REAL is V158() V159() V160() V164() V194() V195() V197() set
NAT is non empty non trivial V29() non finite cardinal limit_cardinal V158() V159() V160() V161() V162() V163() V164() V192() V194() Element of bool REAL
bool REAL is non empty set
NAT is non empty non trivial V29() non finite cardinal limit_cardinal V158() V159() V160() V161() V162() V163() V164() V192() V194() set
bool NAT is non empty non trivial non finite set
bool NAT is non empty non trivial non finite set
2 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
[:2,2:] is non empty Relation-like RAT -valued INT -valued finite V148() V149() V150() V151() set
RAT is V158() V159() V160() V161() V164() set
INT is V158() V159() V160() V161() V162() V164() set
[:[:2,2:],2:] is non empty Relation-like RAT -valued INT -valued finite V148() V149() V150() V151() set
bool [:[:2,2:],2:] is non empty finite V39() set
COMPLEX is V158() V164() set
INT.Ring is non empty non degenerated non trivial non finite left_add-cancelable right_add-cancelable add-cancelable right_complementable strict Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like domRing-like gcd-like Euclidian doubleLoopStr
the carrier of INT.Ring is non empty non trivial non finite V158() V159() V160() V161() V162() set
[: the carrier of INT.Ring,NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V148() V149() V150() V151() set
bool [: the carrier of INT.Ring,NAT:] is non empty non trivial non finite set
{} is empty trivial Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() set
the empty trivial Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() set is empty trivial Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() set
1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
3 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
0 is empty trivial Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V147() V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() Element of NAT
Seg 1 is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() set
Seg 2 is non empty finite 2 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1,2} is non empty finite V39() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() set
{{}} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() set
the non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
R is non empty add-associative right_zeroed left_zeroed addLoopStr
the carrier of R is non empty set
I is Element of the carrier of R
D is Element of the carrier of R
<*I,D*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum <*I,D*> is Element of the carrier of R
I + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (I,D) is Element of the carrier of R
[I,D] is non empty V18() set
{I,D} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,D},{I}} is non empty finite V39() set
the addF of R . [I,D] is set
<*I*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
<*D*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
<*I*> ^ <*D*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite 1 + 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
1 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Sum <*I*> is Element of the carrier of R
Sum <*D*> is Element of the carrier of R
(Sum <*I*>) + (Sum <*D*>) is Element of the carrier of R
the addF of R . ((Sum <*I*>),(Sum <*D*>)) is Element of the carrier of R
[(Sum <*I*>),(Sum <*D*>)] is non empty V18() set
{(Sum <*I*>),(Sum <*D*>)} is non empty finite set
{(Sum <*I*>)} is non empty trivial finite 1 -element set
{{(Sum <*I*>),(Sum <*D*>)},{(Sum <*I*>)}} is non empty finite V39() set
the addF of R . [(Sum <*I*>),(Sum <*D*>)] is set
I + (Sum <*D*>) is Element of the carrier of R
the addF of R . (I,(Sum <*D*>)) is Element of the carrier of R
[I,(Sum <*D*>)] is non empty V18() set
{I,(Sum <*D*>)} is non empty finite set
{{I,(Sum <*D*>)},{I}} is non empty finite V39() set
the addF of R . [I,(Sum <*D*>)] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
R is non empty multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
R is non empty multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
f is Element of the carrier of R
e9 is Element of the carrier of R
e9 * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
f is Element of the carrier of R
e9 is Element of the carrier of R
f * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
D is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
f is Element of the carrier of R
e9 is Element of the carrier of R
e9 * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
R is non empty commutative multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
I is non empty Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
I is non empty (R) Element of bool the carrier of R
the Element of I is Element of I
(0. R) * the Element of I is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
R is non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
I is non empty (R) Element of bool the carrier of R
the Element of I is Element of I
the Element of I * (0. R) is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of I,(0. R)) is right_add-cancelable Element of the carrier of R
[ the Element of I,(0. R)] is non empty V18() set
{ the Element of I,(0. R)} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I,(0. R)},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I,(0. R)] is set
R is non empty right_zeroed addLoopStr
the carrier of R is non empty set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
I is Element of the carrier of R
D is Element of the carrier of R
I + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (I,D) is Element of the carrier of R
[I,D] is non empty V18() set
{I,D} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,D},{I}} is non empty finite V39() set
the addF of R . [I,D] is set
R is non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
D is right_add-cancelable Element of the carrier of R
I is right_add-cancelable Element of the carrier of R
I * D is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,D) is right_add-cancelable Element of the carrier of R
[I,D] is non empty V18() set
{I,D} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,D},{I}} is non empty finite V39() set
the multF of R . [I,D] is set
e is right_add-cancelable Element of the carrier of R
e * D is right_add-cancelable Element of the carrier of R
the multF of R . (e,D) is right_add-cancelable Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
D is left_add-cancelable Element of the carrier of R
I is left_add-cancelable Element of the carrier of R
D * I is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,I) is left_add-cancelable Element of the carrier of R
[D,I] is non empty V18() set
{D,I} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,I},{D}} is non empty finite V39() set
the multF of R . [D,I] is set
e is left_add-cancelable Element of the carrier of R
D * e is left_add-cancelable Element of the carrier of R
the multF of R . (D,e) is left_add-cancelable Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
R is non empty right_zeroed addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
I is Element of bool the carrier of R
R is non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
I is Element of bool the carrier of R
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) Element of bool the carrier of R
I is Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive left-distributive distributive left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left-distributive left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
e is Element of the carrier of R
D is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
D is Element of the carrier of R
e is Element of the carrier of R
D + e is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the addF of R . [D,e] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
D is Element of the carrier of R
e is Element of the carrier of R
D + e is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the addF of R . [D,e] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
e is Element of the carrier of R
D is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
D is Element of the carrier of R
e is Element of the carrier of R
D + e is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the addF of R . [D,e] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
D is set
{D} is non empty trivial finite 1 -element set
R is non empty non trivial left_add-cancelable right_add-cancelable add-cancelable right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty non trivial set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
I is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left-distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
1. R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- (1. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(- (1. R)) * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((- (1. R)),D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(- (1. R)),D] is non empty V18() set
{(- (1. R)),D} is non empty finite set
{(- (1. R))} is non empty trivial finite 1 -element set
{{(- (1. R)),D},{(- (1. R))}} is non empty finite V39() set
the multF of R . [(- (1. R)),D] is set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . ((0. R),D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(0. R),D] is non empty V18() set
{(0. R),D} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),D},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R),D] is set
(1. R) + (- (1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((1. R),(- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(1. R),(- (1. R))] is non empty V18() set
{(1. R),(- (1. R))} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),(- (1. R))},{(1. R)}} is non empty finite V39() set
the addF of R . [(1. R),(- (1. R))] is set
((1. R) + (- (1. R))) * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (((1. R) + (- (1. R))),D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[((1. R) + (- (1. R))),D] is non empty V18() set
{((1. R) + (- (1. R))),D} is non empty finite set
{((1. R) + (- (1. R)))} is non empty trivial finite 1 -element set
{{((1. R) + (- (1. R))),D},{((1. R) + (- (1. R)))}} is non empty finite V39() set
the multF of R . [((1. R) + (- (1. R))),D] is set
(1. R) * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . ((1. R),D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
((1. R) * D) + ((- (1. R)) * D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (((1. R) * D),((- (1. R)) * D)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[((1. R) * D),((- (1. R)) * D)] is non empty V18() set
{((1. R) * D),((- (1. R)) * D)} is non empty finite set
{((1. R) * D)} is non empty trivial finite 1 -element set
{{((1. R) * D),((- (1. R)) * D)},{((1. R) * D)}} is non empty finite V39() set
the addF of R . [((1. R) * D),((- (1. R)) * D)] is set
D + ((- (1. R)) * D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (D,((- (1. R)) * D)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,((- (1. R)) * D)] is non empty V18() set
{D,((- (1. R)) * D)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,((- (1. R)) * D)},{D}} is non empty finite V39() set
the addF of R . [D,((- (1. R)) * D)] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive right_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
1. R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- (1. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D * (- (1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,(- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(- (1. R))] is non empty V18() set
{D,(- (1. R))} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(- (1. R))},{D}} is non empty finite V39() set
the multF of R . [D,(- (1. R))] is set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D * (0. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (D,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(0. R)] is non empty V18() set
{D,(0. R)} is non empty finite set
{{D,(0. R)},{D}} is non empty finite V39() set
the multF of R . [D,(0. R)] is set
(1. R) + (- (1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((1. R),(- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(1. R),(- (1. R))] is non empty V18() set
{(1. R),(- (1. R))} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),(- (1. R))},{(1. R)}} is non empty finite V39() set
the addF of R . [(1. R),(- (1. R))] is set
D * ((1. R) + (- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (D,((1. R) + (- (1. R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,((1. R) + (- (1. R)))] is non empty V18() set
{D,((1. R) + (- (1. R)))} is non empty finite set
{{D,((1. R) + (- (1. R)))},{D}} is non empty finite V39() set
the multF of R . [D,((1. R) + (- (1. R)))] is set
D * (1. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (D,(1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(1. R)] is non empty V18() set
{D,(1. R)} is non empty finite set
{{D,(1. R)},{D}} is non empty finite V39() set
the multF of R . [D,(1. R)] is set
(D * (1. R)) + (D * (- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((D * (1. R)),(D * (- (1. R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(D * (1. R)),(D * (- (1. R)))] is non empty V18() set
{(D * (1. R)),(D * (- (1. R)))} is non empty finite set
{(D * (1. R))} is non empty trivial finite 1 -element set
{{(D * (1. R)),(D * (- (1. R)))},{(D * (1. R))}} is non empty finite V39() set
the addF of R . [(D * (1. R)),(D * (- (1. R)))] is set
D + (D * (- (1. R))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (D,(D * (- (1. R)))) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(D * (- (1. R)))] is non empty V18() set
{D,(D * (- (1. R)))} is non empty finite set
{{D,(D * (- (1. R)))},{D}} is non empty finite V39() set
the addF of R . [D,(D * (- (1. R)))] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed left-distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D - e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D + (- e) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,(- e)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(- e)] is non empty V18() set
{D,(- e)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(- e)},{D}} is non empty finite V39() set
the addF of R . [D,(- e)] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive right_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D - e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
- e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D + (- e) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,(- e)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,(- e)] is non empty V18() set
{D,(- e)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(- e)},{D}} is non empty finite V39() set
the addF of R . [D,(- e)] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is Element of I
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(e + 1) * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
1 * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(1 * D) + (e * D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((1 * D),(e * D)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(1 * D),(e * D)] is non empty V18() set
{(1 * D),(e * D)} is non empty finite set
{(1 * D)} is non empty trivial finite 1 -element set
{{(1 * D),(e * D)},{(1 * D)}} is non empty finite V39() set
the addF of R . [(1 * D),(e * D)] is set
D + (e * D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (D,(e * D)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(e * D)] is non empty V18() set
{D,(e * D)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(e * D)},{D}} is non empty finite V39() set
the addF of R . [D,(e * D)] is set
0 * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is Element of I
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
D |^ e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D |^ (e + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D |^ 1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(D |^ e) * (D |^ 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((D |^ e),(D |^ 1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(D |^ e),(D |^ 1)] is non empty V18() set
{(D |^ e),(D |^ 1)} is non empty finite set
{(D |^ e)} is non empty trivial finite 1 -element set
{{(D |^ e),(D |^ 1)},{(D |^ e)}} is non empty finite V39() set
the multF of R . [(D |^ e),(D |^ 1)] is set
D |^ 1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
I is non empty (R) Element of bool the carrier of R
the addF of R || I is Relation-like Function-like set
[:I,I:] is non empty Relation-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
[:[:I,I:], the carrier of R:] is non empty Relation-like set
bool [:[:I,I:], the carrier of R:] is non empty set
D is non empty Relation-like [:I,I:] -defined the carrier of R -valued Function-like total V21([:I,I:], the carrier of R) Element of bool [:[:I,I:], the carrier of R:]
dom D is non empty Relation-like I -defined I -valued Element of bool [:I,I:]
bool [:I,I:] is non empty set
e is set
D . e is set
e is set
D is set
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
e9 is Element of the carrier of R
f is Element of the carrier of R
F is Element of the carrier of R
i is Element of the carrier of R
F + i is Element of the carrier of R
the addF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
R is non empty multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
I is non empty (R) Element of bool the carrier of R
the multF of R || I is Relation-like Function-like set
[:I,I:] is non empty Relation-like set
the multF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
[:[:I,I:], the carrier of R:] is non empty Relation-like set
bool [:[:I,I:], the carrier of R:] is non empty set
D is non empty Relation-like [:I,I:] -defined the carrier of R -valued Function-like total V21([:I,I:], the carrier of R) Element of bool [:[:I,I:], the carrier of R:]
dom D is non empty Relation-like I -defined I -valued Element of bool [:I,I:]
bool [:I,I:] is non empty set
e is set
D . e is set
e is set
D is set
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
e9 is Element of I
f is Element of I
[e9,f] is non empty V18() Element of [:I,I:]
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
e9 * f is Element of the carrier of R
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
the multF of R . [e9,f] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
(R,I) is non empty Relation-like [:I,I:] -defined I -valued Function-like total V21([:I,I:],I) Element of bool [:[:I,I:],I:]
[:I,I:] is non empty Relation-like set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R || I is Relation-like Function-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
In ((0. R),I) is Element of I
addLoopStr(# I,(R,I),(In ((0. R),I)) #) is non empty strict addLoopStr
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
(R,I) is non empty addLoopStr
(R,I) is non empty Relation-like [:I,I:] -defined I -valued Function-like total V21([:I,I:],I) Element of bool [:[:I,I:],I:]
[:I,I:] is non empty Relation-like set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R || I is Relation-like Function-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
In ((0. R),I) is Element of I
addLoopStr(# I,(R,I),(In ((0. R),I)) #) is non empty strict addLoopStr
dom the addF of R is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
e is set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
dom ( the addF of R || I) is set
e is non empty addLoopStr
the carrier of e is non empty set
e is Element of the carrier of e
D is Element of the carrier of e
e + D is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,D) is Element of the carrier of e
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of e . [e,D] is set
e9 is Element of I
f is Element of I
e9 + f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
[e9,f] is non empty V18() Element of [:I,I:]
e is Element of the carrier of e
D is Element of the carrier of e
e9 is Element of the carrier of e
f is Element of I
F is Element of I
f + F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (f,F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the addF of R . [f,F] is set
i is Element of I
[(f + F),i] is non empty V18() Element of [: the carrier of R,I:]
[: the carrier of R,I:] is non empty Relation-like set
{(f + F),i} is non empty finite set
{(f + F)} is non empty trivial finite 1 -element set
{{(f + F),i},{(f + F)}} is non empty finite V39() set
F + i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (F,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
[f,(F + i)] is non empty V18() Element of [:I, the carrier of R:]
[:I, the carrier of R:] is non empty Relation-like set
{f,(F + i)} is non empty finite set
{{f,(F + i)},{f}} is non empty finite V39() set
e + D is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,D) is Element of the carrier of e
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of e . [e,D] is set
(e + D) + e9 is Element of the carrier of e
the addF of e . ((e + D),e9) is Element of the carrier of e
[(e + D),e9] is non empty V18() set
{(e + D),e9} is non empty finite set
{(e + D)} is non empty trivial finite 1 -element set
{{(e + D),e9},{(e + D)}} is non empty finite V39() set
the addF of e . [(e + D),e9] is set
( the addF of R || I) . [(f + F),i] is set
(f + F) + i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((f + F),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(f + F),i] is non empty V18() set
the addF of R . [(f + F),i] is set
f + (F + i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (f,(F + i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,(F + i)] is non empty V18() set
the addF of R . [f,(F + i)] is set
(R,I) . [f,(F + i)] is set
D + e9 is Element of the carrier of e
the addF of e . (D,e9) is Element of the carrier of e
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of e . [D,e9] is set
e + (D + e9) is Element of the carrier of e
the addF of e . (e,(D + e9)) is Element of the carrier of e
[e,(D + e9)] is non empty V18() set
{e,(D + e9)} is non empty finite set
{{e,(D + e9)},{e}} is non empty finite V39() set
the addF of e . [e,(D + e9)] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
(R,I) is non empty add-associative addLoopStr
(R,I) is non empty Relation-like [:I,I:] -defined I -valued Function-like total V21([:I,I:],I) Element of bool [:[:I,I:],I:]
[:I,I:] is non empty Relation-like set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R || I is Relation-like Function-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
In ((0. R),I) is Element of I
addLoopStr(# I,(R,I),(In ((0. R),I)) #) is non empty strict addLoopStr
dom the addF of R is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
e is set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
dom ( the addF of R || I) is set
e is non empty addLoopStr
the carrier of e is non empty set
e is Element of the carrier of e
D is Element of I
[D,(0. R)] is non empty V18() Element of [:I, the carrier of R:]
[:I, the carrier of R:] is non empty Relation-like set
{D,(0. R)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(0. R)},{D}} is non empty finite V39() set
0. e is zero Element of the carrier of e
the ZeroF of e is Element of the carrier of e
e + (0. e) is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,(0. e)) is Element of the carrier of e
[e,(0. e)] is non empty V18() set
{e,(0. e)} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,(0. e)},{e}} is non empty finite V39() set
the addF of e . [e,(0. e)] is set
( the addF of R || I) . [D,(0. R)] is set
D + (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (D,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(0. R)] is non empty V18() set
the addF of R . [D,(0. R)] is set
R is non empty Abelian doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
(R,I) is non empty addLoopStr
(R,I) is non empty Relation-like [:I,I:] -defined I -valued Function-like total V21([:I,I:],I) Element of bool [:[:I,I:],I:]
[:I,I:] is non empty Relation-like set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R || I is Relation-like Function-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
In ((0. R),I) is Element of I
addLoopStr(# I,(R,I),(In ((0. R),I)) #) is non empty strict addLoopStr
dom the addF of R is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
e is set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
dom ( the addF of R || I) is set
e is non empty addLoopStr
the carrier of e is non empty set
e is Element of the carrier of e
D is Element of the carrier of e
e + D is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,D) is Element of the carrier of e
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of e . [e,D] is set
e9 is Element of I
f is Element of I
e9 + f is Element of the carrier of R
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
[e9,f] is non empty V18() Element of [:I,I:]
e is Element of the carrier of e
D is Element of the carrier of e
e + D is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,D) is Element of the carrier of e
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of e . [e,D] is set
e9 is Element of I
f is Element of I
e9 + f is Element of the carrier of R
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
D + e is Element of the carrier of e
the addF of e . (D,e) is Element of the carrier of e
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the addF of e . [D,e] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed right-distributive left-distributive right_unital distributive left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
(R,I) is non empty Abelian add-associative right_zeroed addLoopStr
(R,I) is non empty Relation-like [:I,I:] -defined I -valued Function-like total V21([:I,I:],I) Element of bool [:[:I,I:],I:]
[:I,I:] is non empty Relation-like set
[:[:I,I:],I:] is non empty Relation-like set
bool [:[:I,I:],I:] is non empty set
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R || I is Relation-like Function-like set
the addF of R | [:I,I:] is Relation-like [:I,I:] -defined [: the carrier of R, the carrier of R:] -defined the carrier of R -valued set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
In ((0. R),I) is Element of I
addLoopStr(# I,(R,I),(In ((0. R),I)) #) is non empty strict addLoopStr
dom the addF of R is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
bool [: the carrier of R, the carrier of R:] is non empty set
e is set
[:I,I:] is non empty Relation-like the carrier of R -defined the carrier of R -valued Element of bool [: the carrier of R, the carrier of R:]
dom ( the addF of R || I) is set
e is non empty addLoopStr
the carrier of e is non empty set
e is Element of the carrier of e
D is Element of the carrier of e
