of the carrier of L
(bg .--> (p9 . bg)) . bg is set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p9 + m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
u is set
(p9 + m) . u is set
p9 . u is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . bg is Element of the carrier of L
(bg .--> (p9 . bg)) . bg is set
p9 . u is Element of the carrier of L
(p9 + m) . u is Element of the carrier of L
m . u is Element of the carrier of L
(p9 . u) + (m . u) is Element of the carrier of L
the addF of L . ((p9 . u),(m . u)) is Element of the carrier of L
[(p9 . u),(m . u)] is set
the addF of L . [(p9 . u),(m . u)] is set
(0. L) + (p9 . bg) is Element of the carrier of L
the addF of L . ((0. L),(p9 . bg)) is Element of the carrier of L
[(0. L),(p9 . bg)] is set
the addF of L . [(0. L),(p9 . bg)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . u is Element of the carrier of L
(0_ (n,L)) . u is Element of the carrier of L
(p9 + m) . u is Element of the carrier of L
p9 . u is Element of the carrier of L
(p9 . u) + (m . u) is Element of the carrier of L
the addF of L . ((p9 . u),(m . u)) is Element of the carrier of L
[(p9 . u),(m . u)] is set
the addF of L . [(p9 . u),(m . u)] is set
p9 . u is Element of the carrier of L
(p9 . u) + (0. L) is Element of the carrier of L
the addF of L . ((p9 . u),(0. L)) is Element of the carrier of L
[(p9 . u),(0. L)] is set
the addF of L . [(p9 . u),(0. L)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (p9 + m) is non empty functional Element of bool (Bags n)
m + p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(m + p9),p) is Element of the carrier of L
(n,L,p9,p) is Element of the carrier of L
(n,L,m,p) is Element of the carrier of L
(n,L,p9,p) + (n,L,m,p) is Element of the carrier of L
the addF of L . ((n,L,p9,p),(n,L,m,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,m,p)] is set
the addF of L . [(n,L,p9,p),(n,L,m,p)] is set
(n,L,p9,p) + (n,L,f,p) is Element of the carrier of L
the addF of L . ((n,L,p9,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,f,p)] is set
the addF of L . [(n,L,p9,p),(n,L,f,p)] is set
((n,L,p9,p) + (n,L,f,p)) + (n,L,m,p) is Element of the carrier of L
the addF of L . (((n,L,p9,p) + (n,L,f,p)),(n,L,m,p)) is Element of the carrier of L
[((n,L,p9,p) + (n,L,f,p)),(n,L,m,p)] is set
the addF of L . [((n,L,p9,p) + (n,L,f,p)),(n,L,m,p)] is set
p9 + f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(p9 + f),p) is Element of the carrier of L
(n,L,(p9 + f),p) + (n,L,m,p) is Element of the carrier of L
the addF of L . ((n,L,(p9 + f),p),(n,L,m,p)) is Element of the carrier of L
[(n,L,(p9 + f),p),(n,L,m,p)] is set
the addF of L . [(n,L,(p9 + f),p),(n,L,m,p)] is set
m + (p9 + f) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(m + (p9 + f)),p) is Element of the carrier of L
(p9 + m) + f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,((p9 + m) + f),p) is Element of the carrier of L
q is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support q is functional finite Element of bool (Bags n)
card (Support q) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(q + f),p) is Element of the carrier of L
(n,L,q,p) is Element of the carrier of L
(n,L,q,p) + (n,L,f,p) is Element of the carrier of L
the addF of L . ((n,L,q,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,q,p),(n,L,f,p)] is set
the addF of L . [(n,L,q,p),(n,L,f,p)] is set
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
0. L is V80(L) Element of the carrier of L
(0. L) + (n,L,f,p) is Element of the carrier of L
the addF of L . ((0. L),(n,L,f,p)) is Element of the carrier of L
[(0. L),(n,L,f,p)] is set
the addF of L . [(0. L),(n,L,f,p)] is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x - f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(x - f),p) is Element of the carrier of L
(n,L,x,p) is Element of the carrier of L
(n,L,f,p) is Element of the carrier of L
(n,L,x,p) - (n,L,f,p) is Element of the carrier of L
- (n,L,f,p) is Element of the carrier of L
(n,L,x,p) + (- (n,L,f,p)) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the addF of L . ((n,L,x,p),(- (n,L,f,p))) is Element of the carrier of L
[(n,L,x,p),(- (n,L,f,p))] is set
the addF of L . [(n,L,x,p),(- (n,L,f,p))] is set
- f is non empty Relation-like Bags n -defined Bags n -defined the carrier of L -valued Function-like total total V27( Bags n, the carrier of L) V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x + (- f) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(x + (- f)),p) is Element of the carrier of L
(n,L,(- f),p) is Element of the carrier of L
(n,L,x,p) + (n,L,(- f),p) is Element of the carrier of L
the addF of L . ((n,L,x,p),(n,L,(- f),p)) is Element of the carrier of L
[(n,L,x,p),(n,L,(- f),p)] is set
the addF of L . [(n,L,x,p),(n,L,(- f),p)] is set
(n,L,x,p) + (- (n,L,f,p)) is Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support f is functional finite Element of bool (Bags n)
x *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{p} is non empty trivial functional finite 1 -element set
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{q} is non empty trivial functional finite 1 -element set
p + q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x *' f) . (p + q) is Element of the carrier of L
decomp (p + q) is non empty Relation-like NAT -defined K257(2,(Bags n)) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like Function-yielding V48() FinSequence-yielding finite-support FinSequence of K257(2,(Bags n))
K257(2,(Bags n)) is M8( Bags n)