e + D is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (e,D) is Element of the carrier of e
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of e . [e,D] is set
e9 is Element of I
f is Element of I
e9 + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
[e9,f] is non empty V18() Element of [:I,I:]
D is Element of the carrier of e
e is non empty (R) (R) Element of bool the carrier of R
e9 is Element of e
- e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is Element of the carrier of e
D + f is Element of the carrier of e
the addF of e is non empty Relation-like [: the carrier of e, the carrier of e:] -defined the carrier of e -valued Function-like total V21([: the carrier of e, the carrier of e:], the carrier of e) Element of bool [:[: the carrier of e, the carrier of e:], the carrier of e:]
[: the carrier of e, the carrier of e:] is non empty Relation-like set
[:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty Relation-like set
bool [:[: the carrier of e, the carrier of e:], the carrier of e:] is non empty set
the addF of e . (D,f) is Element of the carrier of e
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the addF of e . [D,f] is set
e9 + (- e9) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (e9,(- e9)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[e9,(- e9)] is non empty V18() set
{e9,(- e9)} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,(- e9)},{e9}} is non empty finite V39() set
the addF of R . [e9,(- e9)] is set
0. e is zero Element of the carrier of e
the ZeroF of e is Element of the carrier of e
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative 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 R is non empty set
bool the carrier of R is non empty set
I is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{ (I * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R : verum } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,e) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,e] is non empty V18() set
{I,e} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,e},{I}} is non empty finite V39() set
the multF of R . [I,e] is set
1. R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * (1. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,(1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,(1. R)] is non empty V18() set
{I,(1. R)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(1. R)},{I}} is non empty finite V39() set
the multF of R . [I,(1. R)] is set
e is non empty Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e is non empty Element of bool the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,f] is non empty V18() set
{I,f} is non empty finite set
{{I,f},{I}} is non empty finite V39() set
the multF of R . [I,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,F) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,F] is non empty V18() set
{I,F} is non empty finite set
{{I,F},{I}} is non empty finite V39() set
the multF of R . [I,F] is set
D + e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . (D,e9) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f + F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (f,F) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the addF of R . [f,F] is set
I * (f + F) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,(f + F)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,(f + F)] is non empty V18() set
{I,(f + F)} is non empty finite set
{{I,(f + F)},{I}} is non empty finite V39() set
the multF of R . [I,(f + F)] is set
e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,f] is non empty V18() set
{I,f} is non empty finite set
{{I,f},{I}} is non empty finite V39() set
the multF of R . [I,f] is set
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D * e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (D,e9) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
D * f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (D,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
(D * f) * I is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . ((D * f),I) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(D * f),I] is non empty V18() set
{(D * f),I} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),I},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),I] is set
e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,f] is non empty V18() set
{I,f} is non empty finite set
{{I,f},{I}} is non empty finite V39() set
the multF of R . [I,f] is set
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e9 * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (e9,D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (f,D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[f,D] is non empty V18() set
{f,D} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,D},{f}} is non empty finite V39() set
the multF of R . [f,D] is set
I * (f * D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,(f * D)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,(f * D)] is non empty V18() set
{I,(f * D)} is non empty finite set
{{I,(f * D)},{I}} is non empty finite V39() set
the multF of R . [I,(f * D)] is set
R is non empty right_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
I is non empty (R) Element of bool the carrier of R
D is set
e is Element of the carrier of R
e * (1. R) is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,(1. R)) is Element of the carrier of R
[e,(1. R)] is non empty V18() set
{e,(1. R)} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,(1. R)},{e}} is non empty finite V39() set
the multF of R . [e,(1. R)] is set
D is set
R is non empty unital right_unital left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
D is set
e is Element of the carrier of R
(1. R) * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),e) is Element of the carrier of R
[(1. R),e] is non empty V18() set
{(1. R),e} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),e},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),e] is set
D is set
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
D is Element of the carrier of R
e is Element of the carrier of R
R is non empty right_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
D is set
e is Element of the carrier of R
e * (1. R) is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,(1. R)) is Element of the carrier of R
[e,(1. R)] is non empty V18() set
{e,(1. R)} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,(1. R)},{e}} is non empty finite V39() set
the multF of R . [e,(1. R)] is set
D is set
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
D is Element of the carrier of R
e is Element of the carrier of R
R is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative 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 R is non empty non trivial set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
I is non empty (R) (R) (R) Element of bool the carrier of R
D is non empty non degenerated non trivial left_add-cancelable right_add-cancelable add-cancelable right_complementable almost_left_invertible Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like Euclidian doubleLoopStr
the carrier of D is non empty non trivial set
0. D is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
the ZeroF of D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
e is set
e is Element of I
{(0. D)} is non empty trivial finite 1 -element (D) (D) (D) Element of bool the carrier of D
bool the carrier of D is non empty set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
1. D is non zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
the OneF of D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
e * e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
the multF of D is non empty Relation-like [: the carrier of D, the carrier of D:] -defined the carrier of D -valued Function-like total V21([: the carrier of D, the carrier of D:], the carrier of D) Element of bool [:[: the carrier of D, the carrier of D:], the carrier of D:]
[: the carrier of D, the carrier of D:] is non empty Relation-like set
[:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty Relation-like set
bool [:[: the carrier of D, the carrier of D:], the carrier of D:] is non empty set
the multF of D . (e,e) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of D
[e,e] is non empty V18() set
{e,e} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,e},{e}} is non empty finite V39() set
the multF of D . [e,e] is set
I is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{ (D * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R : verum } is set
1. R is non zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * (1. R) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,(1. R)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,(1. R)] is non empty V18() set
{I,(1. R)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(1. R)},{I}} is non empty finite V39() set
the multF of R . [I,(1. R)] is set
e is non empty (R) (R) (R) Element of bool the carrier of R
e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
I * e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (I,e) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[I,e] is non empty V18() set
{I,e} is non empty finite set
{{I,e},{I}} is non empty finite V39() set
the multF of R . [I,e] is set
I is non empty (R) (R) (R) Element of bool the carrier of R
D is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
dom (<*> the carrier of R) is empty trivial proper Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() Element of bool NAT
e is set
(<*> the carrier of R) /. e is Element of the carrier of R
the Element of I is Element of I
the Element of I * the Element of I is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of I, the Element of I) is Element of the carrier of R
[ the Element of I, the Element of I] is non empty V18() set
{ the Element of I, the Element of I} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I, the Element of I},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I, the Element of I] is set
e is Element of the carrier of R
<*e*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom <*e*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e9 is set
<*e*> /. e9 is Element of the carrier of R
e is Element of the carrier of R
e * the Element of I is Element of the carrier of R
the multF of R . (e, the Element of I) is Element of the carrier of R
[e, the Element of I] is non empty V18() set
{e, the Element of I} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e, the Element of I},{e}} is non empty finite V39() set
the multF of R . [e, the Element of I] is set
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*e*> . e9 is set
the Element of I is Element of I
the Element of I * the Element of I is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of I, the Element of I) is Element of the carrier of R
[ the Element of I, the Element of I] is non empty V18() set
{ the Element of I, the Element of I} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I, the Element of I},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I, the Element of I] is set
e is Element of the carrier of R
<*e*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom <*e*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e9 is set
<*e*> /. e9 is Element of the carrier of R
e is Element of the carrier of R
the Element of I * e is Element of the carrier of R
the multF of R . ( the Element of I,e) is Element of the carrier of R
[ the Element of I,e] is non empty V18() set
{ the Element of I,e} is non empty finite set
{{ the Element of I,e},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I,e] is set
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*e*> . e9 is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
the Element of I is Element of I
the Element of the carrier of R is Element of the carrier of R
the Element of the carrier of R * the Element of I is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of the carrier of R, the Element of I) is Element of the carrier of R
[ the Element of the carrier of R, the Element of I] is non empty V18() set
{ the Element of the carrier of R, the Element of I} is non empty finite set
{ the Element of the carrier of R} is non empty trivial finite 1 -element set
{{ the Element of the carrier of R, the Element of I},{ the Element of the carrier of R}} is non empty finite V39() set
the multF of R . [ the Element of the carrier of R, the Element of I] is set
( the Element of the carrier of R * the Element of I) * the Element of the carrier of R is Element of the carrier of R
the multF of R . (( the Element of the carrier of R * the Element of I), the Element of the carrier of R) is Element of the carrier of R
[( the Element of the carrier of R * the Element of I), the Element of the carrier of R] is non empty V18() set
{( the Element of the carrier of R * the Element of I), the Element of the carrier of R} is non empty finite set
{( the Element of the carrier of R * the Element of I)} is non empty trivial finite 1 -element set
{{( the Element of the carrier of R * the Element of I), the Element of the carrier of R},{( the Element of the carrier of R * the Element of I)}} is non empty finite V39() set
the multF of R . [( the Element of the carrier of R * the Element of I), the Element of the carrier of R] is set
<*(( the Element of the carrier of R * the Element of I) * the Element of the carrier of R)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
D /. e9 is Element of the carrier of R
f is Element of the carrier of R
i is Element of I
f * i is Element of the carrier of R
the multF of R . (f,i) is Element of the carrier of R
[f,i] is non empty V18() set
{f,i} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,i},{f}} is non empty finite V39() set
the multF of R . [f,i] is set
F is Element of the carrier of R
(f * i) * F is Element of the carrier of R
the multF of R . ((f * i),F) is Element of the carrier of R
[(f * i),F] is non empty V18() set
{(f * i),F} is non empty finite set
{(f * i)} is non empty trivial finite 1 -element set
{{(f * i),F},{(f * i)}} is non empty finite V39() set
the multF of R . [(f * i),F] is set
the Element of I is Element of I
the Element of the carrier of R is Element of the carrier of R
the Element of the carrier of R * the Element of I is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of the carrier of R, the Element of I) is Element of the carrier of R
[ the Element of the carrier of R, the Element of I] is non empty V18() set
{ the Element of the carrier of R, the Element of I} is non empty finite set
{ the Element of the carrier of R} is non empty trivial finite 1 -element set
{{ the Element of the carrier of R, the Element of I},{ the Element of the carrier of R}} is non empty finite V39() set
the multF of R . [ the Element of the carrier of R, the Element of I] is set
<*( the Element of the carrier of R * the Element of I)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e /. D is Element of the carrier of R
e9 is Element of the carrier of R
f is Element of I
e9 * f is Element of the carrier of R
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
the Element of I is Element of I
the Element of the carrier of R is Element of the carrier of R
the Element of I * the Element of the carrier of R is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of I, the Element of the carrier of R) is Element of the carrier of R
[ the Element of I, the Element of the carrier of R] is non empty V18() set
{ the Element of I, the Element of the carrier of R} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I, the Element of the carrier of R},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I, the Element of the carrier of R] is set
<*( the Element of I * the Element of the carrier of R)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e /. D is Element of the carrier of R
f is Element of I
e9 is Element of the carrier of R
f * e9 is Element of the carrier of R
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
e ^ e is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I \/ D is non empty Element of bool the carrier of R
e ^ e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom (e ^ e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e ^ e) /. e9 is Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. f is Element of the carrier of R
e . f is set
(e ^ e) . f is set
F is Element of the carrier of R
k is Element of I
F * k is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (F,k) is Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the multF of R . [F,k] is set
i is Element of the carrier of R
(F * k) * i is Element of the carrier of R
the multF of R . ((F * k),i) is Element of the carrier of R
[(F * k),i] is non empty V18() set
{(F * k),i} is non empty finite set
{(F * k)} is non empty trivial finite 1 -element set
{{(F * k),i},{(F * k)}} is non empty finite V39() set
the multF of R . [(F * k),i] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
(len e) + F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. F is Element of the carrier of R
e . F is set
(e ^ e) . f is set
i is Element of the carrier of R
i is Element of D
i * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,i) is Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
k is Element of the carrier of R
(i * i) * k is Element of the carrier of R
the multF of R . ((i * i),k) is Element of the carrier of R
[(i * i),k] is non empty V18() set
{(i * i),k} is non empty finite set
{(i * i)} is non empty trivial finite 1 -element set
{{(i * i),k},{(i * i)}} is non empty finite V39() set
the multF of R . [(i * i),k] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
R is non empty associative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
D * e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (D * e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(D * e) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
D is Element of the carrier of R
f is Element of I
D * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e9 is Element of the carrier of R
(D * f) * e9 is Element of the carrier of R
the multF of R . ((D * f),e9) is Element of the carrier of R
[(D * f),e9] is non empty V18() set
{(D * f),e9} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),e9},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),e9] is set
D * D is Element of the carrier of R
the multF of R . (D,D) is Element of the carrier of R
[D,D] is non empty V18() set
{D,D} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,D},{D}} is non empty finite V39() set
the multF of R . [D,D] is set
F is Element of the carrier of R
F * f is Element of the carrier of R
the multF of R . (F,f) is Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the multF of R . [F,f] is set
(F * f) * e9 is Element of the carrier of R
the multF of R . ((F * f),e9) is Element of the carrier of R
[(F * f),e9] is non empty V18() set
{(F * f),e9} is non empty finite set
{(F * f)} is non empty trivial finite 1 -element set
{{(F * f),e9},{(F * f)}} is non empty finite V39() set
the multF of R . [(F * f),e9] is set
D * (e /. e) is Element of the carrier of R
the multF of R . (D,(e /. e)) is Element of the carrier of R
[D,(e /. e)] is non empty V18() set
{D,(e /. e)} is non empty finite set
{{D,(e /. e)},{D}} is non empty finite V39() set
the multF of R . [D,(e /. e)] is set
D * (D * f) is Element of the carrier of R
the multF of R . (D,(D * f)) is Element of the carrier of R
[D,(D * f)] is non empty V18() set
{D,(D * f)} is non empty finite set
{{D,(D * f)},{D}} is non empty finite V39() set
the multF of R . [D,(D * f)] is set
(D * (D * f)) * e9 is Element of the carrier of R
the multF of R . ((D * (D * f)),e9) is Element of the carrier of R
[(D * (D * f)),e9] is non empty V18() set
{(D * (D * f)),e9} is non empty finite set
{(D * (D * f))} is non empty trivial finite 1 -element set
{{(D * (D * f)),e9},{(D * (D * f))}} is non empty finite V39() set
the multF of R . [(D * (D * f)),e9] is set
R is non empty associative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e * D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (e * D) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e * D) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
D is Element of the carrier of R
f is Element of I
D * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e9 is Element of the carrier of R
(D * f) * e9 is Element of the carrier of R
the multF of R . ((D * f),e9) is Element of the carrier of R
[(D * f),e9] is non empty V18() set
{(D * f),e9} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),e9},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),e9] is set
e9 * D is Element of the carrier of R
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
F is Element of the carrier of R
(D * f) * F is Element of the carrier of R
the multF of R . ((D * f),F) is Element of the carrier of R
[(D * f),F] is non empty V18() set
{(D * f),F} is non empty finite set
{{(D * f),F},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),F] is set
(e /. e) * D is Element of the carrier of R
the multF of R . ((e /. e),D) is Element of the carrier of R
[(e /. e),D] is non empty V18() set
{(e /. e),D} is non empty finite set
{(e /. e)} is non empty trivial finite 1 -element set
{{(e /. e),D},{(e /. e)}} is non empty finite V39() set
the multF of R . [(e /. e),D] is set
f * e9 is Element of the carrier of R
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
D * (f * e9) is Element of the carrier of R
the multF of R . (D,(f * e9)) is Element of the carrier of R
[D,(f * e9)] is non empty V18() set
{D,(f * e9)} is non empty finite set
{{D,(f * e9)},{D}} is non empty finite V39() set
the multF of R . [D,(f * e9)] is set
(D * (f * e9)) * D is Element of the carrier of R
the multF of R . ((D * (f * e9)),D) is Element of the carrier of R
[(D * (f * e9)),D] is non empty V18() set
{(D * (f * e9)),D} is non empty finite set
{(D * (f * e9))} is non empty trivial finite 1 -element set
{{(D * (f * e9)),D},{(D * (f * e9))}} is non empty finite V39() set
the multF of R . [(D * (f * e9)),D] is set
(f * e9) * D is Element of the carrier of R
the multF of R . ((f * e9),D) is Element of the carrier of R
[(f * e9),D] is non empty V18() set
{(f * e9),D} is non empty finite set
{(f * e9)} is non empty trivial finite 1 -element set
{{(f * e9),D},{(f * e9)}} is non empty finite V39() set
the multF of R . [(f * e9),D] is set
D * ((f * e9) * D) is Element of the carrier of R
the multF of R . (D,((f * e9) * D)) is Element of the carrier of R
[D,((f * e9) * D)] is non empty V18() set
{D,((f * e9) * D)} is non empty finite set
{{D,((f * e9) * D)},{D}} is non empty finite V39() set
the multF of R . [D,((f * e9) * D)] is set
f * (e9 * D) is Element of the carrier of R
the multF of R . (f,(e9 * D)) is Element of the carrier of R
[f,(e9 * D)] is non empty V18() set
{f,(e9 * D)} is non empty finite set
{{f,(e9 * D)},{f}} is non empty finite V39() set
the multF of R . [f,(e9 * D)] is set
D * (f * (e9 * D)) is Element of the carrier of R
the multF of R . (D,(f * (e9 * D))) is Element of the carrier of R
[D,(f * (e9 * D))] is non empty V18() set
{D,(f * (e9 * D))} is non empty finite set
{{D,(f * (e9 * D))},{D}} is non empty finite V39() set
the multF of R . [D,(f * (e9 * D))] is set
R is non empty right_unital multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is set
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
e is Element of the carrier of R
D is Element of I
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
(e * D) * (1. R) is Element of the carrier of R
the multF of R . ((e * D),(1. R)) is Element of the carrier of R
[(e * D),(1. R)] is non empty V18() set
{(e * D),(1. R)} is non empty finite set
{(e * D)} is non empty trivial finite 1 -element set
{{(e * D),(1. R)},{(e * D)}} is non empty finite V39() set
the multF of R . [(e * D),(1. R)] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
e ^ e is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I \/ D is non empty Element of bool the carrier of R
e ^ e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom (e ^ e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e ^ e) /. e9 is Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. f is Element of the carrier of R
e . f is set
(e ^ e) . f is set
F is Element of the carrier of R
i is Element of I
F * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the multF of R . [F,i] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
(len e) + F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. F is Element of the carrier of R
e . F is set
(e ^ e) . f is set
i is Element of the carrier of R
k is Element of D
i * k is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,k) is Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the multF of R . [i,k] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
R is non empty associative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
D * e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (D * e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(D * e) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
D is Element of the carrier of R
e9 is Element of I
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
D * D is Element of the carrier of R
the multF of R . (D,D) is Element of the carrier of R
[D,D] is non empty V18() set
{D,D} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,D},{D}} is non empty finite V39() set
the multF of R . [D,D] is set
f is Element of the carrier of R
f * e9 is Element of the carrier of R
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
D * (e /. e) is Element of the carrier of R
the multF of R . (D,(e /. e)) is Element of the carrier of R
[D,(e /. e)] is non empty V18() set
{D,(e /. e)} is non empty finite set
{{D,(e /. e)},{D}} is non empty finite V39() set
the multF of R . [D,(e /. e)] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e * D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (e * D) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e * D) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
e9 is Element of the carrier of R
f is Element of I
e9 * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
D is Element of the carrier of R
(e9 * f) * D is Element of the carrier of R
the multF of R . ((e9 * f),D) is Element of the carrier of R
[(e9 * f),D] is non empty V18() set
{(e9 * f),D} is non empty finite set
{(e9 * f)} is non empty trivial finite 1 -element set
{{(e9 * f),D},{(e9 * f)}} is non empty finite V39() set
the multF of R . [(e9 * f),D] is set
R is non empty left_unital multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is set
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
D is Element of I
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
(1. R) * D is Element of the carrier of R
the multF of R . ((1. R),D) is Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
((1. R) * D) * e is Element of the carrier of R
the multF of R . (((1. R) * D),e) is Element of the carrier of R
[((1. R) * D),e] is non empty V18() set
{((1. R) * D),e} is non empty finite set
{((1. R) * D)} is non empty trivial finite 1 -element set
{{((1. R) * D),e},{((1. R) * D)}} is non empty finite V39() set
the multF of R . [((1. R) * D),e] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
e ^ e is Relation-like NAT -defined Function-like finite FinSequence-like FinSubsequence-like set
I \/ D is non empty Element of bool the carrier of R
e ^ e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom (e ^ e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e ^ e) /. e9 is Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. f is Element of the carrier of R
e . f is set
(e ^ e) . f is set
i is Element of I
F is Element of the carrier of R
i * F is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,F) is Element of the carrier of R
[i,F] is non empty V18() set
{i,F} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,F},{i}} is non empty finite V39() set
the multF of R . [i,F] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
(len e) + F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. F is Element of the carrier of R
e . F is set
(e ^ e) . f is set
k is Element of D
i is Element of the carrier of R
k * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (k,i) is Element of the carrier of R
[k,i] is non empty V18() set
{k,i} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,i},{k}} is non empty finite V39() set
the multF of R . [k,i] is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
R is non empty associative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e * D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (e * D) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(e * D) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
e9 is Element of I
D is Element of the carrier of R
e9 * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
D * D is Element of the carrier of R
the multF of R . (D,D) is Element of the carrier of R
[D,D] is non empty V18() set
{D,D} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,D},{D}} is non empty finite V39() set
the multF of R . [D,D] is set
f is Element of the carrier of R
e9 * f is Element of the carrier of R
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
(e /. e) * D is Element of the carrier of R
the multF of R . ((e /. e),D) is Element of the carrier of R
[(e /. e),D] is non empty V18() set
{(e /. e),D} is non empty finite set
{(e /. e)} is non empty trivial finite 1 -element set
{{(e /. e),D},{(e /. e)}} is non empty finite V39() set
the multF of R . [(e /. e),D] is set
R is non empty associative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
D * e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is set
dom (D * e) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(D * e) /. e is Element of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. e is Element of the carrier of R
f is Element of I
e9 is Element of the carrier of R
f * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
D is Element of the carrier of R
D * f is Element of the carrier of R
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
(D * f) * e9 is Element of the carrier of R
the multF of R . ((D * f),e9) is Element of the carrier of R
[(D * f),e9] is non empty V18() set
{(D * f),e9} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),e9},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),e9] is set
D * (e /. e) is Element of the carrier of R
the multF of R . (D,(e /. e)) is Element of the carrier of R
[D,(e /. e)] is non empty V18() set
{D,(e /. e)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(e /. e)},{D}} is non empty finite V39() set
the multF of R . [D,(e /. e)] is set
R is non empty associative commutative multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is set
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
e is Element of the carrier of R
e9 is Element of I
e * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,e9) is Element of the carrier of R
[e,e9] is non empty V18() set
{e,e9} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,e9},{e}} is non empty finite V39() set
the multF of R . [e,e9] is set
D is Element of the carrier of R
(e * e9) * D is Element of the carrier of R
the multF of R . ((e * e9),D) is Element of the carrier of R
[(e * e9),D] is non empty V18() set
{(e * e9),D} is non empty finite set
{(e * e9)} is non empty trivial finite 1 -element set
{{(e * e9),D},{(e * e9)}} is non empty finite V39() set
the multF of R . [(e * e9),D] is set
e * D is Element of the carrier of R
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
(e * D) * e9 is Element of the carrier of R
the multF of R . ((e * D),e9) is Element of the carrier of R
[(e * D),e9] is non empty V18() set
{(e * D),e9} is non empty finite set
{(e * D)} is non empty trivial finite 1 -element set
{{(e * D),e9},{(e * D)}} is non empty finite V39() set
the multF of R . [(e * D),e9] is set
e is set
D /. e is Element of the carrier of R
e is Element of the carrier of R
e9 is Element of I
e * e9 is Element of the carrier of R
the multF of R . (e,e9) is Element of the carrier of R
[e,e9] is non empty V18() set
{e,e9} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,e9},{e}} is non empty finite V39() set
the multF of R . [e,e9] is set
D is Element of the carrier of R
(e * e9) * D is Element of the carrier of R
the multF of R . ((e * e9),D) is Element of the carrier of R
[(e * e9),D] is non empty V18() set
{(e * e9),D} is non empty finite set
{(e * e9)} is non empty trivial finite 1 -element set
{{(e * e9),D},{(e * e9)}} is non empty finite V39() set
the multF of R . [(e * e9),D] is set
e * D is Element of the carrier of R
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
e9 * (e * D) is Element of the carrier of R
the multF of R . (e9,(e * D)) is Element of the carrier of R
[e9,(e * D)] is non empty V18() set
{e9,(e * D)} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,(e * D)},{e9}} is non empty finite V39() set
the multF of R . [e9,(e * D)] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is Element of the carrier of R
<*e*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
f is Element of the carrier of R
i is Element of I
f * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,i) is Element of the carrier of R
[f,i] is non empty V18() set
{f,i} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,i},{f}} is non empty finite V39() set
the multF of R . [f,i] is set
F is Element of the carrier of R
(f * i) * F is Element of the carrier of R
the multF of R . ((f * i),F) is Element of the carrier of R
[(f * i),F] is non empty V18() set
{(f * i),F} is non empty finite set
{(f * i)} is non empty trivial finite 1 -element set
{{(f * i),F},{(f * i)}} is non empty finite V39() set
the multF of R . [(f * i),F] is set
e /. D is Element of the carrier of R
e . D is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is set
k is Element of the carrier of R
i is Element of I
k * i is Element of the carrier of R
the multF of R . (k,i) is Element of the carrier of R
[k,i] is non empty V18() set
{k,i} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,i},{k}} is non empty finite V39() set
the multF of R . [k,i] is set
i is Element of the carrier of R
(k * i) * i is Element of the carrier of R
the multF of R . ((k * i),i) is Element of the carrier of R
[(k * i),i] is non empty V18() set
{(k * i),i} is non empty finite set
{(k * i)} is non empty trivial finite 1 -element set
{{(k * i),i},{(k * i)}} is non empty finite V39() set
the multF of R . [(k * i),i] is set
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(len D) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom <*e*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*e*> /. e9 is Element of the carrier of R
D /. (len D) is Element of the carrier of R
D . (len D) is set
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
f is Element of the carrier of R
i is Element of I
f * i is Element of the carrier of R
the multF of R . (f,i) is Element of the carrier of R
[f,i] is non empty V18() set
{f,i} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,i},{f}} is non empty finite V39() set
the multF of R . [f,i] is set
F is Element of the carrier of R
(f * i) * F is Element of the carrier of R
the multF of R . ((f * i),F) is Element of the carrier of R
[(f * i),F] is non empty V18() set
{(f * i),F} is non empty finite set
{(f * i)} is non empty trivial finite 1 -element set
{{(f * i),F},{(f * i)}} is non empty finite V39() set
the multF of R . [(f * i),F] is set
<*e*> . e9 is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is Element of the carrier of R
<*e*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
f is Element of the carrier of R
F is Element of I
f * F is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
e /. D is Element of the carrier of R
e . D is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is set
i is Element of the carrier of R
k is Element of I
i * k is Element of the carrier of R
the multF of R . (i,k) is Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the multF of R . [i,k] is set
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(len D) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom <*e*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*e*> /. e9 is Element of the carrier of R
D /. (len D) is Element of the carrier of R
D . (len D) is set
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
f is Element of the carrier of R
F is Element of I
f * F is Element of the carrier of R
the multF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
<*e*> . e9 is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e is Element of the carrier of R
<*e*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
D is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
F is Element of I
f is Element of the carrier of R
F * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (F,f) is Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the multF of R . [F,f] is set
e /. D is Element of the carrier of R
e . D is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is set
k is Element of I
i is Element of the carrier of R
k * i is Element of the carrier of R
the multF of R . (k,i) is Element of the carrier of R
[k,i] is non empty V18() set
{k,i} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,i},{k}} is non empty finite V39() set
the multF of R . [k,i] is set
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(len D) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D ^ <*e*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
e9 is set
dom <*e*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*e*> /. e9 is Element of the carrier of R
D /. (len D) is Element of the carrier of R
D . (len D) is set
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
F is Element of I
f is Element of the carrier of R
F * f is Element of the carrier of R
the multF of R . (F,f) is Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the multF of R . [F,f] is set
<*e*> . e9 is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
[: the carrier of R, the carrier of R, the carrier of R:] is non empty set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
[: the carrier of R, the carrier of R, the carrier of R:] is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len D) is finite len D -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
D is Element of the carrier of R
f is Element of I
D * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e9 is Element of the carrier of R
(D * f) * e9 is Element of the carrier of R
the multF of R . ((D * f),e9) is Element of the carrier of R
[(D * f),e9] is non empty V18() set
{(D * f),e9} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),e9},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),e9] is set
F is Element of the carrier of R
[D,F,e9] is V18() V19() Element of [: the carrier of R, the carrier of R, the carrier of R:]
[D,F] is non empty V18() set
{D,F} is non empty finite set
{{D,F},{D}} is non empty finite V39() set
[[D,F],e9] is non empty V18() set
{[D,F],e9} is non empty finite set
{[D,F]} is non empty trivial Relation-like finite 1 -element set
{{[D,F],e9},{[D,F]}} is non empty finite V39() set
i is Element of [: the carrier of R, the carrier of R, the carrier of R:]
k is Element of [: the carrier of R, the carrier of R, the carrier of R:]
e is Relation-like NAT -defined [: the carrier of R, the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R, the carrier of R:]
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D is set
D . D is set
e /. D is Element of [: the carrier of R, the carrier of R, the carrier of R:]
(e /. D) `1_3 is Element of the carrier of R
(e /. D) `1 is set
((e /. D) `1) `1 is set
(e /. D) `2_3 is Element of the carrier of R
((e /. D) `1) `2 is set
((e /. D) `1_3) * ((e /. D) `2_3) is Element of the carrier of R
the multF of R . (((e /. D) `1_3),((e /. D) `2_3)) is Element of the carrier of R
[((e /. D) `1_3),((e /. D) `2_3)] is non empty V18() set
{((e /. D) `1_3),((e /. D) `2_3)} is non empty finite set
{((e /. D) `1_3)} is non empty trivial finite 1 -element set
{{((e /. D) `1_3),((e /. D) `2_3)},{((e /. D) `1_3)}} is non empty finite V39() set
the multF of R . [((e /. D) `1_3),((e /. D) `2_3)] is set
(e /. D) `3_3 is Element of the carrier of R
(((e /. D) `1_3) * ((e /. D) `2_3)) * ((e /. D) `3_3) is Element of the carrier of R
the multF of R . ((((e /. D) `1_3) * ((e /. D) `2_3)),((e /. D) `3_3)) is Element of the carrier of R
[(((e /. D) `1_3) * ((e /. D) `2_3)),((e /. D) `3_3)] is non empty V18() set
{(((e /. D) `1_3) * ((e /. D) `2_3)),((e /. D) `3_3)} is non empty finite set
{(((e /. D) `1_3) * ((e /. D) `2_3))} is non empty trivial finite 1 -element set
{{(((e /. D) `1_3) * ((e /. D) `2_3)),((e /. D) `3_3)},{(((e /. D) `1_3) * ((e /. D) `2_3))}} is non empty finite V39() set
the multF of R . [(((e /. D) `1_3) * ((e /. D) `2_3)),((e /. D) `3_3)] is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. e9 is Element of [: the carrier of R, the carrier of R, the carrier of R:]
D /. e9 is Element of the carrier of R
f is Element of the carrier of R
F is Element of the carrier of R
i is Element of the carrier of R
[f,F,i] is V18() V19() Element of [: the carrier of R, the carrier of R, the carrier of R:]
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
[[f,F],i] is non empty V18() set
{[f,F],i} is non empty finite set
{[f,F]} is non empty trivial Relation-like finite 1 -element set
{{[f,F],i},{[f,F]}} is non empty finite V39() set
f * F is Element of the carrier of R
the multF of R . (f,F) is Element of the carrier of R
the multF of R . [f,F] is set
(f * F) * i is Element of the carrier of R
the multF of R . ((f * F),i) is Element of the carrier of R
[(f * F),i] is non empty V18() set
{(f * F),i} is non empty finite set
{(f * F)} is non empty trivial finite 1 -element set
{{(f * F),i},{(f * F)}} is non empty finite V39() set
the multF of R . [(f * F),i] is set
[f,F,i] `3_3 is Element of the carrier of R
[f,F,i] `1_3 is Element of the carrier of R
[f,F,i] `1 is set
([f,F,i] `1) `1 is set
[f,F,i] `2_3 is Element of the carrier of R
([f,F,i] `1) `2 is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty multLoopStr
the carrier of I is non empty set
bool the carrier of I is non empty set
[: the carrier of R, the carrier of I:] is non empty Relation-like set
bool [: the carrier of R, the carrier of I:] is non empty set
[: the carrier of R, the carrier of R, the carrier of R:] is non empty set
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is non empty Element of bool the carrier of I
D is non empty Relation-like the carrier of R -defined the carrier of I -valued Function-like total V21( the carrier of R, the carrier of I) Element of bool [: the carrier of R, the carrier of I:]
D .: D is Element of bool the carrier of I
e9 is Relation-like NAT -defined [: the carrier of R, the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R, the carrier of R:]
f is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of I
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
F is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty Element of bool the carrier of R
e9 /. F is Element of [: the carrier of R, the carrier of R, the carrier of R:]
(e9 /. F) `2_3 is Element of the carrier of R
(e9 /. F) `1 is set
((e9 /. F) `1) `2 is set
D . ((e9 /. F) `2_3) is Element of the carrier of I
(e9 /. F) `1_3 is Element of the carrier of R
((e9 /. F) `1) `1 is set
D . ((e9 /. F) `1_3) is Element of the carrier of I
(e9 /. F) `3_3 is Element of the carrier of R
D . ((e9 /. F) `3_3) is Element of the carrier of I
k is Element of the carrier of I
i is Element of the carrier of I
i is Element of e
f . F is set
f /. F is Element of the carrier of I
i is Element of the carrier of I
c1 is Element of e
i * c1 is Element of the carrier of I
the multF of I is non empty Relation-like [: the carrier of I, the carrier of I:] -defined the carrier of I -valued Function-like total V21([: the carrier of I, the carrier of I:], the carrier of I) Element of bool [:[: the carrier of I, the carrier of I:], the carrier of I:]
[: the carrier of I, the carrier of I:] is non empty Relation-like set
[:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty Relation-like set
bool [:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty set
the multF of I . (i,c1) is Element of the carrier of I
[i,c1] is non empty V18() set
{i,c1} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c1},{i}} is non empty finite V39() set
the multF of I . [i,c1] is set
c is Element of the carrier of I
(i * c1) * c is Element of the carrier of I
the multF of I . ((i * c1),c) is Element of the carrier of I
[(i * c1),c] is non empty V18() set
{(i * c1),c} is non empty finite set
{(i * c1)} is non empty trivial finite 1 -element set
{{(i * c1),c},{(i * c1)}} is non empty finite V39() set
the multF of I . [(i * c1),c] is set
F is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like (I,e)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is set
F . i is set
e9 /. i is Element of [: the carrier of R, the carrier of R, the carrier of R:]
(e9 /. i) `1_3 is Element of the carrier of R
(e9 /. i) `1 is set
((e9 /. i) `1) `1 is set
D . ((e9 /. i) `1_3) is Element of the carrier of I
(e9 /. i) `2_3 is Element of the carrier of R
((e9 /. i) `1) `2 is set
D . ((e9 /. i) `2_3) is Element of the carrier of I
(D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3)) is Element of the carrier of I
the multF of I . ((D . ((e9 /. i) `1_3)),(D . ((e9 /. i) `2_3))) is Element of the carrier of I
[(D . ((e9 /. i) `1_3)),(D . ((e9 /. i) `2_3))] is non empty V18() set
{(D . ((e9 /. i) `1_3)),(D . ((e9 /. i) `2_3))} is non empty finite set
{(D . ((e9 /. i) `1_3))} is non empty trivial finite 1 -element set
{{(D . ((e9 /. i) `1_3)),(D . ((e9 /. i) `2_3))},{(D . ((e9 /. i) `1_3))}} is non empty finite V39() set
the multF of I . [(D . ((e9 /. i) `1_3)),(D . ((e9 /. i) `2_3))] is set
(e9 /. i) `3_3 is Element of the carrier of R
D . ((e9 /. i) `3_3) is Element of the carrier of I
((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))) * (D . ((e9 /. i) `3_3)) is Element of the carrier of I
the multF of I . (((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))),(D . ((e9 /. i) `3_3))) is Element of the carrier of I
[((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))),(D . ((e9 /. i) `3_3))] is non empty V18() set
{((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))),(D . ((e9 /. i) `3_3))} is non empty finite set
{((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3)))} is non empty trivial finite 1 -element set
{{((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))),(D . ((e9 /. i) `3_3))},{((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3)))}} is non empty finite V39() set
the multF of I . [((D . ((e9 /. i) `1_3)) * (D . ((e9 /. i) `2_3))),(D . ((e9 /. i) `3_3))] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
[: the carrier of R, the carrier of R:] is non empty Relation-like set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
[: the carrier of R, the carrier of R:] is non empty Relation-like set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len D) is finite len D -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
D is Element of the carrier of R
e9 is Element of I
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is Element of the carrier of R
[D,f] is non empty V18() Element of [: the carrier of R, the carrier of R:]
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
F is Element of [: the carrier of R, the carrier of R:]
i is Element of [: the carrier of R, the carrier of R:]
e is Relation-like NAT -defined [: the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R:]
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D is set
D . D is set
e /. D is Element of [: the carrier of R, the carrier of R:]
(e /. D) `1 is Element of the carrier of R
(e /. D) `2 is Element of the carrier of R
((e /. D) `1) * ((e /. D) `2) is Element of the carrier of R
the multF of R . (((e /. D) `1),((e /. D) `2)) is Element of the carrier of R
[((e /. D) `1),((e /. D) `2)] is non empty V18() set
{((e /. D) `1),((e /. D) `2)} is non empty finite set
{((e /. D) `1)} is non empty trivial finite 1 -element set
{{((e /. D) `1),((e /. D) `2)},{((e /. D) `1)}} is non empty finite V39() set
the multF of R . [((e /. D) `1),((e /. D) `2)] is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. e9 is Element of [: the carrier of R, the carrier of R:]
D /. e9 is Element of the carrier of R
f is Element of the carrier of R
F is Element of the carrier of R
[f,F] is non empty V18() Element of [: the carrier of R, the carrier of R:]
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
f * F is Element of the carrier of R
the multF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
the multF of R . [f,F] is set
[f,F] `1 is Element of the carrier of R
[f,F] `2 is Element of the carrier of R
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty multLoopStr
the carrier of I is non empty set
bool the carrier of I is non empty set
[: the carrier of R, the carrier of I:] is non empty Relation-like set
bool [: the carrier of R, the carrier of I:] is non empty set
[: the carrier of R, the carrier of R:] is non empty Relation-like set
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is non empty Element of bool the carrier of I
D is non empty Relation-like the carrier of R -defined the carrier of I -valued Function-like total V21( the carrier of R, the carrier of I) Element of bool [: the carrier of R, the carrier of I:]
D .: D is Element of bool the carrier of I
e9 is Relation-like NAT -defined [: the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R:]
f is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of I
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
F is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty Element of bool the carrier of R
e9 /. F is Element of [: the carrier of R, the carrier of R:]
(e9 /. F) `2 is Element of the carrier of R
D . ((e9 /. F) `2) is Element of the carrier of I
(e9 /. F) `1 is Element of the carrier of R
D . ((e9 /. F) `1) is Element of the carrier of I
k is Element of the carrier of I
i is Element of e
f . F is set
f /. F is Element of the carrier of I
i is Element of the carrier of I
i is Element of e
i * i is Element of the carrier of I
the multF of I is non empty Relation-like [: the carrier of I, the carrier of I:] -defined the carrier of I -valued Function-like total V21([: the carrier of I, the carrier of I:], the carrier of I) Element of bool [:[: the carrier of I, the carrier of I:], the carrier of I:]
[: the carrier of I, the carrier of I:] is non empty Relation-like set
[:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty Relation-like set
bool [:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty set
the multF of I . (i,i) is Element of the carrier of I
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of I . [i,i] is set
F is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like (I,e)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is set
F . i is set
e9 /. i is Element of [: the carrier of R, the carrier of R:]
(e9 /. i) `1 is Element of the carrier of R
D . ((e9 /. i) `1) is Element of the carrier of I
(e9 /. i) `2 is Element of the carrier of R
D . ((e9 /. i) `2) is Element of the carrier of I
(D . ((e9 /. i) `1)) * (D . ((e9 /. i) `2)) is Element of the carrier of I
the multF of I . ((D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))) is Element of the carrier of I
[(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))] is non empty V18() set
{(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))} is non empty finite set
{(D . ((e9 /. i) `1))} is non empty trivial finite 1 -element set
{{(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))},{(D . ((e9 /. i) `1))}} is non empty finite V39() set
the multF of I . [(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
[: the carrier of R, the carrier of R:] is non empty Relation-like set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
[: the carrier of R, the carrier of R:] is non empty Relation-like set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len D) is finite len D -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. e is Element of the carrier of R
e9 is Element of I
D is Element of the carrier of R
e9 * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f is Element of the carrier of R
[f,D] is non empty V18() Element of [: the carrier of R, the carrier of R:]
{f,D} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,D},{f}} is non empty finite V39() set
F is Element of [: the carrier of R, the carrier of R:]
i is Element of [: the carrier of R, the carrier of R:]
e is Relation-like NAT -defined [: the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R:]
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D is set
D . D is set
e /. D is Element of [: the carrier of R, the carrier of R:]
(e /. D) `1 is Element of the carrier of R
(e /. D) `2 is Element of the carrier of R
((e /. D) `1) * ((e /. D) `2) is Element of the carrier of R
the multF of R . (((e /. D) `1),((e /. D) `2)) is Element of the carrier of R
[((e /. D) `1),((e /. D) `2)] is non empty V18() set
{((e /. D) `1),((e /. D) `2)} is non empty finite set
{((e /. D) `1)} is non empty trivial finite 1 -element set
{{((e /. D) `1),((e /. D) `2)},{((e /. D) `1)}} is non empty finite V39() set