len (decomp (p + q)) is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 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 p9 is Element of the carrier of L
len p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p9 is finite Element of bool NAT
q9 is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(x *' f),q9) is Element of the carrier of L
(n,L,x,q9) is Element of the carrier of L
(n,L,f,q9) is Element of the carrier of L
(n,L,x,q9) * (n,L,f,q9) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((n,L,x,q9),(n,L,f,q9)) is Element of the carrier of L
[(n,L,x,q9),(n,L,f,q9)] is set
the multF of L . [(n,L,x,q9),(n,L,f,q9)] is set
x . p is Element of the carrier of L
f . q is Element of the carrier of L
(x . p) * (f . q) is Element of the carrier of L
the multF of L . ((x . p),(f . q)) is Element of the carrier of L
[(x . p),(f . q)] is set
the multF of L . [(x . p),(f . q)] is set
(n,p,L,q9) is Element of the carrier of L
(n,q,L,q9) is Element of the carrier of L
(n,p,L,q9) * (n,q,L,q9) is Element of the carrier of L
the multF of L . ((n,p,L,q9),(n,q,L,q9)) is Element of the carrier of L
[(n,p,L,q9),(n,q,L,q9)] is set
the multF of L . [(n,p,L,q9),(n,q,L,q9)] is set
((x . p) * (f . q)) * ((n,p,L,q9) * (n,q,L,q9)) is Element of the carrier of L
the multF of L . (((x . p) * (f . q)),((n,p,L,q9) * (n,q,L,q9))) is Element of the carrier of L
[((x . p) * (f . q)),((n,p,L,q9) * (n,q,L,q9))] is set
the multF of L . [((x . p) * (f . q)),((n,p,L,q9) * (n,q,L,q9))] is set
((x . p) * (f . q)) * (n,p,L,q9) is Element of the carrier of L
the multF of L . (((x . p) * (f . q)),(n,p,L,q9)) is Element of the carrier of L
[((x . p) * (f . q)),(n,p,L,q9)] is set
the multF of L . [((x . p) * (f . q)),(n,p,L,q9)] is set
(((x . p) * (f . q)) * (n,p,L,q9)) * (n,q,L,q9) is Element of the carrier of L
the multF of L . ((((x . p) * (f . q)) * (n,p,L,q9)),(n,q,L,q9)) is Element of the carrier of L
[(((x . p) * (f . q)) * (n,p,L,q9)),(n,q,L,q9)] is set
the multF of L . [(((x . p) * (f . q)) * (n,p,L,q9)),(n,q,L,q9)] is set
(x . p) * (n,p,L,q9) is Element of the carrier of L
the multF of L . ((x . p),(n,p,L,q9)) is Element of the carrier of L
[(x . p),(n,p,L,q9)] is set
the multF of L . [(x . p),(n,p,L,q9)] is set
((x . p) * (n,p,L,q9)) * (f . q) is Element of the carrier of L
the multF of L . (((x . p) * (n,p,L,q9)),(f . q)) is Element of the carrier of L
[((x . p) * (n,p,L,q9)),(f . q)] is set
the multF of L . [((x . p) * (n,p,L,q9)),(f . q)] is set
(((x . p) * (n,p,L,q9)) * (f . q)) * (n,q,L,q9) is Element of the carrier of L
the multF of L . ((((x . p) * (n,p,L,q9)) * (f . q)),(n,q,L,q9)) is Element of the carrier of L
[(((x . p) * (n,p,L,q9)) * (f . q)),(n,q,L,q9)] is set
the multF of L . [(((x . p) * (n,p,L,q9)) * (f . q)),(n,q,L,q9)] is set
(f . q) * (n,q,L,q9) is Element of the carrier of L
the multF of L . ((f . q),(n,q,L,q9)) is Element of the carrier of L
[(f . q),(n,q,L,q9)] is set
the multF of L . [(f . q),(n,q,L,q9)] is set
((x . p) * (n,p,L,q9)) * ((f . q) * (n,q,L,q9)) is Element of the carrier of L
the multF of L . (((x . p) * (n,p,L,q9)),((f . q) * (n,q,L,q9))) is Element of the carrier of L
[((x . p) * (n,p,L,q9)),((f . q) * (n,q,L,q9))] is set
the multF of L . [((x . p) * (n,p,L,q9)),((f . q) * (n,q,L,q9))] is set
(n,L,x,q9) * ((f . q) * (n,q,L,q9)) is Element of the carrier of L
the multF of L . ((n,L,x,q9),((f . q) * (n,q,L,q9))) is Element of the carrier of L
[(n,L,x,q9),((f . q) * (n,q,L,q9))] is set
the multF of L . [(n,L,x,q9),((f . q) * (n,q,L,q9))] is set
0. L is V80(L) Element of the carrier of L
p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
f . p is Element of the carrier of L
p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
x . p is Element of the carrier of L
Support (x *' f) is functional finite Element of bool (Bags n)
{(p + q)} is non empty trivial functional finite 1 -element set
p is set
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
(x *' f) . q is Element of the carrier of L
decomp q is non empty Relation-like NAT -defined K257(2,(Bags n)) -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like Function-yielding V48() FinSequence-yielding finite-support FinSequence of K257(2,(Bags n))
len (decomp q) is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 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 p9 is Element of the carrier of L
len p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
dom p9 is finite Element of bool NAT
Seg (len p9) is finite len p9 -element Element of bool NAT
dom (decomp q) is non empty finite Element of bool NAT
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 /. bg is Element of the carrier of L
divisors q is non empty Relation-like NAT -defined Bags n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
p9 is non empty functional finite Element of bool (Bags n)
SgmX ((BagOrder n),p9) is non empty Relation-like NAT -defined Bags n -valued Function-like one-to-one finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
rng (divisors q) is non empty finite set
(decomp q) /. bg is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support Element of K257(2,(Bags n))
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
<*p9,p9*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
x . p9 is Element of the carrier of L
f . p9 is Element of the carrier of L
(x . p9) * (f . p9) is Element of the carrier of L
the multF of L . ((x . p9),(f . p9)) is Element of the carrier of L
[(x . p9),(f . p9)] is set
the multF of L . [(x . p9),(f . p9)] is set