the multF of R . [((e /. D) `1),((e /. D) `2)] is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. e9 is Element of [: the carrier of R, the carrier of R:]
D /. e9 is Element of the carrier of R
f is Element of the carrier of R
F is Element of the carrier of R
[f,F] is non empty V18() Element of [: the carrier of R, the carrier of R:]
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
f * F is Element of the carrier of R
the multF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
the multF of R . [f,F] is set
[f,F] `1 is Element of the carrier of R
[f,F] `2 is Element of the carrier of R
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty multLoopStr
the carrier of I is non empty set
bool the carrier of I is non empty set
[: the carrier of R, the carrier of I:] is non empty Relation-like set
bool [: the carrier of R, the carrier of I:] is non empty set
[: the carrier of R, the carrier of R:] is non empty Relation-like set
D is non empty Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,D)
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is non empty Element of bool the carrier of I
D is non empty Relation-like the carrier of R -defined the carrier of I -valued Function-like total V21( the carrier of R, the carrier of I) Element of bool [: the carrier of R, the carrier of I:]
D .: D is Element of bool the carrier of I
e9 is Relation-like NAT -defined [: the carrier of R, the carrier of R:] -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of [: the carrier of R, the carrier of R:]
f is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of I
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
F is set
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is non empty Element of bool the carrier of R
e9 /. F is Element of [: the carrier of R, the carrier of R:]
(e9 /. F) `1 is Element of the carrier of R
D . ((e9 /. F) `1) is Element of the carrier of I
(e9 /. F) `2 is Element of the carrier of R
D . ((e9 /. F) `2) is Element of the carrier of I
k is Element of the carrier of I
i is Element of e
f . F is set
f /. F is Element of the carrier of I
i is Element of e
i is Element of the carrier of I
i * i is Element of the carrier of I
the multF of I is non empty Relation-like [: the carrier of I, the carrier of I:] -defined the carrier of I -valued Function-like total V21([: the carrier of I, the carrier of I:], the carrier of I) Element of bool [:[: the carrier of I, the carrier of I:], the carrier of I:]
[: the carrier of I, the carrier of I:] is non empty Relation-like set
[:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty Relation-like set
bool [:[: the carrier of I, the carrier of I:], the carrier of I:] is non empty set
the multF of I . (i,i) is Element of the carrier of I
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of I . [i,i] is set
F is Relation-like NAT -defined the carrier of I -valued Function-like finite FinSequence-like FinSubsequence-like (I,e)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is set
F . i is set
e9 /. i is Element of [: the carrier of R, the carrier of R:]
(e9 /. i) `1 is Element of the carrier of R
D . ((e9 /. i) `1) is Element of the carrier of I
(e9 /. i) `2 is Element of the carrier of R
D . ((e9 /. i) `2) is Element of the carrier of I
(D . ((e9 /. i) `1)) * (D . ((e9 /. i) `2)) is Element of the carrier of I
the multF of I . ((D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))) is Element of the carrier of I
[(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))] is non empty V18() set
{(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))} is non empty finite set
{(D . ((e9 /. i) `1))} is non empty trivial finite 1 -element set
{{(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))},{(D . ((e9 /. i) `1))}} is non empty finite V39() set
the multF of I . [(D . ((e9 /. i) `1)),(D . ((e9 /. i) `2))] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg e is finite e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D | (Seg e) is Relation-like NAT -defined Seg e -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
e9 is Element of the carrier of R
F is Element of I
e9 * F is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,F) is Element of the carrier of R
[e9,F] is non empty V18() set
{e9,F} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,F},{e9}} is non empty finite V39() set
the multF of R . [e9,F] is set
f is Element of the carrier of R
(e9 * F) * f is Element of the carrier of R
the multF of R . ((e9 * F),f) is Element of the carrier of R
[(e9 * F),f] is non empty V18() set
{(e9 * F),f} is non empty finite set
{(e9 * F)} is non empty trivial finite 1 -element set
{{(e9 * F),f},{(e9 * F)}} is non empty finite V39() set
the multF of R . [(e9 * F),f] is set
e /. D is Element of the carrier of R
e . D is set
D . D is set
i is Element of the carrier of R
i is Element of I
i * i is Element of the carrier of R
the multF of R . (i,i) is Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
k is Element of the carrier of R
(i * i) * k is Element of the carrier of R
the multF of R . ((i * i),k) is Element of the carrier of R
[(i * i),k] is non empty V18() set
{(i * i),k} is non empty finite set
{(i * i)} is non empty trivial finite 1 -element set
{{(i * i),k},{(i * i)}} is non empty finite V39() set
the multF of R . [(i * i),k] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg e is finite e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D | (Seg e) is Relation-like NAT -defined Seg e -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
e9 is Element of the carrier of R
f is Element of I
e9 * f is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
e /. D is Element of the carrier of R
e . D is set
D . D is set
F is Element of the carrier of R
i is Element of I
F * i is Element of the carrier of R
the multF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the multF of R . [F,i] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg e is finite e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D | (Seg e) is Relation-like NAT -defined Seg e -defined NAT -defined the carrier of R -valued Function-like finite FinSubsequence-like Element of bool [:NAT, the carrier of R:]
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. D is Element of the carrier of R
f is Element of I
e9 is Element of the carrier of R
f * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,e9) is Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
e /. D is Element of the carrier of R
e . D is set
D . D is set
i is Element of I
F is Element of the carrier of R
i * F is Element of the carrier of R
the multF of R . (i,F) is Element of the carrier of R
[i,F] is non empty V18() set
{i,F} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,F},{i}} is non empty finite V39() set
the multF of R . [i,F] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
{ b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) (R) Element of bool the carrier of R ) } is set
meet { b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) (R) Element of bool the carrier of R ) } is set
e is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is set
F is Element of bool the carrier of R
{(D + e9)} is non empty trivial finite 1 -element Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is set
F is Element of bool the carrier of R
{(D * e9)} is non empty trivial finite 1 -element Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
e9 * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f is set
F is Element of bool the carrier of R
{(e9 * D)} is non empty trivial finite 1 -element Element of bool the carrier of R
D is non empty (R) (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) (R) Element of bool the carrier of R
D is non empty (R) (R) (R) Element of bool the carrier of R
e is non empty (R) (R) (R) Element of bool the carrier of R
{ b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) Element of bool the carrier of R ) } is set
meet { b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) Element of bool the carrier of R ) } is set
e is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is set
F is Element of bool the carrier of R
{(D + e9)} is non empty trivial finite 1 -element Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is set
F is Element of bool the carrier of R
{(D * e9)} is non empty trivial finite 1 -element Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
e is non empty (R) (R) Element of bool the carrier of R
{ b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) Element of bool the carrier of R ) } is set
meet { b1 where b1 is Element of bool the carrier of R : ( I c= b1 & b1 is non empty (R) (R) Element of bool the carrier of R ) } is set
e is set
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is set
F is Element of bool the carrier of R
{(D + e9)} is non empty trivial finite 1 -element Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
e9 * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f is set
F is Element of bool the carrier of R
{(e9 * D)} is non empty trivial finite 1 -element Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) Element of bool the carrier of R
e9 is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
e is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed right-distributive left-distributive distributive left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(0. R)}) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(0. R)}) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(0. R)}) is non empty (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(0. R)}) is non empty (R) (R) Element of bool the carrier of R
R is non empty unital right_unital well-unital left_unital doubleLoopStr
the carrier of R is non empty set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
{(1. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(1. R)}) is non empty (R) (R) (R) Element of bool the carrier of R
I is set
D is Element of the carrier of R
D * (1. R) is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,(1. R)) is Element of the carrier of R
[D,(1. R)] is non empty V18() set
{D,(1. R)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(1. R)},{D}} is non empty finite V39() set
the multF of R . [D,(1. R)] is set
R is non empty right_unital doubleLoopStr
the carrier of R is non empty set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
{(1. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(1. R)}) is non empty (R) (R) Element of bool the carrier of R
I is set
D is Element of the carrier of R
D * (1. R) is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,(1. R)) is Element of the carrier of R
[D,(1. R)] is non empty V18() set
{D,(1. R)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(1. R)},{D}} is non empty finite V39() set
the multF of R . [D,(1. R)] is set
R is non empty left_unital doubleLoopStr
the carrier of R is non empty set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
{(1. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{(1. R)}) is non empty (R) (R) Element of bool the carrier of R
I is set
D is Element of the carrier of R
(1. R) * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),D) is Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
R is non empty doubleLoopStr
[#] R is non empty non proper Element of bool the carrier of R
the carrier of R is non empty set
bool the carrier of R is non empty set
(R,([#] R)) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
[#] R is non empty non proper Element of bool the carrier of R
the carrier of R is non empty set
bool the carrier of R is non empty set
(R,([#] R)) is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
[#] R is non empty non proper Element of bool the carrier of R
the carrier of R is non empty set
bool the carrier of R is non empty set
(R,([#] R)) is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
(R,D) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
(R,D) is non empty (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
(R,D) is non empty (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
{ b1 where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ex b2 being Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I) st Sum b2 = b1 } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*D*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
f is set
dom <*D*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*D*> . f is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(1. R) * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
((1. R) * D) * (1. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (((1. R) * D),(1. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((1. R) * D),(1. R)] is non empty V18() set
{((1. R) * D),(1. R)} is non empty finite set
{((1. R) * D)} is non empty trivial finite 1 -element set
{{((1. R) * D),(1. R)},{((1. R) * D)}} is non empty finite V39() set
the multF of R . [((1. R) * D),(1. R)] is set
<*D*> /. f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*k*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
i ^ <*k*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len <*k*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + (len <*k*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
dom <*k*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*k*> /. 1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c is Element of I
i * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,c] is non empty V18() set
{i,c} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c},{i}} is non empty finite V39() set
the multF of R . [i,c] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(i * c) * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((i * c),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i * c),i] is non empty V18() set
{(i * c),i} is non empty finite set
{(i * c)} is non empty trivial finite 1 -element set
{{(i * c),i},{(i * c)}} is non empty finite V39() set
the multF of R . [(i * c),i] is set
<*k*> . 1 is set
Sum <*k*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(Sum i) + (Sum <*k*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((Sum i),(Sum <*k*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(Sum <*k*>)] is non empty V18() set
{(Sum i),(Sum <*k*>)} is non empty finite set
{(Sum i)} is non empty trivial finite 1 -element set
{{(Sum i),(Sum <*k*>)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(Sum <*k*>)] is set
the Element of I is Element of I
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(0. R) * the Element of I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 + f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(R,I,I,i,i) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I \/ I)
I \/ I is non empty Element of bool the carrier of R
Sum (R,I,I,i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (F,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the addF of R . [F,k] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f * e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 * f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 * i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
{ b1 where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ex b2 being Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I) st Sum b2 = b1 } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*D*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
f is set
dom <*D*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*D*> . f is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(1. R) * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
<*D*> /. f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*k*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
i ^ <*k*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len <*k*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + (len <*k*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
dom <*k*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*k*> /. 1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Element of I
i * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
<*k*> . 1 is set
Sum <*k*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(Sum i) + (Sum <*k*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((Sum i),(Sum <*k*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(Sum <*k*>)] is non empty V18() set
{(Sum i),(Sum <*k*>)} is non empty finite set
{(Sum i)} is non empty trivial finite 1 -element set
{{(Sum i),(Sum <*k*>)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(Sum <*k*>)] is set
the Element of I is Element of I
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(0. R) * the Element of I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 + f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(R,I,I,i,i) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I \/ I)
I \/ I is non empty Element of bool the carrier of R
Sum (R,I,I,i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (F,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the addF of R . [F,k] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 * f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the multF of R . [e9,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 * i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) Element of bool the carrier of R
{ b1 where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ex b2 being Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I) st Sum b2 = b1 } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*D*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
f is set
dom <*D*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*D*> . f is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * (1. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,(1. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(1. R)] is non empty V18() set
{D,(1. R)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(1. R)},{D}} is non empty finite V39() set
the multF of R . [D,(1. R)] is set
<*D*> /. f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*k*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
i ^ <*k*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len <*k*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + (len <*k*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
dom <*k*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*k*> /. 1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Element of I
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
<*k*> . 1 is set
Sum <*k*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(Sum i) + (Sum <*k*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((Sum i),(Sum <*k*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(Sum <*k*>)] is non empty V18() set
{(Sum i),(Sum <*k*>)} is non empty finite set
{(Sum i)} is non empty trivial finite 1 -element set
{{(Sum i),(Sum <*k*>)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(Sum <*k*>)] is set
the Element of I is Element of I
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the Element of I * (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ( the Element of I,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[ the Element of I,(0. R)] is non empty V18() set
{ the Element of I,(0. R)} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I,(0. R)},{ the Element of I}} is non empty finite V39() set
the multF of R . [ the Element of I,(0. R)] is set
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 + f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(R,I,I,i,i) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I \/ I)
I \/ I is non empty Element of bool the carrier of R
Sum (R,I,I,i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (F,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the addF of R . [F,k] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f * e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the multF of R . [f,e9] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{I} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
{ (I * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : verum } is set
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I}) : verum } is set
e9 is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I})
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
F . (len f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F . i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,k] is non empty V18() set
{I,k} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,k},{I}} is non empty finite V39() set
the multF of R . [I,k] is set
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,k] is non empty V18() set
{I,k} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,k},{I}} is non empty finite V39() set
the multF of R . [I,k] is set
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
f /. (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c is Element of {I}
i * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,c] is non empty V18() set
{i,c} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c},{i}} is non empty finite V39() set
the multF of R . [i,c] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(i * c) * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((i * c),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i * c),i] is non empty V18() set
{(i * c),i} is non empty finite set
{(i * c)} is non empty trivial finite 1 -element set
{{(i * c),i},{(i * c)}} is non empty finite V39() set
the multF of R . [(i * c),i] is set
f . (i + 1) is set
(F . i) + (f /. (i + 1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((F . i),(f /. (i + 1))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(F . i),(f /. (i + 1))] is non empty V18() set
{(F . i),(f /. (i + 1))} is non empty finite set
{(F . i)} is non empty trivial finite 1 -element set
{{(F . i),(f /. (i + 1))},{(F . i)}} is non empty finite V39() set
the addF of R . [(F . i),(f /. (i + 1))] is set
i * I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,I) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,I] is non empty V18() set
{i,I} is non empty finite set
{{i,I},{i}} is non empty finite V39() set
the multF of R . [i,I] is set
(i * I) * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((i * I),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i * I),i] is non empty V18() set
{(i * I),i} is non empty finite set
{(i * I)} is non empty trivial finite 1 -element set
{{(i * I),i},{(i * I)}} is non empty finite V39() set
the multF of R . [(i * I),i] is set
(I * k) + ((i * I) * i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * k),((i * I) * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * k),((i * I) * i)] is non empty V18() set
{(I * k),((i * I) * i)} is non empty finite set
{(I * k)} is non empty trivial finite 1 -element set
{{(I * k),((i * I) * i)},{(I * k)}} is non empty finite V39() set
the addF of R . [(I * k),((i * I) * i)] is set
i * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
I * (i * i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,(i * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(i * i)] is non empty V18() set
{I,(i * i)} is non empty finite set
{{I,(i * i)},{I}} is non empty finite V39() set
the multF of R . [I,(i * i)] is set
(I * k) + (I * (i * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * k),(I * (i * i))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * k),(I * (i * i))] is non empty V18() set
{(I * k),(I * (i * i))} is non empty finite set
{{(I * k),(I * (i * i))},{(I * k)}} is non empty finite V39() set
the addF of R . [(I * k),(I * (i * i))] is set
k + (i * i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (k,(i * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[k,(i * i)] is non empty V18() set
{k,(i * i)} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,(i * i)},{k}} is non empty finite V39() set
the addF of R . [k,(i * i)] is set
I * (k + (i * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,(k + (i * i))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(k + (i * i))] is non empty V18() set
{I,(k + (i * i))} is non empty finite set
{{I,(k + (i * i))},{I}} is non empty finite V39() set
the multF of R . [I,(k + (i * i))] is set
I * (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(0. R)] is non empty V18() set
{I,(0. R)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(0. R)},{I}} is non empty finite V39() set
the multF of R . [I,(0. R)] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,i] is non empty V18() set
{I,i} is non empty finite set
{{I,i},{I}} is non empty finite V39() set
the multF of R . [I,i] is set
e9 is set
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I})
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I})
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,f] is non empty V18() set
{I,f} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,f},{I}} is non empty finite V39() set
the multF of R . [I,f] is set
<*(I * f)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom <*(I * f)*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
i is set
<*(I * f)*> /. i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len <*(I * f)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e is Element of {I}
f * e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (f,e) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,e] is non empty V18() set
{f,e} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e},{f}} is non empty finite V39() set
the multF of R . [f,e] is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(f * e) * (1. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((f * e),(1. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(f * e),(1. R)] is non empty V18() set
{(f * e),(1. R)} is non empty finite set
{(f * e)} is non empty trivial finite 1 -element set
{{(f * e),(1. R)},{(f * e)}} is non empty finite V39() set
the multF of R . [(f * e),(1. R)] is set
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I})
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{I,D} is non empty finite Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I,D}) is non empty (R) (R) (R) Element of bool the carrier of R
{ ((I * b1) + (D * b2)) where b1, b2 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : verum } is set
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I,D}) : verum } is set
F is set
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I,D})
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
len i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
k . (len i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k . i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