(divisors q) /. bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
dom (divisors q) is non empty finite Element of bool NAT
(divisors q) . bg is Relation-like Function-like set
<*p,q*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
<*p,q*> . 2 is set
q -' p is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
<*p,(q -' p)*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
<*p,(q -' p)*> . 2 is set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 /. bg is Element of the carrier of L
p9 /. 1 is Element of the carrier of L
dom (decomp (p + q)) is non empty finite Element of bool NAT
<*p,q*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
p is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(decomp (p + q)) /. p is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support Element of K257(2,(Bags n))
Seg (len p9) is finite len p9 -element Element of bool NAT
p9 /. p is Element of the carrier of L
q is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
<*q,p9*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
x . q is Element of the carrier of L
f . p9 is Element of the carrier of L
(x . q) * (f . p9) is Element of the carrier of L
the multF of L . ((x . q),(f . p9)) is Element of the carrier of L
[(x . q),(f . p9)] is set
the multF of L . [(x . q),(f . p9)] is set
<*p,q*> . 2 is set
bg is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 /. bg is Element of the carrier of L
(decomp (p + q)) /. bg is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support Element of K257(2,(Bags n))
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
<*p9,p9*> is non empty Relation-like NAT -defined Function-like finite 2 -element FinSequence-like FinSubsequence-like finite-support set
x . p9 is Element of the carrier of L
f . p9 is Element of the carrier of L
(x . p9) * (f . p9) is Element of the carrier of L
the multF of L . ((x . p9),(f . p9)) is Element of the carrier of L
[(x . p9),(f . p9)] is set
the multF of L . [(x . p9),(f . p9)] is set
(decomp (p + q)) . bg is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
(decomp (p + q)) . p is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like finite-support set
<*q,p9*> . 1 is set
the Relation-like n -defined Function-like Element of Support (x *' f) is Relation-like n -defined Function-like Element of Support (x *' f)
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
bg is set
(n,(p + q),L,q9) is Element of the carrier of L
((x *' f) . (p + q)) * (n,(p + q),L,q9) is Element of the carrier of L
the multF of L . (((x *' f) . (p + q)),(n,(p + q),L,q9)) is Element of the carrier of L
[((x *' f) . (p + q)),(n,(p + q),L,q9)] is set
the multF of L . [((x *' f) . (p + q)),(n,(p + q),L,q9)] is set
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
f is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
{f} is non empty trivial functional finite 1 -element set
p is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p *' x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
q is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(p *' x),q) is Element of the carrier of L
(n,L,p,q) is Element of the carrier of L
(n,L,x,q) is Element of the carrier of L
(n,L,p,q) * (n,L,x,q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((n,L,p,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p,q),(n,L,x,q)] is set
the multF of L . [(n,L,p,q),(n,L,x,q)] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
q9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support q9 is functional finite Element of bool (Bags n)
card (Support q9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q9 *' x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(q9 *' x),q) is Element of the carrier of L
(n,L,q9,q) is Element of the carrier of L
(n,L,q9,q) * (n,L,x,q) is Element of the carrier of L
the multF of L . ((n,L,q9,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,q9,q),(n,L,x,q)] is set
the multF of L . [(n,L,q9,q),(n,L,x,q)] is set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support q9)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support q9))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((BagOrder n),(Support q9))) /. (len (SgmX ((BagOrder n),(Support q9)))) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Seg (len (SgmX ((BagOrder n),(Support q9)))) is finite len (SgmX ((BagOrder n),(Support q9))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support q9))) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support q9))) . (len (SgmX ((BagOrder n),(Support q9)))) is Relation-like Function-like set
rng (SgmX ((BagOrder n),(Support q9))) is finite set
q9 . ((SgmX ((BagOrder n),(Support q9))) /. (len (SgmX ((BagOrder n),(Support q9))))) is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(0_ (n,L)) +* (((SgmX ((BagOrder n),(Support q9))) /. (len (SgmX ((BagOrder n),(Support q9))))),(q9 . ((SgmX ((BagOrder n),(Support q9))) /. (len (SgmX ((BagOrder n),(Support q9))))))) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
q9 +* (((SgmX ((BagOrder n),(Support q9))) /. (len (SgmX ((BagOrder n),(Support q9))))),(0. L)) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
dom q9 is non empty functional Element of bool (Bags n)
p9 is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
p9 .--> (0. L) is Relation-like {p9} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{p9} is non empty trivial functional finite 1 -element set
{p9} --> (0. L) is non empty Relation-like {p9} -defined the carrier of L -valued {(0. L)} -valued Function-like constant total V27({p9},{(0. L)}) finite finite-support Element of bool [:{p9},{(0. L)}:]
{(0. L)} is non empty trivial finite 1 -element set
[:{p9},{(0. L)}:] is non empty Relation-like finite set
bool [:{p9},{(0. L)}:] is non empty finite V38() set
q9 +* (p9 .--> (0. L)) is Relation-like Function-like set
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional Element of bool (Bags n)
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (p9 .--> (0. L)) is trivial functional finite Element of bool {p9}
bool {p9} is non empty finite V38() set
p9 . m is Element of the carrier of L
(p9 .--> (0. L)) . p9 is set
q9 . m is Element of the carrier of L
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support m is functional finite Element of bool (Bags n)
(Support m) \/ {p9} is non empty finite set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
q9 . m is Element of the carrier of L
dom (p9 .--> (0. L)) is trivial functional finite Element of bool {p9}
bool {p9} is non empty finite V38() set
m . m is Element of the carrier of L
dom (p9 .--> (0. L)) is trivial functional finite Element of bool {p9}
bool {p9} is non empty finite V38() set
m . p9 is Element of the carrier of L
(p9 .--> (0. L)) . p9 is set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m . m is Element of the carrier of L
q9 . m is Element of the carrier of L
card (Support m) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (Support m)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
dom (0_ (n,L)) is non empty functional Element of bool (Bags n)
q9 . p9 is Element of the carrier of L
p9 .--> (q9 . p9) is Relation-like {p9} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{p9} --> (q9 . p9) is non empty Relation-like {p9} -defined the carrier of L -valued {(q9 . p9)} -valued Function-like constant total V27({p9},{(q9 . p9)}) finite finite-support Element of bool [:{p9},{(q9 . p9)}:]
{(q9 . p9)} is non empty trivial finite 1 -element set
[:{p9},{(q9 . p9)}:] is non empty Relation-like finite set
bool [:{p9},{(q9 . p9)}:] is non empty finite V38() set
(0_ (n,L)) +* (p9 .--> (q9 . p9)) is Relation-like Function-like set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support m is functional Element of bool (Bags n)
m is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m . u is Element of the carrier of L
dom (p9 .--> (q9 . p9)) is trivial functional finite Element of bool {p9}
(0_ (n,L)) . u is Element of the carrier of L
m is set
dom (p9 .--> (q9 . p9)) is trivial functional finite Element of bool {p9}
m . p9 is Element of the carrier of L
(p9 .--> (q9 . p9)) . p9 is set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
u is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
m + u is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
u is set
(m + u) . u is set
q9 . u is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (p9 .--> (q9 . p9)) is trivial functional finite Element of bool {p9}
u . p9 is Element of the carrier of L
(p9 .--> (q9 . p9)) . p9 is set
m . u is Element of the carrier of L
(m + u) . u is Element of the carrier of L
u . u is Element of the carrier of L
(m . u) + (u . u) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((m . u),(u . u)) is Element of the carrier of L
[(m . u),(u . u)] is set
the addF of L . [(m . u),(u . u)] is set
(0. L) + (q9 . p9) is Element of the carrier of L
the addF of L . ((0. L),(q9 . p9)) is Element of the carrier of L
[(0. L),(q9 . p9)] is set
the addF of L . [(0. L),(q9 . p9)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (p9 .--> (q9 . p9)) is trivial functional finite Element of bool {p9}
u . u is Element of the carrier of L
(0_ (n,L)) . u is Element of the carrier of L
(m + u) . u is Element of the carrier of L
m . u is Element of the carrier of L
(m . u) + (u . u) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((m . u),(u . u)) is Element of the carrier of L
[(m . u),(u . u)] is set
the addF of L . [(m . u),(u . u)] is set
q9 . u is Element of the carrier of L
(q9 . u) + (0. L) is Element of the carrier of L
the addF of L . ((q9 . u),(0. L)) is Element of the carrier of L
[(q9 . u),(0. L)] is set
the addF of L . [(q9 . u),(0. L)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (m + u) is non empty functional Element of bool (Bags n)
u + m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(u + m),q) is Element of the carrier of L
(n,L,m,q) is Element of the carrier of L
(n,L,u,q) is Element of the carrier of L
(n,L,m,q) + (n,L,u,q) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((n,L,m,q),(n,L,u,q)) is Element of the carrier of L
[(n,L,m,q),(n,L,u,q)] is set
the addF of L . [(n,L,m,q),(n,L,u,q)] is set
(n,L,m,q) * (n,L,x,q) is Element of the carrier of L
the multF of L . ((n,L,m,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,m,q),(n,L,x,q)] is set
the multF of L . [(n,L,m,q),(n,L,x,q)] is set
(n,L,u,q) * (n,L,x,q) is Element of the carrier of L
the multF of L . ((n,L,u,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,u,q),(n,L,x,q)] is set
the multF of L . [(n,L,u,q),(n,L,x,q)] is set
((n,L,m,q) * (n,L,x,q)) + ((n,L,u,q) * (n,L,x,q)) is Element of the carrier of L
the addF of L . (((n,L,m,q) * (n,L,x,q)),((n,L,u,q) * (n,L,x,q))) is Element of the carrier of L
[((n,L,m,q) * (n,L,x,q)),((n,L,u,q) * (n,L,x,q))] is set
the addF of L . [((n,L,m,q) * (n,L,x,q)),((n,L,u,q) * (n,L,x,q))] is set
m *' x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(m *' x),q) is Element of the carrier of L