k . (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len i) is finite len i -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom i is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i /. (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i . (i + 1) is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,i] is non empty V18() set
{I,i} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,i},{I}} is non empty finite V39() set
the multF of R . [I,i] is set
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,c] is non empty V18() set
{D,c} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,c},{D}} is non empty finite V39() set
the multF of R . [D,c] is set
(I * i) + (D * c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((I * i),(D * c)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(D * c)] is non empty V18() set
{(I * i),(D * c)} is non empty finite set
{(I * i)} is non empty trivial finite 1 -element set
{{(I * i),(D * c)},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(D * c)] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,i] is non empty V18() set
{I,i} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,i},{I}} is non empty finite V39() set
the multF of R . [I,i] is set
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,c] is non empty V18() set
{D,c} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,c},{D}} is non empty finite V39() set
the multF of R . [D,c] is set
(I * i) + (D * c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((I * i),(D * c)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(D * c)] is non empty V18() set
{(I * i),(D * c)} is non empty finite set
{(I * i)} is non empty trivial finite 1 -element set
{{(I * i),(D * c)},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(D * c)] is set
c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
l is Element of {I,D}
c1 * l is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c1,l) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,l] is non empty V18() set
{c1,l} is non empty finite set
{c1} is non empty trivial finite 1 -element set
{{c1,l},{c1}} is non empty finite V39() set
the multF of R . [c1,l] is set
r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(c1 * l) * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((c1 * l),r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(c1 * l),r] is non empty V18() set
{(c1 * l),r} is non empty finite set
{(c1 * l)} is non empty trivial finite 1 -element set
{{(c1 * l),r},{(c1 * l)}} is non empty finite V39() set
the multF of R . [(c1 * l),r] is set
c1 * I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c1,I) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,I] is non empty V18() set
{c1,I} is non empty finite set
{{c1,I},{c1}} is non empty finite V39() set
the multF of R . [c1,I] is set
(c1 * I) * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((c1 * I),r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(c1 * I),r] is non empty V18() set
{(c1 * I),r} is non empty finite set
{(c1 * I)} is non empty trivial finite 1 -element set
{{(c1 * I),r},{(c1 * I)}} is non empty finite V39() set
the multF of R . [(c1 * I),r] is set
((I * i) + (D * c)) + ((c1 * I) * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((I * i) + (D * c)),((c1 * I) * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((I * i) + (D * c)),((c1 * I) * r)] is non empty V18() set
{((I * i) + (D * c)),((c1 * I) * r)} is non empty finite set
{((I * i) + (D * c))} is non empty trivial finite 1 -element set
{{((I * i) + (D * c)),((c1 * I) * r)},{((I * i) + (D * c))}} is non empty finite V39() set
the addF of R . [((I * i) + (D * c)),((c1 * I) * r)] is set
I * c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,c1] is non empty V18() set
{I,c1} is non empty finite set
{{I,c1},{I}} is non empty finite V39() set
the multF of R . [I,c1] is set
(I * c1) * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((I * c1),r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * c1),r] is non empty V18() set
{(I * c1),r} is non empty finite set
{(I * c1)} is non empty trivial finite 1 -element set
{{(I * c1),r},{(I * c1)}} is non empty finite V39() set
the multF of R . [(I * c1),r] is set
(I * i) + ((I * c1) * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),((I * c1) * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),((I * c1) * r)] is non empty V18() set
{(I * i),((I * c1) * r)} is non empty finite set
{{(I * i),((I * c1) * r)},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),((I * c1) * r)] is set
((I * i) + ((I * c1) * r)) + (D * c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((I * i) + ((I * c1) * r)),(D * c)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((I * i) + ((I * c1) * r)),(D * c)] is non empty V18() set
{((I * i) + ((I * c1) * r)),(D * c)} is non empty finite set
{((I * i) + ((I * c1) * r))} is non empty trivial finite 1 -element set
{{((I * i) + ((I * c1) * r)),(D * c)},{((I * i) + ((I * c1) * r))}} is non empty finite V39() set
the addF of R . [((I * i) + ((I * c1) * r)),(D * c)] is set
c1 * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c1,r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,r] is non empty V18() set
{c1,r} is non empty finite set
{{c1,r},{c1}} is non empty finite V39() set
the multF of R . [c1,r] is set
I * (c1 * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,(c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(c1 * r)] is non empty V18() set
{I,(c1 * r)} is non empty finite set
{{I,(c1 * r)},{I}} is non empty finite V39() set
the multF of R . [I,(c1 * r)] is set
(I * i) + (I * (c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),(I * (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(I * (c1 * r))] is non empty V18() set
{(I * i),(I * (c1 * r))} is non empty finite set
{{(I * i),(I * (c1 * r))},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(I * (c1 * r))] is set
((I * i) + (I * (c1 * r))) + (D * c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((I * i) + (I * (c1 * r))),(D * c)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((I * i) + (I * (c1 * r))),(D * c)] is non empty V18() set
{((I * i) + (I * (c1 * r))),(D * c)} is non empty finite set
{((I * i) + (I * (c1 * r)))} is non empty trivial finite 1 -element set
{{((I * i) + (I * (c1 * r))),(D * c)},{((I * i) + (I * (c1 * r)))}} is non empty finite V39() set
the addF of R . [((I * i) + (I * (c1 * r))),(D * c)] is set
i + (c1 * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (i,(c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,(c1 * r)] is non empty V18() set
{i,(c1 * r)} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,(c1 * r)},{i}} is non empty finite V39() set
the addF of R . [i,(c1 * r)] is set
I * (i + (c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,(i + (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(i + (c1 * r))] is non empty V18() set
{I,(i + (c1 * r))} is non empty finite set
{{I,(i + (c1 * r))},{I}} is non empty finite V39() set
the multF of R . [I,(i + (c1 * r))] is set
(I * (i + (c1 * r))) + (D * c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * (i + (c1 * r))),(D * c)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * (i + (c1 * r))),(D * c)] is non empty V18() set
{(I * (i + (c1 * r))),(D * c)} is non empty finite set
{(I * (i + (c1 * r)))} is non empty trivial finite 1 -element set
{{(I * (i + (c1 * r))),(D * c)},{(I * (i + (c1 * r)))}} is non empty finite V39() set
the addF of R . [(I * (i + (c1 * r))),(D * c)] is set
c1 * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c1,D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,D] is non empty V18() set
{c1,D} is non empty finite set
{{c1,D},{c1}} is non empty finite V39() set
the multF of R . [c1,D] is set
(c1 * D) * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((c1 * D),r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(c1 * D),r] is non empty V18() set
{(c1 * D),r} is non empty finite set
{(c1 * D)} is non empty trivial finite 1 -element set
{{(c1 * D),r},{(c1 * D)}} is non empty finite V39() set
the multF of R . [(c1 * D),r] is set
((I * i) + (D * c)) + ((c1 * D) * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((I * i) + (D * c)),((c1 * D) * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((I * i) + (D * c)),((c1 * D) * r)] is non empty V18() set
{((I * i) + (D * c)),((c1 * D) * r)} is non empty finite set
{((I * i) + (D * c))} is non empty trivial finite 1 -element set
{{((I * i) + (D * c)),((c1 * D) * r)},{((I * i) + (D * c))}} is non empty finite V39() set
the addF of R . [((I * i) + (D * c)),((c1 * D) * r)] is set
D * c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,c1] is non empty V18() set
{D,c1} is non empty finite set
{{D,c1},{D}} is non empty finite V39() set
the multF of R . [D,c1] is set
(D * c1) * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((D * c1),r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(D * c1),r] is non empty V18() set
{(D * c1),r} is non empty finite set
{(D * c1)} is non empty trivial finite 1 -element set
{{(D * c1),r},{(D * c1)}} is non empty finite V39() set
the multF of R . [(D * c1),r] is set
(D * c) + ((D * c1) * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((D * c),((D * c1) * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(D * c),((D * c1) * r)] is non empty V18() set
{(D * c),((D * c1) * r)} is non empty finite set
{(D * c)} is non empty trivial finite 1 -element set
{{(D * c),((D * c1) * r)},{(D * c)}} is non empty finite V39() set
the addF of R . [(D * c),((D * c1) * r)] is set
(I * i) + ((D * c) + ((D * c1) * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),((D * c) + ((D * c1) * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),((D * c) + ((D * c1) * r))] is non empty V18() set
{(I * i),((D * c) + ((D * c1) * r))} is non empty finite set
{{(I * i),((D * c) + ((D * c1) * r))},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),((D * c) + ((D * c1) * r))] is set
c1 * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c1,r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,r] is non empty V18() set
{c1,r} is non empty finite set
{{c1,r},{c1}} is non empty finite V39() set
the multF of R . [c1,r] is set
D * (c1 * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,(c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(c1 * r)] is non empty V18() set
{D,(c1 * r)} is non empty finite set
{{D,(c1 * r)},{D}} is non empty finite V39() set
the multF of R . [D,(c1 * r)] is set
(D * c) + (D * (c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((D * c),(D * (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(D * c),(D * (c1 * r))] is non empty V18() set
{(D * c),(D * (c1 * r))} is non empty finite set
{{(D * c),(D * (c1 * r))},{(D * c)}} is non empty finite V39() set
the addF of R . [(D * c),(D * (c1 * r))] is set
(I * i) + ((D * c) + (D * (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),((D * c) + (D * (c1 * r)))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),((D * c) + (D * (c1 * r)))] is non empty V18() set
{(I * i),((D * c) + (D * (c1 * r)))} is non empty finite set
{{(I * i),((D * c) + (D * (c1 * r)))},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),((D * c) + (D * (c1 * r)))] is set
c + (c1 * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (c,(c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c,(c1 * r)] is non empty V18() set
{c,(c1 * r)} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,(c1 * r)},{c}} is non empty finite V39() set
the addF of R . [c,(c1 * r)] is set
D * (c + (c1 * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,(c + (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(c + (c1 * r))] is non empty V18() set
{D,(c + (c1 * r))} is non empty finite set
{{D,(c + (c1 * r))},{D}} is non empty finite V39() set
the multF of R . [D,(c + (c1 * r))] is set
(I * i) + (D * (c + (c1 * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),(D * (c + (c1 * r)))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(D * (c + (c1 * r)))] is non empty V18() set
{(I * i),(D * (c + (c1 * r)))} is non empty finite set
{{(I * i),(D * (c + (c1 * r)))},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(D * (c + (c1 * r)))] is set
I * (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,(0. R)] is non empty V18() set
{I,(0. R)} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,(0. R)},{I}} is non empty finite V39() set
the multF of R . [I,(0. R)] is set
(I * (0. R)) + (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((I * (0. R)),(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * (0. R)),(0. R)] is non empty V18() set
{(I * (0. R)),(0. R)} is non empty finite set
{(I * (0. R))} is non empty trivial finite 1 -element set
{{(I * (0. R)),(0. R)},{(I * (0. R))}} is non empty finite V39() set
the addF of R . [(I * (0. R)),(0. R)] is set
D * (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,(0. R)] is non empty V18() set
{D,(0. R)} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,(0. R)},{D}} is non empty finite V39() set
the multF of R . [D,(0. R)] is set
(I * (0. R)) + (D * (0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * (0. R)),(D * (0. R))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * (0. R)),(D * (0. R))] is non empty V18() set
{(I * (0. R)),(D * (0. R))} is non empty finite set
{{(I * (0. R)),(D * (0. R))},{(I * (0. R))}} is non empty finite V39() set
the addF of R . [(I * (0. R)),(D * (0. R))] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,i] is non empty V18() set
{I,i} is non empty finite set
{{I,i},{I}} is non empty finite V39() set
the multF of R . [I,i] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,i] is non empty V18() set
{D,i} is non empty finite set
{{D,i},{D}} is non empty finite V39() set
the multF of R . [D,i] is set
(I * i) + (D * i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((I * i),(D * i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(D * i)] is non empty V18() set
{(I * i),(D * i)} is non empty finite set
{(I * i)} is non empty trivial finite 1 -element set
{{(I * i),(D * i)},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(D * i)] is set
F is set
(R,{I,D}) is non empty (R) (R) (R) Element of bool the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I,D})
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I,D})
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,i] is non empty V18() set
{I,i} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,i},{I}} is non empty finite V39() set
the multF of R . [I,i] is set
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,k] is non empty V18() set
{D,k} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,k},{D}} is non empty finite V39() set
the multF of R . [D,k] is set
(I * i) + (D * k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((I * i),(D * k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * i),(D * k)] is non empty V18() set
{(I * i),(D * k)} is non empty finite set
{(I * i)} is non empty trivial finite 1 -element set
{{(I * i),(D * k)},{(I * i)}} is non empty finite V39() set
the addF of R . [(I * i),(D * k)] is set
<*(I * i),(D * k)*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite 2 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom <*(I * i),(D * k)*> is non empty finite 2 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
i is set
<*(I * i),(D * k)*> /. i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len <*(I * i),(D * k)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len <*(I * i),(D * k)*>) is non empty finite len <*(I * i),(D * k)*> -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1,2} is non empty finite V39() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*(I * i),(D * k)*> . 1 is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is Element of {I,D}
(1. R) * e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((1. R),e) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(1. R),e] is non empty V18() set
{(1. R),e} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),e},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),e] is set
((1. R) * e) * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (((1. R) * e),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((1. R) * e),i] is non empty V18() set
{((1. R) * e),i} is non empty finite set
{((1. R) * e)} is non empty trivial finite 1 -element set
{{((1. R) * e),i},{((1. R) * e)}} is non empty finite V39() set
the multF of R . [((1. R) * e),i] is set
<*(I * i),(D * k)*> . 2 is set
1. R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the OneF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is Element of {I,D}
(1. R) * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((1. R),D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(1. R),D] is non empty V18() set
{(1. R),D} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),D},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),D] is set
((1. R) * D) * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (((1. R) * D),k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((1. R) * D),k] is non empty V18() set
{((1. R) * D),k} is non empty finite set
{((1. R) * D)} is non empty trivial finite 1 -element set
{{((1. R) * D),k},{((1. R) * D)}} is non empty finite V39() set
the multF of R . [((1. R) * D),k] is set
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,{I,D})
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
I is Element of the carrier of R
{I} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed unital associative commutative 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 R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{D} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{D}) is non empty (R) (R) (R) Element of bool the carrier of R
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e is set
{ (D * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R : verum } is set
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,D] is non empty V18() set
{D,D} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,D},{D}} is non empty finite V39() set
the multF of R . [D,D] is set
e * D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom (e * D) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
f is set
(e * D) /. f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e /. f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
k is Element of I
F * k is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (F,k) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the multF of R . [F,k] is set
i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(F * k) * i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . ((F * k),i) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(F * k),i] is non empty V18() set
{(F * k),i} is non empty finite set
{(F * k)} is non empty trivial finite 1 -element set
{{(F * k),i},{(F * k)}} is non empty finite V39() set
the multF of R . [(F * k),i] is set
((F * k) * i) * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (((F * k) * i),D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[((F * k) * i),D] is non empty V18() set
{((F * k) * i),D} is non empty finite set
{((F * k) * i)} is non empty trivial finite 1 -element set
{{((F * k) * i),D},{((F * k) * i)}} is non empty finite V39() set
the multF of R . [((F * k) * i),D] is set
i * D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . (i,D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[i,D] is non empty V18() set
{i,D} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,D},{i}} is non empty finite V39() set
the multF of R . [i,D] is set
(F * k) * (i * D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R . ((F * k),(i * D)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(F * k),(i * D)] is non empty V18() set
{(F * k),(i * D)} is non empty finite set
{{(F * k),(i * D)},{(F * k)}} is non empty finite V39() set
the multF of R . [(F * k),(i * D)] is set
Sum (e * D) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
R is set
{R} is non empty trivial finite 1 -element set
I is set
{R,I} is non empty finite set
D is set
R is non empty doubleLoopStr
the carrier of R is non empty set
I is Element of the carrier of R
D is Element of the carrier of R
{I,D} is non empty finite Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I,D}) is non empty (R) (R) (R) Element of bool the carrier of R
{I} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
{D} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{D}) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
e is non empty doubleLoopStr
the carrier of e is non empty set
I is Element of the carrier of R
{I} is non empty trivial finite 1 -element Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
D is Element of the carrier of R
{I,D} is non empty finite Element of bool the carrier of R
(R,{I,D}) is non empty (R) (R) (R) Element of bool the carrier of R
D is Element of the carrier of e
{D} is non empty trivial finite 1 -element Element of bool the carrier of e
bool the carrier of e is non empty set
(e,{D}) is non empty (e) (e) (e) Element of bool the carrier of e
e is Element of the carrier of e
{e,D} is non empty finite Element of bool the carrier of e
(e,{e,D}) is non empty (e) (e) (e) Element of bool the carrier of e
R is non empty multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
D is Element of the carrier of R
I is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
e is non empty set
the Element of e is Element of e
e9 is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
e is set
e is non empty set
the Element of e is Element of e
e9 is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
{ (D * b1) where b1 is Element of the carrier of R : b1 in e } is set
f is non empty set
F is set
i is Element of the carrier of R
D * i is Element of the carrier of R
the multF of R . (D,i) is Element of the carrier of R
[D,i] is non empty V18() set
{D,i} is non empty finite set
{{D,i},{D}} is non empty finite V39() set
the multF of R . [D,i] is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is Element of the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
the Element of I is Element of I
D * the Element of I is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D, the Element of I) is Element of the carrier of R
[D, the Element of I] is non empty V18() set
{D, the Element of I} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D, the Element of I},{D}} is non empty finite V39() set
the multF of R . [D, the Element of I] is set
R is non empty right-distributive left-distributive distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is Element of the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
e9 is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is Element of the carrier of R
D * f is Element of the carrier of R
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e9 + f is Element of the carrier of R
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is Element of the carrier of R
D * F is Element of the carrier of R
the multF of R . (D,F) is Element of the carrier of R
[D,F] is non empty V18() set
{D,F} is non empty finite set
{{D,F},{D}} is non empty finite V39() set
the multF of R . [D,F] is set
R is non empty associative doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is Element of the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
e9 is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
e9 * e is Element of the carrier of R
the multF of R . (e9,e) is Element of the carrier of R
[e9,e] is non empty V18() set
{e9,e} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,e},{e9}} is non empty finite V39() set
the multF of R . [e9,e] is set
D * (e9 * e) is Element of the carrier of R
the multF of R . (D,(e9 * e)) is Element of the carrier of R
[D,(e9 * e)] is non empty V18() set
{D,(e9 * e)} is non empty finite set
{{D,(e9 * e)},{D}} is non empty finite V39() set
the multF of R . [D,(e9 * e)] is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
I is non empty Element of bool the carrier of R
(R,I,(0. R)) is non empty Element of bool the carrier of R
{ ((0. R) * b1) where b1 is Element of the carrier of R : b1 in I } is set
the Element of I is Element of I
e is set
(0. R) * the Element of I is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
D is set
e is left_add-cancelable Element of the carrier of R
(0. R) * e is left_add-cancelable Element of the carrier of R
the multF of R . ((0. R),e) is left_add-cancelable Element of the carrier of R
[(0. R),e] is non empty V18() set
{(0. R),e} is non empty finite set
{{(0. R),e},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R),e] is set
R is non empty left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
I is Element of bool the carrier of R
(R,I,(1. R)) is Element of bool the carrier of R
{ ((1. R) * b1) where b1 is Element of the carrier of R : b1 in I } is set
D is set
e is Element of the carrier of R
(1. R) * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),e) is Element of the carrier of R
[(1. R),e] is non empty V18() set
{(1. R),e} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),e},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),e] is set
D is set
e is Element of the carrier of R
(1. R) * e is Element of the carrier of R
the multF of R . ((1. R),e) is Element of the carrier of R
[(1. R),e] is non empty V18() set
{(1. R),e} is non empty finite set
{{(1. R),e},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),e] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
e is non empty set
the Element of e is Element of e
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
e is set
e is non empty set
the Element of e is Element of e
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
e is set
D is non empty set
the Element of D is Element of D
e is non empty set
the Element of e is Element of e
F is Element of the carrier of R
i is Element of the carrier of R
F + i is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in e ) } is set
k is non empty set
i is set
i is Element of the carrier of R
c is Element of the carrier of R
i + c is Element of the carrier of R
the addF of R . (i,c) is Element of the carrier of R
[i,c] is non empty V18() set
{i,c} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c},{i}} is non empty finite V39() set
the addF of R . [i,c] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
the Element of D is Element of D
the Element of I is Element of I
the Element of I + the Element of D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ( the Element of I, the Element of D) is Element of the carrier of R
[ the Element of I, the Element of D] is non empty V18() set
{ the Element of I, the Element of D} is non empty finite set
{ the Element of I} is non empty trivial finite 1 -element set
{{ the Element of I, the Element of D},{ the Element of I}} is non empty finite V39() set
the addF of R . [ the Element of I, the Element of D] is set