(n,L,(m *' x),q) + ((n,L,u,q) * (n,L,x,q)) is Element of the carrier of L
the addF of L . ((n,L,(m *' x),q),((n,L,u,q) * (n,L,x,q))) is Element of the carrier of L
[(n,L,(m *' x),q),((n,L,u,q) * (n,L,x,q))] is set
the addF of L . [(n,L,(m *' x),q),((n,L,u,q) * (n,L,x,q))] is set
u *' x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(u *' x),q) is Element of the carrier of L
(n,L,(m *' x),q) + (n,L,(u *' x),q) is Element of the carrier of L
the addF of L . ((n,L,(m *' x),q),(n,L,(u *' x),q)) is Element of the carrier of L
[(n,L,(m *' x),q),(n,L,(u *' x),q)] is set
the addF of L . [(n,L,(m *' x),q),(n,L,(u *' x),q)] is set
(m *' x) + (u *' x) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,((m *' x) + (u *' x)),q) is Element of the carrier of L
x *' (m + u) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(x *' (m + u)),q) is Element of the carrier of L
Support p is functional finite Element of bool (Bags n)
card (Support p) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional finite Element of bool (Bags n)
card (Support p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 *' x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(p9 *' x),q) is Element of the carrier of L
(n,L,p9,q) is Element of the carrier of L
(n,L,p9,q) * (n,L,x,q) is Element of the carrier of L
the multF of L . ((n,L,p9,q),(n,L,x,q)) is Element of the carrier of L
[(n,L,p9,q),(n,L,x,q)] is set
the multF of L . [(n,L,p9,q),(n,L,x,q)] is set
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
0. L is V80(L) Element of the carrier of L
(0. L) * (n,L,x,q) is Element of the carrier of L
the multF of L . ((0. L),(n,L,x,q)) is Element of the carrier of L
[(0. L),(n,L,x,q)] is set
the multF of L . [(0. L),(n,L,x,q)] is set
p9 is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is epsilon-transitive epsilon-connected ordinal set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
x is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
x *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,(x *' f),p) is Element of the carrier of L
(n,L,x,p) is Element of the carrier of L
(n,L,f,p) is Element of the carrier of L
(n,L,x,p) * (n,L,f,p) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . ((n,L,x,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,x,p),(n,L,f,p)] is set
the multF of L . [(n,L,x,p),(n,L,f,p)] is set
Support x is functional finite Element of bool (Bags n)
bool (Bags n) is non empty set
card (Support x) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional finite Element of bool (Bags n)
card (Support p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
p9 *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(p9 *' f),p) is Element of the carrier of L
(n,L,p9,p) is Element of the carrier of L
(n,L,p9,p) * (n,L,f,p) is Element of the carrier of L
the multF of L . ((n,L,p9,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,f,p)] is set
the multF of L . [(n,L,p9,p),(n,L,f,p)] is set
BagOrder n is Relation-like Bags n -defined Bags n -valued total V27( Bags n, Bags n) reflexive antisymmetric transitive being_linear-order Element of bool [:(Bags n),(Bags n):]
[:(Bags n),(Bags n):] is non empty Relation-like set
bool [:(Bags n),(Bags n):] is non empty set
SgmX ((BagOrder n),(Support p9)) is Relation-like NAT -defined Bags n -valued Function-like finite FinSequence-like FinSubsequence-like Function-yielding V48() finite-support FinSequence of Bags n
len (SgmX ((BagOrder n),(Support p9))) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9)))) is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
Seg (len (SgmX ((BagOrder n),(Support p9)))) is finite len (SgmX ((BagOrder n),(Support p9))) -element Element of bool NAT
dom (SgmX ((BagOrder n),(Support p9))) is finite Element of bool NAT
(SgmX ((BagOrder n),(Support p9))) . (len (SgmX ((BagOrder n),(Support p9)))) is Relation-like Function-like set
rng (SgmX ((BagOrder n),(Support p9))) is finite set
p9 . ((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))) is Element of the carrier of L
0. L is V80(L) Element of the carrier of L
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(0_ (n,L)) +* (((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))),(p9 . ((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))))) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
p9 +* (((SgmX ((BagOrder n),(Support p9))) /. (len (SgmX ((BagOrder n),(Support p9))))),(0. L)) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
dom p9 is non empty functional Element of bool (Bags n)
bg is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
bg .--> (0. L) is Relation-like {bg} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{bg} is non empty trivial functional finite 1 -element set
{bg} --> (0. L) is non empty Relation-like {bg} -defined the carrier of L -valued {(0. L)} -valued Function-like constant total V27({bg},{(0. L)}) finite finite-support Element of bool [:{bg},{(0. L)}:]
{(0. L)} is non empty trivial finite 1 -element set
[:{bg},{(0. L)}:] is non empty Relation-like finite set
bool [:{bg},{(0. L)}:] is non empty finite V38() set
p9 +* (bg .--> (0. L)) is Relation-like Function-like set
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional Element of bool (Bags n)
p9 is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . m is Element of the carrier of L
(bg .--> (0. L)) . bg is set
p9 . m is Element of the carrier of L
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support p9 is functional finite Element of bool (Bags n)
(Support p9) \/ {bg} is non empty finite set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
p9 . m is Element of the carrier of L
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . m is Element of the carrier of L
dom (bg .--> (0. L)) is trivial functional finite Element of bool {bg}
bool {bg} is non empty finite V38() set
p9 . bg is Element of the carrier of L
(bg .--> (0. L)) . bg is set
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
p9 . m is Element of the carrier of L
p9 . m is Element of the carrier of L
card (Support p9) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
(card (Support p9)) + 1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real positive non negative Element of NAT
dom (0_ (n,L)) is non empty functional Element of bool (Bags n)
p9 . bg is Element of the carrier of L
bg .--> (p9 . bg) is Relation-like {bg} -defined the carrier of L -valued Function-like one-to-one finite finite-support set
{bg} --> (p9 . bg) is non empty Relation-like {bg} -defined the carrier of L -valued {(p9 . bg)} -valued Function-like constant total V27({bg},{(p9 . bg)}) finite finite-support Element of bool [:{bg},{(p9 . bg)}:]
{(p9 . bg)} is non empty trivial finite 1 -element set
[:{bg},{(p9 . bg)}:] is non empty Relation-like finite set
bool [:{bg},{(p9 . bg)}:] is non empty finite V38() set
(0_ (n,L)) +* (bg .--> (p9 . bg)) is Relation-like Function-like set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) Element of bool [:(Bags n), the carrier of L:]
Support m is functional Element of bool (Bags n)
m is set
m is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support Element of Bags n
m . m is Element of the carrier of L
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
(0_ (n,L)) . m is Element of the carrier of L
m is set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . bg is Element of the carrier of L
(bg .--> (p9 . bg)) . bg is set
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p9 + m is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
u is set
(p9 + m) . u is set
p9 . u is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . bg is Element of the carrier of L
(bg .--> (p9 . bg)) . bg is set
p9 . u is Element of the carrier of L
(p9 + m) . u is Element of the carrier of L
m . u is Element of the carrier of L
(p9 . u) + (m . u) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((p9 . u),(m . u)) is Element of the carrier of L
[(p9 . u),(m . u)] is set
the addF of L . [(p9 . u),(m . u)] is set
(0. L) + (p9 . bg) is Element of the carrier of L
the addF of L . ((0. L),(p9 . bg)) is Element of the carrier of L
[(0. L),(p9 . bg)] is set
the addF of L . [(0. L),(p9 . bg)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (bg .--> (p9 . bg)) is trivial functional finite Element of bool {bg}
m . u is Element of the carrier of L
(0_ (n,L)) . u is Element of the carrier of L
(p9 + m) . u is Element of the carrier of L
p9 . u is Element of the carrier of L
(p9 . u) + (m . u) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((p9 . u),(m . u)) is Element of the carrier of L
[(p9 . u),(m . u)] is set
the addF of L . [(p9 . u),(m . u)] is set
p9 . u is Element of the carrier of L
(p9 . u) + (0. L) is Element of the carrier of L
the addF of L . ((p9 . u),(0. L)) is Element of the carrier of L
[(p9 . u),(0. L)] is set
the addF of L . [(p9 . u),(0. L)] is set
u is Relation-like n -defined RAT -valued Function-like total V211() V212() V213() V214() finite-support set
dom (p9 + m) is non empty functional Element of bool (Bags n)
m + p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(m + p9),p) is Element of the carrier of L
(n,L,p9,p) is Element of the carrier of L
(n,L,m,p) is Element of the carrier of L
(n,L,p9,p) + (n,L,m,p) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
the addF of L . ((n,L,p9,p),(n,L,m,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,m,p)] is set
the addF of L . [(n,L,p9,p),(n,L,m,p)] is set
(n,L,p9,p) * (n,L,f,p) is Element of the carrier of L
the multF of L . ((n,L,p9,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,p9,p),(n,L,f,p)] is set
the multF of L . [(n,L,p9,p),(n,L,f,p)] is set
(n,L,m,p) * (n,L,f,p) is Element of the carrier of L
the multF of L . ((n,L,m,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,m,p),(n,L,f,p)] is set
the multF of L . [(n,L,m,p),(n,L,f,p)] is set
((n,L,p9,p) * (n,L,f,p)) + ((n,L,m,p) * (n,L,f,p)) is Element of the carrier of L
the addF of L . (((n,L,p9,p) * (n,L,f,p)),((n,L,m,p) * (n,L,f,p))) is Element of the carrier of L
[((n,L,p9,p) * (n,L,f,p)),((n,L,m,p) * (n,L,f,p))] is set
the addF of L . [((n,L,p9,p) * (n,L,f,p)),((n,L,m,p) * (n,L,f,p))] is set
p9 *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(p9 *' f),p) is Element of the carrier of L
(n,L,(p9 *' f),p) + ((n,L,m,p) * (n,L,f,p)) is Element of the carrier of L
the addF of L . ((n,L,(p9 *' f),p),((n,L,m,p) * (n,L,f,p))) is Element of the carrier of L
[(n,L,(p9 *' f),p),((n,L,m,p) * (n,L,f,p))] is set
the addF of L . [(n,L,(p9 *' f),p),((n,L,m,p) * (n,L,f,p))] is set
m *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(m *' f),p) is Element of the carrier of L
(n,L,(p9 *' f),p) + (n,L,(m *' f),p) is Element of the carrier of L
the addF of L . ((n,L,(p9 *' f),p),(n,L,(m *' f),p)) is Element of the carrier of L
[(n,L,(p9 *' f),p),(n,L,(m *' f),p)] is set
the addF of L . [(n,L,(p9 *' f),p),(n,L,(m *' f),p)] is set
(p9 *' f) + (m *' f) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,((p9 *' f) + (m *' f)),p) is Element of the carrier of L