R is non empty Abelian addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is set
e is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,e,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in e & b2 in e ) } is set
(R,e,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in e & b2 in e ) } is set
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
D is set
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
e is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,e,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in e & b2 in e ) } is set
(R,e,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in e & b2 in e ) } is set
R is non empty Abelian add-associative addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
e is Element of the carrier of R
D is Element of the carrier of R
e + D is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is Element of the carrier of R
i is Element of the carrier of R
F + i is Element of the carrier of R
the addF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
e9 + F is Element of the carrier of R
the addF of R . (e9,F) is Element of the carrier of R
[e9,F] is non empty V18() set
{e9,F} is non empty finite set
{{e9,F},{e9}} is non empty finite V39() set
the addF of R . [e9,F] is set
f + i is Element of the carrier of R
the addF of R . (f,i) is Element of the carrier of R
[f,i] is non empty V18() set
{f,i} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,i},{f}} is non empty finite V39() set
the addF of R . [f,i] is set
(e9 + F) + (f + i) is Element of the carrier of R
the addF of R . ((e9 + F),(f + i)) is Element of the carrier of R
[(e9 + F),(f + i)] is non empty V18() set
{(e9 + F),(f + i)} is non empty finite set
{(e9 + F)} is non empty trivial finite 1 -element set
{{(e9 + F),(f + i)},{(e9 + F)}} is non empty finite V39() set
the addF of R . [(e9 + F),(f + i)] is set
(e9 + F) + f is Element of the carrier of R
the addF of R . ((e9 + F),f) is Element of the carrier of R
[(e9 + F),f] is non empty V18() set
{(e9 + F),f} is non empty finite set
{{(e9 + F),f},{(e9 + F)}} is non empty finite V39() set
the addF of R . [(e9 + F),f] is set
((e9 + F) + f) + i is Element of the carrier of R
the addF of R . (((e9 + F) + f),i) is Element of the carrier of R
[((e9 + F) + f),i] is non empty V18() set
{((e9 + F) + f),i} is non empty finite set
{((e9 + F) + f)} is non empty trivial finite 1 -element set
{{((e9 + F) + f),i},{((e9 + F) + f)}} is non empty finite V39() set
the addF of R . [((e9 + F) + f),i] is set
F + e is Element of the carrier of R
the addF of R . (F,e) is Element of the carrier of R
[F,e] is non empty V18() set
{F,e} is non empty finite set
{{F,e},{F}} is non empty finite V39() set
the addF of R . [F,e] is set
(F + e) + i is Element of the carrier of R
the addF of R . ((F + e),i) is Element of the carrier of R
[(F + e),i] is non empty V18() set
{(F + e),i} is non empty finite set
{(F + e)} is non empty trivial finite 1 -element set
{{(F + e),i},{(F + e)}} is non empty finite V39() set
the addF of R . [(F + e),i] is set
R is non empty left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
D is Element of the carrier of R
e is Element of the carrier of R
D * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e) is Element of the carrier of R
[D,e] is non empty V18() set
{D,e} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e},{D}} is non empty finite V39() set
the multF of R . [D,e] is set
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
e9 * e is Element of the carrier of R
the multF of R . (e9,e) is Element of the carrier of R
[e9,e] is non empty V18() set
{e9,e} is non empty finite set
{{e9,e},{e9}} is non empty finite V39() set
the multF of R . [e9,e] is set
f * e is Element of the carrier of R
the multF of R . (f,e) is Element of the carrier of R
[f,e] is non empty V18() set
{f,e} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e},{f}} is non empty finite V39() set
the multF of R . [f,e] is set
(e9 * e) + (f * e) is Element of the carrier of R
the addF of R . ((e9 * e),(f * e)) is Element of the carrier of R
[(e9 * e),(f * e)] is non empty V18() set
{(e9 * e),(f * e)} is non empty finite set
{(e9 * e)} is non empty trivial finite 1 -element set
{{(e9 * e),(f * e)},{(e9 * e)}} is non empty finite V39() set
the addF of R . [(e9 * e),(f * e)] is set
R is non empty right-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
D is Element of the carrier of R
e is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
e9 is Element of the carrier of R
f is Element of the carrier of R
e9 + f is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . (e9,f) is Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
e * e9 is Element of the carrier of R
the multF of R . (e,e9) is Element of the carrier of R
[e,e9] is non empty V18() set
{e,e9} is non empty finite set
{{e,e9},{e}} is non empty finite V39() set
the multF of R . [e,e9] is set
e * f is Element of the carrier of R
the multF of R . (e,f) is Element of the carrier of R
[e,f] is non empty V18() set
{e,f} is non empty finite set
{{e,f},{e}} is non empty finite V39() set
the multF of R . [e,f] is set
(e * e9) + (e * f) is Element of the carrier of R
the addF of R . ((e * e9),(e * f)) is Element of the carrier of R
[(e * e9),(e * f)] is non empty V18() set
{(e * e9),(e * f)} is non empty finite set
{(e * e9)} is non empty trivial finite 1 -element set
{{(e * e9),(e * f)},{(e * e9)}} is non empty finite V39() set
the addF of R . [(e * e9),(e * f)] is set
R is non empty add-associative addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,D,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in e ) } is set
(R,I,(R,D,e)) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in (R,D,e) ) } is set
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
(R,(R,I,D),e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in (R,I,D) & b2 in e ) } is set
e is set
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is Element of the carrier of R
F is Element of the carrier of R
f + F is Element of the carrier of R
the addF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the addF of R . [f,F] is set
F + e9 is Element of the carrier of R
the addF of R . (F,e9) is Element of the carrier of R
[F,e9] is non empty V18() set
{F,e9} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,e9},{F}} is non empty finite V39() set
the addF of R . [F,e9] is set
f + (F + e9) is Element of the carrier of R
the addF of R . (f,(F + e9)) is Element of the carrier of R
[f,(F + e9)] is non empty V18() set
{f,(F + e9)} is non empty finite set
{{f,(F + e9)},{f}} is non empty finite V39() set
the addF of R . [f,(F + e9)] is set
e is set
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is Element of the carrier of R
F is Element of the carrier of R
f + F is Element of the carrier of R
the addF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the addF of R . [f,F] is set
D + f is Element of the carrier of R
the addF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the addF of R . [D,f] is set
(D + f) + F is Element of the carrier of R
the addF of R . ((D + f),F) is Element of the carrier of R
[(D + f),F] is non empty V18() set
{(D + f),F} is non empty finite set
{(D + f)} is non empty trivial finite 1 -element set
{{(D + f),F},{(D + f)}} is non empty finite V39() set
the addF of R . [(D + f),F] is set
R is non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is non empty (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
e is set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
e is Element of I
e + (0. R) is right_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,(0. R)) is right_add-cancelable Element of the carrier of R
[e,(0. R)] is non empty V18() set
{e,(0. R)} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,(0. R)},{e}} is non empty finite V39() set
the addF of R . [e,(0. R)] is set
R is non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is non empty (R) Element of bool the carrier of R
I is non empty (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
e is set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
e is Element of D
(0. R) + e is right_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((0. R),e) is right_add-cancelable Element of the carrier of R
[(0. R),e] is non empty V18() set
{(0. R),e} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),e},{(0. R)}} is non empty finite V39() set
the addF of R . [(0. R),e] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
e is (R) Element of bool the carrier of R
e is set
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
{I,D} is non empty finite Element of bool the carrier of R
bool the carrier of R is non empty set
(R,{I,D}) is non empty (R) (R) (R) Element of bool the carrier of R
{I} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{I}) is non empty (R) (R) (R) Element of bool the carrier of R
{D} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{D}) is non empty (R) (R) (R) Element of bool the carrier of R
(R,(R,{I}),(R,{D})) is non empty (R) (R) (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in (R,{I}) & b2 in (R,{D}) ) } is set
e is set
{ ((I * b1) + (D * b2)) where b1, b2 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : verum } is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (I,e) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,e] is non empty V18() set
{I,e} is non empty finite set
{I} is non empty trivial finite 1 -element set
{{I,e},{I}} is non empty finite V39() set
the multF of R . [I,e] is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,D] is non empty V18() set
{D,D} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,D},{D}} is non empty finite V39() set
the multF of R . [D,D] is set
(I * e) + (D * D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((I * e),(D * D)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(I * e),(D * D)] is non empty V18() set
{(I * e),(D * D)} is non empty finite set
{(I * e)} is non empty trivial finite 1 -element set
{{(I * e),(D * D)},{(I * e)}} is non empty finite V39() set
the addF of R . [(I * e),(D * D)] is set
{ (D * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : verum } is set
{ (I * b1) where b1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : verum } is set
f is Element of (R,{I})
e9 is Element of (R,{D})
f + e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (f,e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,e9] is non empty V18() set
{f,e9} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,e9},{f}} is non empty finite V39() set
the addF of R . [f,e9] is set
{ (b1 + b2) where b1, b2 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ( b1 in (R,{I}) & b2 in (R,{D}) ) } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e + D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (e,D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
I * e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (I,e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[I,e9] is non empty V18() set
{I,e9} is non empty finite set
{{I,e9},{I}} is non empty finite V39() set
the multF of R . [I,e9] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D * f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (D,f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
R is non empty 1-sorted
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
I /\ D is set
I /\ D is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
{ b1 where b1 is Element of the carrier of R : ( S2[b1] & S1[b1] ) } is set
{ b1 where b1 is Element of the carrier of R : S2[b1] } is set
{ b1 where b1 is Element of the carrier of R : S1[b1] } is set
{ b1 where b1 is Element of the carrier of R : S2[b1] } /\ { b1 where b1 is Element of the carrier of R : S1[b1] } is set
e9 is Element of bool the carrier of R
f is Element of bool the carrier of R
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is non empty (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
e is Element of bool the carrier of R
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is Element of the carrier of R
F is Element of the carrier of R
F + f is Element of the carrier of R
the addF of R . (F,f) is Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the addF of R . [F,f] is set
D is Element of bool the carrier of R
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
e is Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
D * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is Element of the carrier of R
D * f is Element of the carrier of R
the multF of R . (D,f) is Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
D is Element of bool the carrier of R
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
e is Element of bool the carrier of R
e9 is Element of the carrier of R
D is Element of the carrier of R
e9 * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f is Element of the carrier of R
f * D is Element of the carrier of R
the multF of R . (f,D) is Element of the carrier of R
[f,D] is non empty V18() set
{f,D} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,D},{f}} is non empty finite V39() set
the multF of R . [f,D] is set
D is Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable Abelian add-associative right_zeroed left-distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is Element of bool the carrier of R
e is non empty Element of bool the carrier of R
(R,D,e) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in e ) } is set
(R,I,(R,D,e)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in (R,D,e) ) } is set
(R,I,e) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in e ) } is set
(R,D,(R,I,e)) is Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in (R,I,e) ) } is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
D + e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is Element of e
D + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (D,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the addF of R . [D,f] is set
k is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
F is Element of I
i is Element of I
F + i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (F,i) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
e9 + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (e9,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[e9,f] is non empty V18() set
{e9,f} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,f},{e9}} is non empty finite V39() set
the addF of R . [e9,f] is set
F is Element of I
- F is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
F + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (F,f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the addF of R . [F,f] is set
(F + f) + (- F) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((F + f),(- F)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(F + f),(- F)] is non empty V18() set
{(F + f),(- F)} is non empty finite set
{(F + f)} is non empty trivial finite 1 -element set
{{(F + f),(- F)},{(F + f)}} is non empty finite V39() set
the addF of R . [(F + f),(- F)] is set
F + (- F) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (F,(- F)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[F,(- F)] is non empty V18() set
{F,(- F)} is non empty finite set
{{F,(- F)},{F}} is non empty finite V39() set
the addF of R . [F,(- F)] is set
(F + (- F)) + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((F + (- F)),f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(F + (- F)),f] is non empty V18() set
{(F + (- F)),f} is non empty finite set
{(F + (- F))} is non empty trivial finite 1 -element set
{{(F + (- F)),f},{(F + (- F))}} is non empty finite V39() set
the addF of R . [(F + (- F)),f] is set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(0. R) + f is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((0. R),f) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(0. R),f] is non empty V18() set
{(0. R),f} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),f},{(0. R)}} is non empty finite V39() set
the addF of R . [(0. R),f] is set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
e is set
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum D is Element of the carrier of R
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
len (<*> the carrier of R) is empty trivial Relation-like non-empty empty-yielding RAT -valued functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSequence-membered V106() complex ext-real non positive non negative V147() V148() V149() V150() V151() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() Element of NAT
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(<*> the carrier of R) . D is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
Sum (<*> the carrier of R) is Element of the carrier of R
R is non empty commutative doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is set
e is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,e,e) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in e & b4 in e ) ) } is set
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e9 is Element of the carrier of R
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 . f is set
F is Element of the carrier of R
i is Element of the carrier of R
F * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the multF of R . [F,i] is set
(R,e,e) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in e & b4 in e ) ) } is set
D is set
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e9 is Element of the carrier of R
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 . f is set
F is Element of the carrier of R
i is Element of the carrier of R
F * i is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (F,i) is Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the multF of R . [F,i] is set
e is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,e,e) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in e & b4 in e ) ) } is set
(R,e,e) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in e & b4 in e ) ) } is set
R is non empty add-associative right_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e9 is Element of the carrier of R
D + e9 is Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum f is Element of the carrier of R
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum F is Element of the carrier of R
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f ^ F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len (f ^ F) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(f ^ F) . k is set
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
f . k is set
k - (len f) is V34() V106() complex ext-real set
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(len f) - (len f) is V34() V106() complex ext-real set
(len f) + (len F) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
((len f) + (len F)) - (len f) is V34() V106() complex ext-real set
F . i is set
Sum (f ^ F) is Element of the carrier of R
R is non empty left_add-cancelable right_zeroed associative left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
e is non empty Element of bool the carrier of R
e9 is left_add-cancelable Element of the carrier of R
D is left_add-cancelable Element of the carrier of R
e9 * D is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,D) is left_add-cancelable Element of the carrier of R
[e9,D] is non empty V18() set
{e9,D} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,D},{e9}} is non empty finite V39() set
the multF of R . [e9,D] is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum f is left_add-cancelable Element of the carrier of R
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f * D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len (f * D) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len (f * D)) is finite len (f * D) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom (f * D) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f . i is set
k is left_add-cancelable Element of the carrier of R
i is left_add-cancelable Element of the carrier of R
k * i is left_add-cancelable Element of the carrier of R
the multF of R . (k,i) is left_add-cancelable Element of the carrier of R
[k,i] is non empty V18() set
{k,i} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,i},{k}} is non empty finite V39() set
the multF of R . [k,i] is set
f /. i is left_add-cancelable Element of the carrier of R
(f * D) . i is set
(f * D) /. i is left_add-cancelable Element of the carrier of R
(k * i) * D is left_add-cancelable Element of the carrier of R
the multF of R . ((k * i),D) is left_add-cancelable Element of the carrier of R
[(k * i),D] is non empty V18() set
{(k * i),D} is non empty finite set
{(k * i)} is non empty trivial finite 1 -element set
{{(k * i),D},{(k * i)}} is non empty finite V39() set
the multF of R . [(k * i),D] is set
i * D is left_add-cancelable Element of the carrier of R
the multF of R . (i,D) is left_add-cancelable Element of the carrier of R
[i,D] is non empty V18() set
{i,D} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,D},{i}} is non empty finite V39() set
the multF of R . [i,D] is set
k * (i * D) is left_add-cancelable Element of the carrier of R
the multF of R . (k,(i * D)) is left_add-cancelable Element of the carrier of R
[k,(i * D)] is non empty V18() set
{k,(i * D)} is non empty finite set
{{k,(i * D)},{k}} is non empty finite V39() set
the multF of R . [k,(i * D)] is set
Sum (f * D) is left_add-cancelable Element of the carrier of R
(Sum f) * D is left_add-cancelable Element of the carrier of R
the multF of R . ((Sum f),D) is left_add-cancelable Element of the carrier of R
[(Sum f),D] is non empty V18() set
{(Sum f),D} is non empty finite set
{(Sum f)} is non empty trivial finite 1 -element set
{{(Sum f),D},{(Sum f)}} is non empty finite V39() set
the multF of R . [(Sum f),D] is set
R is non empty right_add-cancelable associative right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
e is non empty Element of bool the carrier of R
e9 is right_add-cancelable Element of the carrier of R
D is right_add-cancelable Element of the carrier of R
D * e9 is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is right_add-cancelable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum f is right_add-cancelable Element of the carrier of R
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D * f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len (D * f) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len (D * f)) is finite len (D * f) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom (D * f) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f . i is set
k is right_add-cancelable Element of the carrier of R
i is right_add-cancelable Element of the carrier of R
k * i is right_add-cancelable Element of the carrier of R
the multF of R . (k,i) is right_add-cancelable Element of the carrier of R
[k,i] is non empty V18() set
{k,i} is non empty finite set
{k} is non empty trivial finite 1 -element set
{{k,i},{k}} is non empty finite V39() set
the multF of R . [k,i] is set
f /. i is right_add-cancelable Element of the carrier of R
(D * f) . i is set
(D * f) /. i is right_add-cancelable Element of the carrier of R
D * (k * i) is right_add-cancelable Element of the carrier of R
the multF of R . (D,(k * i)) is right_add-cancelable Element of the carrier of R
[D,(k * i)] is non empty V18() set
{D,(k * i)} is non empty finite set
{{D,(k * i)},{D}} is non empty finite V39() set
the multF of R . [D,(k * i)] is set
D * k is right_add-cancelable Element of the carrier of R
the multF of R . (D,k) is right_add-cancelable Element of the carrier of R
[D,k] is non empty V18() set
{D,k} is non empty finite set
{{D,k},{D}} is non empty finite V39() set
the multF of R . [D,k] is set
(D * k) * i is right_add-cancelable Element of the carrier of R
the multF of R . ((D * k),i) is right_add-cancelable Element of the carrier of R
[(D * k),i] is non empty V18() set
{(D * k),i} is non empty finite set
{(D * k)} is non empty trivial finite 1 -element set
{{(D * k),i},{(D * k)}} is non empty finite V39() set
the multF of R . [(D * k),i] is set
Sum (D * f) is right_add-cancelable Element of the carrier of R
D * (Sum f) is right_add-cancelable Element of the carrier of R
the multF of R . (D,(Sum f)) is right_add-cancelable Element of the carrier of R
[D,(Sum f)] is non empty V18() set
{D,(Sum f)} is non empty finite set
{{D,(Sum f)},{D}} is non empty finite V39() set
the multF of R . [D,(Sum f)] is set
R is non empty left_add-cancelable right_zeroed left-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
I is non empty Element of bool the carrier of R
(R,{(0. R)},I) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in {(0. R)} & b4 in I ) ) } is set
D is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e is left_add-cancelable Element of the carrier of R
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. e is left_add-cancelable Element of the carrier of R
e . e is set
D is left_add-cancelable Element of the carrier of R
e9 is left_add-cancelable Element of the carrier of R
D * e9 is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is left_add-cancelable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e /. e is left_add-cancelable Element of the carrier of R
e /. 1 is left_add-cancelable Element of the carrier of R
the Element of I is Element of I
e is set
(0. R) * the Element of I is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
<*((0. R) * the Element of I)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len <*((0. R) * the Element of I)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
<*((0. R) * the Element of I)*> . 1 is set
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
<*((0. R) * the Element of I)*> . e9 is set
D is Element of {(0. R)}
D * the Element of I is left_add-cancelable Element of the carrier of R
the multF of R . (D, the Element of I) is left_add-cancelable Element of the carrier of R
[D, the Element of I] is non empty V18() set
{D, the Element of I} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D, the Element of I},{D}} is non empty finite V39() set
the multF of R . [D, the Element of I] is set
Sum <*((0. R) * the Element of I)*> is left_add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_zeroed right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
(R,I,D) is (R) Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
e is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
D . (len e) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (e9 + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e . (e9 + 1) is set
e /. (e9 + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f * F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