f *' (p9 + m) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(f *' (p9 + m)),p) is Element of the carrier of L
q is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Support q is functional finite Element of bool (Bags n)
card (Support q) is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
q *' f is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,(q *' f),p) is Element of the carrier of L
(n,L,q,p) is Element of the carrier of L
(n,L,q,p) * (n,L,f,p) is Element of the carrier of L
the multF of L . ((n,L,q,p),(n,L,f,p)) is Element of the carrier of L
[(n,L,q,p),(n,L,f,p)] is set
the multF of L . [(n,L,q,p),(n,L,f,p)] is set
0_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
0. L is V80(L) Element of the carrier of L
(0. L) * (n,L,f,p) is Element of the carrier of L
the multF of L . ((0. L),(n,L,f,p)) is Element of the carrier of L
[(0. L),(n,L,f,p)] is set
the multF of L . [(0. L),(n,L,f,p)] is set
q is epsilon-transitive epsilon-connected ordinal natural V11() V12() finite cardinal V45() ext-real non negative Element of NAT
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of (Polynom-Ring (n,L)) is non empty set
[: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty Relation-like set
bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
f is set
p is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,p,x) is Element of the carrier of L
f is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
p is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
q is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p . q is set
(n,L,q,x) is Element of the carrier of L
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,p9,x) is Element of the carrier of L
f is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
p is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
q9 is set
q is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
p is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
q . p is set
(n,L,p,x) is Element of the carrier of L
p9 is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
p9 . p is set
q . q9 is set
p9 . q9 is set
dom p9 is non empty Element of bool the carrier of (Polynom-Ring (n,L))
bool the carrier of (Polynom-Ring (n,L)) is non empty set
dom q is non empty Element of bool the carrier of (Polynom-Ring (n,L))
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative Abelian add-associative right_zeroed right-distributive right_unital well-unital left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative Abelian add-associative right_zeroed right-distributive right_unital well-unital left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,x) is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
the carrier of (Polynom-Ring (n,L)) is non empty set
[: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty Relation-like set
bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty set
1_ (Polynom-Ring (n,L)) is Element of the carrier of (Polynom-Ring (n,L))
1. (Polynom-Ring (n,L)) is Element of the carrier of (Polynom-Ring (n,L))
(n,L,x) . (1_ (Polynom-Ring (n,L))) is Element of the carrier of L
1_ (n,L) is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
(n,L,x) . (1_ (n,L)) is set
(n,L,(1_ (n,L)),x) is Element of the carrier of L
1_ L is Element of the carrier of L
1. L is V80(L) Element of the carrier of L
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict Abelian add-associative right_zeroed left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,x) is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
the carrier of (Polynom-Ring (n,L)) is non empty set
[: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty Relation-like set
bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty set
p is Element of the carrier of (Polynom-Ring (n,L))
q is Element of the carrier of (Polynom-Ring (n,L))
p + q is Element of the carrier of (Polynom-Ring (n,L))
the addF of (Polynom-Ring (n,L)) is non empty Relation-like [: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):] -defined the carrier of (Polynom-Ring (n,L)) -valued Function-like total V27([: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L))) Element of bool [:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):]
[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):] is non empty Relation-like set
[:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):] is non empty Relation-like set
bool [:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):] is non empty set
the addF of (Polynom-Ring (n,L)) . (p,q) is Element of the carrier of (Polynom-Ring (n,L))
[p,q] is set
the addF of (Polynom-Ring (n,L)) . [p,q] is set
(n,L,x) . (p + q) is Element of the carrier of L
(n,L,x) . p is Element of the carrier of L
(n,L,x) . q is Element of the carrier of L
((n,L,x) . p) + ((n,L,x) . q) is Element of the carrier of L
the addF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the addF of L . (((n,L,x) . p),((n,L,x) . q)) is Element of the carrier of L
[((n,L,x) . p),((n,L,x) . q)] is set
the addF of L . [((n,L,x) . p),((n,L,x) . q)] is set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
p is Element of the carrier of (Polynom-Ring (n,L))
(n,L,x) . p is Element of the carrier of L
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,p9,x) is Element of the carrier of L
q is Element of the carrier of (Polynom-Ring (n,L))
p + q is Element of the carrier of (Polynom-Ring (n,L))
the addF of (Polynom-Ring (n,L)) . (p,q) is Element of the carrier of (Polynom-Ring (n,L))
[p,q] is set