(D . e9) + (e /. (e9 + 1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((D . e9),(e /. (e9 + 1))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(D . e9),(e /. (e9 + 1))] is non empty V18() set
{(D . e9),(e /. (e9 + 1))} is non empty finite set
{(D . e9)} is non empty trivial finite 1 -element set
{{(D . e9),(e /. (e9 + 1))},{(D . e9)}} is non empty finite V39() set
the addF of R . [(D . e9),(e /. (e9 + 1))] is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed associative right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is non empty (R) Element of bool the carrier of R
e is non empty (R) Element of bool the carrier of R
(R,D,e) is non empty (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in e ) } is set
(R,I,(R,D,e)) is non empty (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in (R,D,e) ) ) } is set
(R,I,D) is non empty (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
(R,I,e) is non empty (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in e ) ) } is set
(R,(R,I,D),(R,I,e)) is non empty (R) (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in (R,I,D) & b2 in (R,I,e) ) } is set
e is set
D is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
e9 . (len D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 . f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e9 . (f + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (F,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (F,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the addF of R . [F,i] is set
k is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (f + 1) is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,c] is non empty V18() set
{i,c} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c},{i}} is non empty finite V39() set
the multF of R . [i,c] is set
c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c1 + r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (c1,r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c1,r] is non empty V18() set
{c1,r} is non empty finite set
{c1} is non empty trivial finite 1 -element set
{{c1,r},{c1}} is non empty finite V39() set
the addF of R . [c1,r] is set
i * c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,c1] is non empty V18() set
{i,c1} is non empty finite set
{{i,c1},{i}} is non empty finite V39() set
the multF of R . [i,c1] is set
<*(i * c1)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
i ^ <*(i * c1)*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
i * r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,r] is non empty V18() set
{i,r} is non empty finite set
{{i,r},{i}} is non empty finite V39() set
the multF of R . [i,r] is set
<*(i * r)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
k ^ <*(i * r)*> is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Seg (len D) is finite len D -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
D /. (f + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len (k ^ <*(i * r)*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
len <*(i * r)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len k) + (len <*(i * r)*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len k) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(k ^ <*(i * r)*>) . k is set
k . k is set
a is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
b is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
a * b is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (a,b) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[a,b] is non empty V18() set
{a,b} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,b},{a}} is non empty finite V39() set
the multF of R . [a,b] is set
Seg (len k) is finite len k -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom k is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
Sum (k ^ <*(i * r)*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len (i ^ <*(i * c1)*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
len <*(i * c1)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + (len <*(i * c1)*>) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(len i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(i ^ <*(i * c1)*>) . k is set
i . k is set
a is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
b is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
a * b is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (a,b) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[a,b] is non empty V18() set
{a,b} is non empty finite set
{a} is non empty trivial finite 1 -element set
{{a,b},{a}} is non empty finite V39() set
the multF of R . [a,b] is set
Seg (len i) is finite len i -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom i is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
Sum (i ^ <*(i * c1)*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * (c1 + r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,(c1 + r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,(c1 + r)] is non empty V18() set
{i,(c1 + r)} is non empty finite set
{{i,(c1 + r)},{i}} is non empty finite V39() set
the multF of R . [i,(c1 + r)] is set
(i * c1) + (i * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((i * c1),(i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i * c1),(i * r)] is non empty V18() set
{(i * c1),(i * r)} is non empty finite set
{(i * c1)} is non empty trivial finite 1 -element set
{{(i * c1),(i * r)},{(i * c1)}} is non empty finite V39() set
the addF of R . [(i * c1),(i * r)] is set
(Sum (i ^ <*(i * c1)*>)) + (Sum (k ^ <*(i * r)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum (i ^ <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum (i ^ <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))] is non empty V18() set
{(Sum (i ^ <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))} is non empty finite set
{(Sum (i ^ <*(i * c1)*>))} is non empty trivial finite 1 -element set
{{(Sum (i ^ <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))},{(Sum (i ^ <*(i * c1)*>))}} is non empty finite V39() set
the addF of R . [(Sum (i ^ <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))] is set
Sum <*(i * c1)*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(Sum i) + (Sum <*(i * c1)*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum i),(Sum <*(i * c1)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(Sum <*(i * c1)*>)] is non empty V18() set
{(Sum i),(Sum <*(i * c1)*>)} is non empty finite set
{(Sum i)} is non empty trivial finite 1 -element set
{{(Sum i),(Sum <*(i * c1)*>)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(Sum <*(i * c1)*>)] is set
((Sum i) + (Sum <*(i * c1)*>)) + (Sum (k ^ <*(i * r)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((Sum i) + (Sum <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((Sum i) + (Sum <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))] is non empty V18() set
{((Sum i) + (Sum <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))} is non empty finite set
{((Sum i) + (Sum <*(i * c1)*>))} is non empty trivial finite 1 -element set
{{((Sum i) + (Sum <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))},{((Sum i) + (Sum <*(i * c1)*>))}} is non empty finite V39() set
the addF of R . [((Sum i) + (Sum <*(i * c1)*>)),(Sum (k ^ <*(i * r)*>))] is set
(Sum i) + (i * c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum i),(i * c1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(i * c1)] is non empty V18() set
{(Sum i),(i * c1)} is non empty finite set
{{(Sum i),(i * c1)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(i * c1)] is set
((Sum i) + (i * c1)) + (Sum (k ^ <*(i * r)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((Sum i) + (i * c1)),(Sum (k ^ <*(i * r)*>))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((Sum i) + (i * c1)),(Sum (k ^ <*(i * r)*>))] is non empty V18() set
{((Sum i) + (i * c1)),(Sum (k ^ <*(i * r)*>))} is non empty finite set
{((Sum i) + (i * c1))} is non empty trivial finite 1 -element set
{{((Sum i) + (i * c1)),(Sum (k ^ <*(i * r)*>))},{((Sum i) + (i * c1))}} is non empty finite V39() set
the addF of R . [((Sum i) + (i * c1)),(Sum (k ^ <*(i * r)*>))] is set
Sum <*(i * r)*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(Sum k) + (Sum <*(i * r)*>) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum k),(Sum <*(i * r)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum k),(Sum <*(i * r)*>)] is non empty V18() set
{(Sum k),(Sum <*(i * r)*>)} is non empty finite set
{(Sum k)} is non empty trivial finite 1 -element set
{{(Sum k),(Sum <*(i * r)*>)},{(Sum k)}} is non empty finite V39() set
the addF of R . [(Sum k),(Sum <*(i * r)*>)] is set
((Sum i) + (i * c1)) + ((Sum k) + (Sum <*(i * r)*>)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((Sum i) + (i * c1)),((Sum k) + (Sum <*(i * r)*>))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((Sum i) + (i * c1)),((Sum k) + (Sum <*(i * r)*>))] is non empty V18() set
{((Sum i) + (i * c1)),((Sum k) + (Sum <*(i * r)*>))} is non empty finite set
{{((Sum i) + (i * c1)),((Sum k) + (Sum <*(i * r)*>))},{((Sum i) + (i * c1))}} is non empty finite V39() set
the addF of R . [((Sum i) + (i * c1)),((Sum k) + (Sum <*(i * r)*>))] is set
(Sum k) + (i * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum k),(i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum k),(i * r)] is non empty V18() set
{(Sum k),(i * r)} is non empty finite set
{{(Sum k),(i * r)},{(Sum k)}} is non empty finite V39() set
the addF of R . [(Sum k),(i * r)] is set
((Sum i) + (i * c1)) + ((Sum k) + (i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((Sum i) + (i * c1)),((Sum k) + (i * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((Sum i) + (i * c1)),((Sum k) + (i * r))] is non empty V18() set
{((Sum i) + (i * c1)),((Sum k) + (i * r))} is non empty finite set
{{((Sum i) + (i * c1)),((Sum k) + (i * r))},{((Sum i) + (i * c1))}} is non empty finite V39() set
the addF of R . [((Sum i) + (i * c1)),((Sum k) + (i * r))] is set
((Sum i) + (i * c1)) + (Sum k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((Sum i) + (i * c1)),(Sum k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((Sum i) + (i * c1)),(Sum k)] is non empty V18() set
{((Sum i) + (i * c1)),(Sum k)} is non empty finite set
{{((Sum i) + (i * c1)),(Sum k)},{((Sum i) + (i * c1))}} is non empty finite V39() set
the addF of R . [((Sum i) + (i * c1)),(Sum k)] is set
(((Sum i) + (i * c1)) + (Sum k)) + (i * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((((Sum i) + (i * c1)) + (Sum k)),(i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(((Sum i) + (i * c1)) + (Sum k)),(i * r)] is non empty V18() set
{(((Sum i) + (i * c1)) + (Sum k)),(i * r)} is non empty finite set
{(((Sum i) + (i * c1)) + (Sum k))} is non empty trivial finite 1 -element set
{{(((Sum i) + (i * c1)) + (Sum k)),(i * r)},{(((Sum i) + (i * c1)) + (Sum k))}} is non empty finite V39() set
the addF of R . [(((Sum i) + (i * c1)) + (Sum k)),(i * r)] is set
(Sum i) + (Sum k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((Sum i),(Sum k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(Sum i),(Sum k)] is non empty V18() set
{(Sum i),(Sum k)} is non empty finite set
{{(Sum i),(Sum k)},{(Sum i)}} is non empty finite V39() set
the addF of R . [(Sum i),(Sum k)] is set
(i * c1) + ((Sum i) + (Sum k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((i * c1),((Sum i) + (Sum k))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i * c1),((Sum i) + (Sum k))] is non empty V18() set
{(i * c1),((Sum i) + (Sum k))} is non empty finite set
{{(i * c1),((Sum i) + (Sum k))},{(i * c1)}} is non empty finite V39() set
the addF of R . [(i * c1),((Sum i) + (Sum k))] is set
((i * c1) + ((Sum i) + (Sum k))) + (i * r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (((i * c1) + ((Sum i) + (Sum k))),(i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((i * c1) + ((Sum i) + (Sum k))),(i * r)] is non empty V18() set
{((i * c1) + ((Sum i) + (Sum k))),(i * r)} is non empty finite set
{((i * c1) + ((Sum i) + (Sum k)))} is non empty trivial finite 1 -element set
{{((i * c1) + ((Sum i) + (Sum k))),(i * r)},{((i * c1) + ((Sum i) + (Sum k)))}} is non empty finite V39() set
the addF of R . [((i * c1) + ((Sum i) + (Sum k))),(i * r)] is set
(e9 . f) + ((i * c1) + (i * r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((e9 . f),((i * c1) + (i * r))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(e9 . f),((i * c1) + (i * r))] is non empty V18() set
{(e9 . f),((i * c1) + (i * r))} is non empty finite set
{(e9 . f)} is non empty trivial finite 1 -element set
{{(e9 . f),((i * c1) + (i * r))},{(e9 . f)}} is non empty finite V39() set
the addF of R . [(e9 . f),((i * c1) + (i * r))] is set
(0. R) + (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((0. R),(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(0. R),(0. R)] is non empty V18() set
{(0. R),(0. R)} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),(0. R)},{(0. R)}} is non empty finite V39() set
the addF of R . [(0. R),(0. R)] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f + F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (f,F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the addF of R . [f,F] is set
e is set
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D + e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f ^ F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len (f ^ F) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(f ^ F) . k is set
Seg (len (f ^ F)) is finite len (f ^ F) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom (f ^ F) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
f . k is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
(0. R) + i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((0. R),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(0. R),i] is non empty V18() set
{(0. R),i} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),i},{(0. R)}} is non empty finite V39() set
the addF of R . [(0. R),i] is set
i * ((0. R) + i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,((0. R) + i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,((0. R) + i)] is non empty V18() set
{i,((0. R) + i)} is non empty finite set
{{i,((0. R) + i)},{i}} is non empty finite V39() set
the multF of R . [i,((0. R) + i)] is set
{ (b1 + b2) where b1, b2 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ( b1 in D & b2 in e ) } is set
Seg (len f) is finite len f -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
(len f) + i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len F) is finite len F -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
F . i is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,c) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,c] is non empty V18() set
{i,c} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,c},{i}} is non empty finite V39() set
the multF of R . [i,c] is set
c + (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (c,(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c,(0. R)] is non empty V18() set
{c,(0. R)} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,(0. R)},{c}} is non empty finite V39() set
the addF of R . [c,(0. R)] is set
{ (b1 + b2) where b1, b2 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R : ( b1 in D & b2 in e ) } is set
i * (c + (0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (i,(c + (0. R))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,(c + (0. R))] is non empty V18() set
{i,(c + (0. R))} is non empty finite set
{{i,(c + (0. R))},{i}} is non empty finite V39() set
the multF of R . [i,(c + (0. R))] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c * c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (c,c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[c,c1] is non empty V18() set
{c,c1} is non empty finite set
{c} is non empty trivial finite 1 -element set
{{c,c1},{c}} is non empty finite V39() set
the multF of R . [c,c1] is set
Sum (f ^ F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed associative commutative right-distributive left-distributive distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty (R) (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
(R,I,D) is non empty (R) (R) Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
(R,(R,I,D),(R,I,D)) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in (R,I,D) ) ) } is set
(R,I,D) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
e is set
(R,I,(R,I,D)) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in (R,I,D) ) ) } is set
(R,D,(R,I,D)) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in D & b4 in (R,I,D) ) ) } is set
(R,(R,I,(R,I,D)),(R,D,(R,I,D))) is non empty (R) (R) (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in (R,I,(R,I,D)) & b2 in (R,D,(R,I,D)) ) } is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e + D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 . f is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (F,i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,i] is non empty V18() set
{F,i} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,i},{F}} is non empty finite V39() set
the multF of R . [F,i] is set
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
f is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f . F is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the multF of R . [i,k] is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
(R,I,D) is non empty (R) (R) Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
(R,(R,I,D),(R,I,D)) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in (R,I,D) ) ) } is set
e is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e is left_add-cancelable Element of the carrier of R
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
D . (len e) is left_add-cancelable Element of the carrier of R
D . 0 is left_add-cancelable Element of the carrier of R
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is left_add-cancelable Element of the carrier of R
e9 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (e9 + 1) is left_add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e . (e9 + 1) is set
e /. (e9 + 1) is left_add-cancelable Element of the carrier of R
f is left_add-cancelable Element of the carrier of R
F is left_add-cancelable Element of the carrier of R
f * F is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,F) is left_add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
(D . e9) + (e /. (e9 + 1)) is left_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((D . e9),(e /. (e9 + 1))) is left_add-cancelable Element of the carrier of R
[(D . e9),(e /. (e9 + 1))] is non empty V18() set
{(D . e9),(e /. (e9 + 1))} is non empty finite set
{(D . e9)} is non empty trivial finite 1 -element set
{{(D . e9),(e /. (e9 + 1))},{(D . e9)}} is non empty finite V39() set
the addF of R . [(D . e9),(e /. (e9 + 1))] is set
R is non empty addLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
R is non empty left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
(R,I,D) is non empty Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
(R,(R,I,D),(R,I,D)) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in (R,I,D) ) ) } is set
e is set
1. R is Element of the carrier of R
the OneF of R is Element of the carrier of R
e is Element of the carrier of R
(1. R) * e is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((1. R),e) is Element of the carrier of R
[(1. R),e] is non empty V18() set
{(1. R),e} is non empty finite set
{(1. R)} is non empty trivial finite 1 -element set
{{(1. R),e},{(1. R)}} is non empty finite V39() set
the multF of R . [(1. R),e] is set
<*((1. R) * e)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len <*((1. R) * e)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
<*((1. R) * e)*> . e9 is set
Sum <*((1. R) * e)*> is Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
D is non empty (R) (R) (R) Element of bool the carrier of R
(R,I,D) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
(R,I,D) is non empty (R) (R) (R) Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
(R,I,D) is non empty (R) (R) (R) Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in I & b2 in D ) } is set
(R,(R,I,D),(R,I,D)) is non empty (R) (R) (R) Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in (R,I,D) ) ) } is set
R is non empty multMagma
the carrier of R is non empty set
bool the carrier of R is non empty set
D is Element of bool the carrier of R
I is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
e is set
D is Element of the carrier of R
(R,D,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in D } is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is non empty (R) Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) Element of bool the carrier of R
e is set
(R,D,(0. R)) is non empty Element of bool the carrier of R
{ ((0. R) * b1) where b1 is Element of the carrier of R : b1 in D } is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
e is non empty Element of bool the carrier of R
D is left_add-cancelable Element of the carrier of R
e9 is left_add-cancelable Element of the carrier of R
D + e9 is left_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (D,e9) is left_add-cancelable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the addF of R . [D,e9] is set
f is left_add-cancelable Element of the carrier of R
(R,D,f) is non empty Element of bool the carrier of R
{ (f * b1) where b1 is Element of the carrier of R : b1 in D } is set
F is left_add-cancelable Element of the carrier of R
(R,D,F) is non empty Element of bool the carrier of R
{ (F * b1) where b1 is Element of the carrier of R : b1 in D } is set
i is set
F + f is left_add-cancelable Element of the carrier of R
the addF of R . (F,f) is left_add-cancelable Element of the carrier of R
[F,f] is non empty V18() set
{F,f} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,f},{F}} is non empty finite V39() set
the addF of R . [F,f] is set
(R,D,(F + f)) is non empty Element of bool the carrier of R
{ ((F + f) * b1) where b1 is Element of the carrier of R : b1 in D } is set
k is left_add-cancelable Element of the carrier of R
(F + f) * k is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . ((F + f),k) is left_add-cancelable Element of the carrier of R
[(F + f),k] is non empty V18() set
{(F + f),k} is non empty finite set
{(F + f)} is non empty trivial finite 1 -element set
{{(F + f),k},{(F + f)}} is non empty finite V39() set
the multF of R . [(F + f),k] is set
f * k is left_add-cancelable Element of the carrier of R
the multF of R . (f,k) is left_add-cancelable Element of the carrier of R
[f,k] is non empty V18() set
{f,k} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,k},{f}} is non empty finite V39() set
the multF of R . [f,k] is set
F * k is left_add-cancelable Element of the carrier of R
the multF of R . (F,k) is left_add-cancelable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{{F,k},{F}} is non empty finite V39() set
the multF of R . [F,k] is set
(F * k) + (f * k) is left_add-cancelable Element of the carrier of R
the addF of R . ((F * k),(f * k)) is left_add-cancelable Element of the carrier of R
[(F * k),(f * k)] is non empty V18() set
{(F * k),(f * k)} is non empty finite set
{(F * k)} is non empty trivial finite 1 -element set
{{(F * k),(f * k)},{(F * k)}} is non empty finite V39() set
the addF of R . [(F * k),(f * k)] is set
R is non empty left_add-cancelable right_zeroed associative commutative left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
e is non empty Element of bool the carrier of R
e9 is left_add-cancelable Element of the carrier of R
D is left_add-cancelable Element of the carrier of R
D * e9 is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,e9) is left_add-cancelable Element of the carrier of R
[D,e9] is non empty V18() set
{D,e9} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,e9},{D}} is non empty finite V39() set
the multF of R . [D,e9] is set
f is left_add-cancelable Element of the carrier of R
(R,D,f) is non empty (R) (R) Element of bool the carrier of R
{ (f * b1) where b1 is Element of the carrier of R : b1 in D } is set
F is set
D * f is left_add-cancelable Element of the carrier of R
the multF of R . (D,f) is left_add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
(R,D,(D * f)) is non empty (R) (R) Element of bool the carrier of R
{ ((D * f) * b1) where b1 is Element of the carrier of R : b1 in D } is set
i is left_add-cancelable Element of the carrier of R
(D * f) * i is left_add-cancelable Element of the carrier of R
the multF of R . ((D * f),i) is left_add-cancelable Element of the carrier of R
[(D * f),i] is non empty V18() set
{(D * f),i} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),i},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),i] is set
D * i is left_add-cancelable Element of the carrier of R
the multF of R . (D,i) is left_add-cancelable Element of the carrier of R
[D,i] is non empty V18() set
{D,i} is non empty finite set
{{D,i},{D}} is non empty finite V39() set
the multF of R . [D,i] is set
f * (D * i) is left_add-cancelable Element of the carrier of R
the multF of R . (f,(D * i)) is left_add-cancelable Element of the carrier of R
[f,(D * i)] is non empty V18() set
{f,(D * i)} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,(D * i)},{f}} is non empty finite V39() set
the multF of R . [f,(D * i)] is set
{ (f * b1) where b1 is left_add-cancelable Element of the carrier of R : b1 in D } is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
e is set
D is set
e is Element of the carrier of R
(R,D,e) is Element of bool the carrier of R
{ (e * b1) where b1 is Element of the carrier of R : b1 in D } is set
e9 is Element of the carrier of R
e * e9 is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,e9) is Element of the carrier of R
[e,e9] is non empty V18() set
{e,e9} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,e9},{e}} is non empty finite V39() set
the multF of R . [e,e9] is set
R is non empty left_add-cancelable right_zeroed left-distributive doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
(R,(R,I,D),D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in D ) ) } is set
e is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e is left_add-cancelable Element of the carrier of R
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero left_add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable Element of the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
D . (len e) is left_add-cancelable Element of the carrier of R
D . 0 is left_add-cancelable Element of the carrier of R
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is left_add-cancelable Element of the carrier of R
e9 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (e9 + 1) is left_add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e . (e9 + 1) is set