the addF of (Polynom-Ring (n,L)) . [p,q] is set
(n,L,x) . (p + q) is Element of the carrier of L
q9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p9 + q9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,x) . (p9 + q9) is set
(n,L,(p9 + q9),x) is Element of the carrier of L
(n,L,q9,x) is Element of the carrier of L
(n,L,p9,x) + (n,L,q9,x) is Element of the carrier of L
the addF of L . ((n,L,p9,x),(n,L,q9,x)) is Element of the carrier of L
[(n,L,p9,x),(n,L,q9,x)] is set
the addF of L . [(n,L,p9,x),(n,L,q9,x)] is set
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative Abelian add-associative right_zeroed right-distributive right_unital well-unital left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,x) is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) unity-preserving additive Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
the carrier of (Polynom-Ring (n,L)) is non empty set
[: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty Relation-like set
bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty set
p is Element of the carrier of (Polynom-Ring (n,L))
q is Element of the carrier of (Polynom-Ring (n,L))
p * q is Element of the carrier of (Polynom-Ring (n,L))
the multF of (Polynom-Ring (n,L)) is non empty Relation-like [: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):] -defined the carrier of (Polynom-Ring (n,L)) -valued Function-like total V27([: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L))) Element of bool [:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):]
[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):] is non empty Relation-like set
[:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):] is non empty Relation-like set
bool [:[: the carrier of (Polynom-Ring (n,L)), the carrier of (Polynom-Ring (n,L)):], the carrier of (Polynom-Ring (n,L)):] is non empty set
the multF of (Polynom-Ring (n,L)) . (p,q) is Element of the carrier of (Polynom-Ring (n,L))
[p,q] is set
the multF of (Polynom-Ring (n,L)) . [p,q] is set
(n,L,x) . (p * q) is Element of the carrier of L
(n,L,x) . p is Element of the carrier of L
(n,L,x) . q is Element of the carrier of L
((n,L,x) . p) * ((n,L,x) . q) is Element of the carrier of L
the multF of L is non empty Relation-like [: the carrier of L, the carrier of L:] -defined the carrier of L -valued Function-like total V27([: the carrier of L, the carrier of L:], the carrier of L) Element of bool [:[: the carrier of L, the carrier of L:], the carrier of L:]
[: the carrier of L, the carrier of L:] is non empty Relation-like set
[:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty Relation-like set
bool [:[: the carrier of L, the carrier of L:], the carrier of L:] is non empty set
the multF of L . (((n,L,x) . p),((n,L,x) . q)) is Element of the carrier of L
[((n,L,x) . p),((n,L,x) . q)] is set
the multF of L . [((n,L,x) . p),((n,L,x) . q)] is set
Bags n is non empty functional Element of bool (Bags n)
Bags n is non empty set
bool (Bags n) is non empty set
[:(Bags n), the carrier of L:] is non empty Relation-like set
bool [:(Bags n), the carrier of L:] is non empty set
p is Element of the carrier of (Polynom-Ring (n,L))
(n,L,x) . p is Element of the carrier of L
p9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,p9,x) is Element of the carrier of L
q is Element of the carrier of (Polynom-Ring (n,L))
p * q is Element of the carrier of (Polynom-Ring (n,L))
the multF of (Polynom-Ring (n,L)) . (p,q) is Element of the carrier of (Polynom-Ring (n,L))
[p,q] is set
the multF of (Polynom-Ring (n,L)) . [p,q] is set
(n,L,x) . (p * q) is Element of the carrier of L
q9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
p9 *' q9 is non empty Relation-like Bags n -defined the carrier of L -valued Function-like total V27( Bags n, the carrier of L) finite-Support Element of bool [:(Bags n), the carrier of L:]
(n,L,x) . (p9 *' q9) is set
(n,L,(p9 *' q9),x) is Element of the carrier of L
(n,L,q9,x) is Element of the carrier of L
(n,L,p9,x) * (n,L,q9,x) is Element of the carrier of L
the multF of L . ((n,L,p9,x),(n,L,q9,x)) is Element of the carrier of L
[(n,L,p9,x),(n,L,q9,x)] is set
the multF of L . [(n,L,p9,x),(n,L,q9,x)] is set
n is epsilon-transitive epsilon-connected ordinal set
L is non empty non degenerated non trivial left_add-cancelable right_add-cancelable right_complementable unital associative commutative Abelian add-associative right_zeroed right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of L is non empty non trivial set
[:n, the carrier of L:] is Relation-like set
bool [:n, the carrier of L:] is non empty set
Polynom-Ring (n,L) is non empty left_add-cancelable right_add-cancelable right_complementable strict unital associative commutative Abelian add-associative right_zeroed right-distributive right_unital well-unital left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
x is Relation-like n -defined the carrier of L -valued Function-like total V27(n, the carrier of L) Element of bool [:n, the carrier of L:]
(n,L,x) is non empty Relation-like the carrier of (Polynom-Ring (n,L)) -defined the carrier of L -valued Function-like total V27( the carrier of (Polynom-Ring (n,L)), the carrier of L) unity-preserving multiplicative additive Element of bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:]
the carrier of (Polynom-Ring (n,L)) is non empty set
[: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty Relation-like set
bool [: the carrier of (Polynom-Ring (n,L)), the carrier of L:] is non empty set