f is left_add-cancelable Element of the carrier of R
F is left_add-cancelable Element of the carrier of R
f * F is left_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,F) is left_add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. (e9 + 1) is left_add-cancelable Element of the carrier of R
(D . e9) + (e /. (e9 + 1)) is left_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((D . e9),(e /. (e9 + 1))) is left_add-cancelable Element of the carrier of R
[(D . e9),(e /. (e9 + 1))] is non empty V18() set
{(D . e9),(e /. (e9 + 1))} is non empty finite set
{(D . e9)} is non empty trivial finite 1 -element set
{{(D . e9),(e /. (e9 + 1))},{(D . e9)}} is non empty finite V39() set
the addF of R . [(D . e9),(e /. (e9 + 1))] is set
i is left_add-cancelable Element of the carrier of R
(R,D,i) is Element of bool the carrier of R
{ (i * b1) where b1 is Element of the carrier of R : b1 in D } is set
R is non empty right_add-cancelable right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
(R,(R,I,D),D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in D ) ) } is set
e is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e is right_add-cancelable Element of the carrier of R
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero right_add-cancelable Element of the carrier of R
the ZeroF of R is right_add-cancelable Element of the carrier of R
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
D . (len e) is right_add-cancelable Element of the carrier of R
D . 0 is right_add-cancelable Element of the carrier of R
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . e9 is right_add-cancelable Element of the carrier of R
e9 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (e9 + 1) is right_add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e . (e9 + 1) is set
f is right_add-cancelable Element of the carrier of R
F is right_add-cancelable Element of the carrier of R
f * F is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (f,F) is right_add-cancelable Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. (e9 + 1) is right_add-cancelable Element of the carrier of R
(D . e9) + (e /. (e9 + 1)) is right_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((D . e9),(e /. (e9 + 1))) is right_add-cancelable Element of the carrier of R
[(D . e9),(e /. (e9 + 1))] is non empty V18() set
{(D . e9),(e /. (e9 + 1))} is non empty finite set
{(D . e9)} is non empty trivial finite 1 -element set
{{(D . e9),(e /. (e9 + 1))},{(D . e9)}} is non empty finite V39() set
the addF of R . [(D . e9),(e /. (e9 + 1))] is set
i is right_add-cancelable Element of the carrier of R
(R,D,i) is Element of bool the carrier of R
{ (i * b1) where b1 is Element of the carrier of R : b1 in D } is set
R is non empty right_add-cancelable associative commutative right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
D is Element of bool the carrier of R
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
e is Element of bool the carrier of R
(R,(R,I,D),e) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,e,b1) c= (R,I,D) } is set
(R,D,e) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in D & b4 in e ) ) } is set
(R,I,(R,D,e)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,(R,D,e),b1) c= I } is set
e is set
D is right_add-cancelable Element of the carrier of R
(R,e,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in e } is set
e9 is set
(R,(R,D,e),D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in (R,D,e) } is set
f is right_add-cancelable Element of the carrier of R
D * f is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is right_add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum F is right_add-cancelable Element of the carrier of R
len F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D * F is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
dom (D * F) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
len (D * F) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len (D * F)) is finite len (D * F) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
Seg (len F) is finite len F -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(D * F) . k is set
F . k is set
i is right_add-cancelable Element of the carrier of R
i is right_add-cancelable Element of the carrier of R
i * i is right_add-cancelable Element of the carrier of R
the multF of R . (i,i) is right_add-cancelable Element of the carrier of R
[i,i] is non empty V18() set
{i,i} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,i},{i}} is non empty finite V39() set
the multF of R . [i,i] is set
D * i is right_add-cancelable Element of the carrier of R
the multF of R . (D,i) is right_add-cancelable Element of the carrier of R
[D,i] is non empty V18() set
{D,i} is non empty finite set
{{D,i},{D}} is non empty finite V39() set
the multF of R . [D,i] is set
{ (D * b1) where b1 is right_add-cancelable Element of the carrier of R : b1 in e } is set
F /. k is right_add-cancelable Element of the carrier of R
(D * F) /. k is right_add-cancelable Element of the carrier of R
D * (i * i) is right_add-cancelable Element of the carrier of R
the multF of R . (D,(i * i)) is right_add-cancelable Element of the carrier of R
[D,(i * i)] is non empty V18() set
{D,(i * i)} is non empty finite set
{{D,(i * i)},{D}} is non empty finite V39() set
the multF of R . [D,(i * i)] is set
(D * i) * i is right_add-cancelable Element of the carrier of R
the multF of R . ((D * i),i) is right_add-cancelable Element of the carrier of R
[(D * i),i] is non empty V18() set
{(D * i),i} is non empty finite set
{(D * i)} is non empty trivial finite 1 -element set
{{(D * i),i},{(D * i)}} is non empty finite V39() set
the multF of R . [(D * i),i] is set
Sum (D * F) is right_add-cancelable Element of the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being right_add-cancelable Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in D ) ) } is set
(R,(R,I,D),D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in (R,I,D) & b4 in D ) ) } is set
e is set
D is right_add-cancelable Element of the carrier of R
(R,(R,D,e),D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in (R,D,e) } is set
e9 is set
(R,e,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in e } is set
f is right_add-cancelable Element of the carrier of R
D * f is right_add-cancelable Element of the carrier of R
the multF of R . (D,f) is right_add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
F is set
(R,D,(D * f)) is Element of bool the carrier of R
{ ((D * f) * b1) where b1 is Element of the carrier of R : b1 in D } is set
i is right_add-cancelable Element of the carrier of R
(D * f) * i is right_add-cancelable Element of the carrier of R
the multF of R . ((D * f),i) is right_add-cancelable Element of the carrier of R
[(D * f),i] is non empty V18() set
{(D * f),i} is non empty finite set
{(D * f)} is non empty trivial finite 1 -element set
{{(D * f),i},{(D * f)}} is non empty finite V39() set
the multF of R . [(D * f),i] is set
i * f is right_add-cancelable Element of the carrier of R
the multF of R . (i,f) is right_add-cancelable Element of the carrier of R
[i,f] is non empty V18() set
{i,f} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,f},{i}} is non empty finite V39() set
the multF of R . [i,f] is set
D * (i * f) is right_add-cancelable Element of the carrier of R
the multF of R . (D,(i * f)) is right_add-cancelable Element of the carrier of R
[D,(i * f)] is non empty V18() set
{D,(i * f)} is non empty finite set
{{D,(i * f)},{D}} is non empty finite V39() set
the multF of R . [D,(i * f)] is set
<*(i * f)*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len <*(i * f)*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
<*(i * f)*> . i is set
<*(i * f)*> . 1 is set
Sum <*(i * f)*> is right_add-cancelable Element of the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being right_add-cancelable Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in D & b4 in e ) ) } is set
{ (D * b1) where b1 is right_add-cancelable Element of the carrier of R : b1 in (R,D,e) } is set
R is non empty multLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
D is Element of bool the carrier of R
e is Element of bool the carrier of R
(R,D,e) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in D & b1 in e ) } is set
I is Element of bool the carrier of R
(R,(R,D,e),I) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,I,b1) c= (R,D,e) } is set
(R,D,I) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,I,b1) c= D } is set
(R,e,I) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,I,b1) c= e } is set
(R,(R,D,I),(R,e,I)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in (R,D,I) & b1 in (R,e,I) ) } is set
e is set
D is Element of the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
e9 is set
f is Element of the carrier of R
e9 is set
f is Element of the carrier of R
e is set
D is Element of the carrier of R
D is Element of the carrier of R
(R,I,D) is Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in I } is set
e9 is Element of the carrier of R
e9 is set
f is Element of the carrier of R
(R,I,f) is Element of bool the carrier of R
{ (f * b1) where b1 is Element of the carrier of R : b1 in I } is set
R is non empty right_add-cancelable right_zeroed right-distributive left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is (R) Element of bool the carrier of R
D is non empty (R) Element of bool the carrier of R
e is non empty (R) Element of bool the carrier of R
(R,D,e) is non empty Element of bool the carrier of R
{ (b1 + b2) where b1, b2 is Element of the carrier of R : ( b1 in D & b2 in e ) } is set
(R,I,(R,D,e)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,(R,D,e),b1) c= I } is set
(R,I,D) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,D,b1) c= I } is set
(R,I,e) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : (R,e,b1) c= I } is set
(R,(R,I,D),(R,I,e)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in (R,I,D) & b1 in (R,I,e) ) } is set
e is set
D is right_add-cancelable Element of the carrier of R
(R,(R,D,e),D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in (R,D,e) } is set
e9 is set
(R,D,D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in D } is set
f is right_add-cancelable Element of the carrier of R
D * f is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is right_add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e9 is set
(R,e,D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in e } is set
f is right_add-cancelable Element of the carrier of R
D * f is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,f) is right_add-cancelable Element of the carrier of R
[D,f] is non empty V18() set
{D,f} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,f},{D}} is non empty finite V39() set
the multF of R . [D,f] is set
e is set
D is right_add-cancelable Element of the carrier of R
D is right_add-cancelable Element of the carrier of R
(R,D,D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in D } is set
e9 is right_add-cancelable Element of the carrier of R
e9 is right_add-cancelable Element of the carrier of R
(R,e,e9) is non empty Element of bool the carrier of R
{ (e9 * b1) where b1 is Element of the carrier of R : b1 in e } is set
f is set
(R,(R,D,e),D) is non empty Element of bool the carrier of R
{ (D * b1) where b1 is Element of the carrier of R : b1 in (R,D,e) } is set
F is right_add-cancelable Element of the carrier of R
D * F is right_add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (D,F) is right_add-cancelable Element of the carrier of R
[D,F] is non empty V18() set
{D,F} is non empty finite set
{D} is non empty trivial finite 1 -element set
{{D,F},{D}} is non empty finite V39() set
the multF of R . [D,F] is set
i is right_add-cancelable Element of the carrier of R
k is right_add-cancelable Element of the carrier of R
i + k is right_add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . (i,k) is right_add-cancelable Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the addF of R . [i,k] is set
D * i is right_add-cancelable Element of the carrier of R
the multF of R . (D,i) is right_add-cancelable Element of the carrier of R
[D,i] is non empty V18() set
{D,i} is non empty finite set
{{D,i},{D}} is non empty finite V39() set
the multF of R . [D,i] is set
e9 * k is right_add-cancelable Element of the carrier of R
the multF of R . (e9,k) is right_add-cancelable Element of the carrier of R
[e9,k] is non empty V18() set
{e9,k} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,k},{e9}} is non empty finite V39() set
the multF of R . [e9,k] is set
(D * i) + (e9 * k) is right_add-cancelable Element of the carrier of R
the addF of R . ((D * i),(e9 * k)) is right_add-cancelable Element of the carrier of R
[(D * i),(e9 * k)] is non empty V18() set
{(D * i),(e9 * k)} is non empty finite set
{(D * i)} is non empty trivial finite 1 -element set
{{(D * i),(e9 * k)},{(D * i)}} is non empty finite V39() set
the addF of R . [(D * i),(e9 * k)] is set
R is non empty unital right_unital well-unital left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in I } is set
e is set
e is Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is Element of the carrier of R
R is non empty unital right_unital well-unital left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in I } is set
the Element of I is Element of I
the Element of I |^ 1 is Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable Abelian add-associative right_zeroed unital associative commutative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in I } is set
e is non empty Element of bool the carrier of R
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e + D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . (e,D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the addF of R . [e,D] is set
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 |^ f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 |^ f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F |^ i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F |^ i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f + i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(e9,F) In_Power (f + i) is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
Sum ((e9,F) In_Power (f + i)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len ((e9,F) In_Power (f + i)) is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
i . (len ((e9,F) In_Power (f + i))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
((e9,F) In_Power (f + i)) . i is set
i - 1 is V34() V106() complex ext-real set
(f + i) - (i - 1) is V34() V106() complex ext-real set
1 - 1 is V34() V106() complex ext-real set
(f + i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
r is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
((f + i) + 1) - 1 is V34() V106() complex ext-real set
r - r is V34() V106() complex ext-real set
(f + i) - r is V34() V106() complex ext-real set
Seg (len ((e9,F) In_Power (f + i))) is finite len ((e9,F) In_Power (f + i)) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom ((e9,F) In_Power (f + i)) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
((e9,F) In_Power (f + i)) /. i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
l is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 |^ l is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(f + i) choose r is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
((f + i) choose r) * (e9 |^ l) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F |^ r is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(((f + i) choose r) * (e9 |^ l)) * (F |^ r) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . ((((f + i) choose r) * (e9 |^ l)),(F |^ r)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(((f + i) choose r) * (e9 |^ l)),(F |^ r)] is non empty V18() set
{(((f + i) choose r) * (e9 |^ l)),(F |^ r)} is non empty finite set
{(((f + i) choose r) * (e9 |^ l))} is non empty trivial finite 1 -element set
{{(((f + i) choose r) * (e9 |^ l)),(F |^ r)},{(((f + i) choose r) * (e9 |^ l))}} is non empty finite V39() set
the multF of R . [(((f + i) choose r) * (e9 |^ l)),(F |^ r)] is set
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
f + k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 |^ k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(e9 |^ f) * (e9 |^ k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((e9 |^ f),(e9 |^ k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(e9 |^ f),(e9 |^ k)] is non empty V18() set
{(e9 |^ f),(e9 |^ k)} is non empty finite set
{(e9 |^ f)} is non empty trivial finite 1 -element set
{{(e9 |^ f),(e9 |^ k)},{(e9 |^ f)}} is non empty finite V39() set
the multF of R . [(e9 |^ f),(e9 |^ k)] is set
f + r is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
- r is V34() V106() complex ext-real non positive set
(f + i) + (- r) is V34() V106() complex ext-real set
((f + i) + (- r)) + r is V34() V106() complex ext-real set
- f is V34() V106() complex ext-real non positive set
(- f) + (f + r) is V34() V106() complex ext-real set
(- f) + (f + i) is V34() V106() complex ext-real set
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
i + k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F |^ k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(F |^ i) * (F |^ k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((F |^ i),(F |^ k)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(F |^ i),(F |^ k)] is non empty V18() set
{(F |^ i),(F |^ k)} is non empty finite set
{(F |^ i)} is non empty trivial finite 1 -element set
{{(F |^ i),(F |^ k)},{(F |^ i)}} is non empty finite V39() set
the multF of R . [(F |^ i),(F |^ k)] is set
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
i . i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
i . (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
Seg (len ((e9,F) In_Power (f + i))) is finite len ((e9,F) In_Power (f + i)) -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom ((e9,F) In_Power (f + i)) is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
((e9,F) In_Power (f + i)) /. (i + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
((e9,F) In_Power (f + i)) . (i + 1) is set
(i . i) + (((e9,F) In_Power (f + i)) /. (i + 1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . ((i . i),(((e9,F) In_Power (f + i)) /. (i + 1))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(i . i),(((e9,F) In_Power (f + i)) /. (i + 1))] is non empty V18() set
{(i . i),(((e9,F) In_Power (f + i)) /. (i + 1))} is non empty finite set
{(i . i)} is non empty trivial finite 1 -element set
{{(i . i),(((e9,F) In_Power (f + i)) /. (i + 1))},{(i . i)}} is non empty finite V39() set
the addF of R . [(i . i),(((e9,F) In_Power (f + i)) /. (i + 1))] is set
e9 + F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R . (e9,F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,F] is non empty V18() set
{e9,F} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,F},{e9}} is non empty finite V39() set
the addF of R . [e9,F] is set
(e9 + F) |^ (f + i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty unital associative commutative right_unital well-unital left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
(R,I) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in I } is set
e is non empty Element of bool the carrier of R
D is Element of the carrier of R
e is Element of the carrier of R
e * D is Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e,D) is Element of the carrier of R
[e,D] is non empty V18() set
{e,D} is non empty finite set
{e} is non empty trivial finite 1 -element set
{{e,D},{e}} is non empty finite V39() set
the multF of R . [e,D] is set
e9 is Element of the carrier of R
F is Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F |^ i is Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F |^ i is Element of the carrier of R
f is Element of the carrier of R
f * F is Element of the carrier of R
the multF of R . (f,F) is Element of the carrier of R
[f,F] is non empty V18() set
{f,F} is non empty finite set
{f} is non empty trivial finite 1 -element set
{{f,F},{f}} is non empty finite V39() set
the multF of R . [f,F] is set
(f * F) |^ i is Element of the carrier of R
f |^ i is Element of the carrier of R
(f |^ i) * (F |^ i) is Element of the carrier of R
the multF of R . ((f |^ i),(F |^ i)) is Element of the carrier of R
[(f |^ i),(F |^ i)] is non empty V18() set
{(f |^ i),(F |^ i)} is non empty finite set
{(f |^ i)} is non empty trivial finite 1 -element set
{{(f |^ i),(F |^ i)},{(f |^ i)}} is non empty finite V39() set
the multF of R . [(f |^ i),(F |^ i)] is set
R is non empty unital associative right_unital well-unital left_unital doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in I } is set
D is Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ e is Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ e is Element of the carrier of R
e is Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is Element of the carrier of R
e * D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ (e * D) is Element of the carrier of R
D |^ 1 is Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ e is Element of the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D |^ e is Element of the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) Element of bool the carrier of R
D is non empty (R) (R) Element of bool the carrier of R
(R,I,D) is non empty Element of bool the carrier of R
{ (Sum b1) where b1 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R : for b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT holds
( not 1 <= b2 or not b2 <= len b1 or ex b3, b4 being Element of the carrier of R st
( b1 . b2 = b3 * b4 & b3 in I & b4 in D ) ) } is set
(R,(R,I,D)) is non empty Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in (R,I,D) } is set
(R,I,D) is (R) Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ( b1 in I & b1 in D ) } is set
(R,(R,I,D)) is Element of bool the carrier of R
{ b1 where b1 is Element of the carrier of R : ex b2 being V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT st b1 |^ b2 in (R,I,D) } is set
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of the carrier of R
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
len e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
f . (len e9) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f . 0 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f . F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
f . (F + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
e9 . (F + 1) is set
Seg (len e9) is finite len e9 -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e9 is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e9 /. (F + 1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(f . F) + (e9 /. (F + 1)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the addF of R . ((f . F),(e9 /. (F + 1))) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(f . F),(e9 /. (F + 1))] is non empty V18() set
{(f . F),(e9 /. (F + 1))} is non empty finite set
{(f . F)} is non empty trivial finite 1 -element set
{{(f . F),(e9 /. (F + 1))},{(f . F)}} is non empty finite V39() set
the addF of R . [(f . F),(e9 /. (F + 1))] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the multF of R . [i,k] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the multF of R . (i,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[i,k] is non empty V18() set
{i,k} is non empty finite set
{i} is non empty trivial finite 1 -element set
{{i,k},{i}} is non empty finite V39() set
the multF of R . [i,k] is set
i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e is set
e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(e |^ D) * (e |^ D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((e |^ D),(e |^ D)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(e |^ D),(e |^ D)] is non empty V18() set
{(e |^ D),(e |^ D)} is non empty finite set
{(e |^ D)} is non empty trivial finite 1 -element set
{{(e |^ D),(e |^ D)},{(e |^ D)}} is non empty finite V39() set
the multF of R . [(e |^ D),(e |^ D)] is set
<*((e |^ D) * (e |^ D))*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
len <*((e |^ D) * (e |^ D))*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
<*((e |^ D) * (e |^ D))*> . f is set
Seg (len <*((e |^ D) * (e |^ D))*>) is non empty finite len <*((e |^ D) * (e |^ D))*> -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
dom <*((e |^ D) * (e |^ D))*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*((e |^ D) * (e |^ D))*> /. f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
<*((e |^ D) * (e |^ D))*> . 1 is set
F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D + D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e |^ (D + D) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
Sum <*((e |^ D) * (e |^ D))*> is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is set
D is Element of the carrier of R
{D} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{D}) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty finite Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
0. R is zero Element of the carrier of R
the ZeroF of R is Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element Element of bool the carrier of R
I is non empty Element of bool the carrier of R
D is set
e is set
D is set
R is non empty left_add-cancelable right_add-cancelable add-cancelable right_complementable add-associative right_zeroed unital right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed add-left-invertible add-right-invertible Loop-like Euclidian doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
(R,{(0. R)}) is non empty (R) (R) (R) (R) Element of bool the carrier of R
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
D is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{ b1 where b1 is Element of I : not b1 = 0. R } is set
e is set
D is Element of I
[: the carrier of R,NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V148() V149() V150() V151() set
bool [: the carrier of R,NAT:] is non empty non trivial non finite set
D is non empty Relation-like the carrier of R -defined NAT -valued Function-like total V21( the carrier of R, NAT ) V148() V149() V150() V151() Element of bool [: the carrier of R,NAT:]
e is non empty Element of bool the carrier of R
{ (D . b1) where b1 is Element of e : verum } is set
the Element of e is Element of e
D . the Element of e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is set
i is Element of e
D . i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is non empty V158() V159() V160() V161() V162() V163() V192() V194() Element of bool NAT
min F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() set
k is Element of e
D . k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{i} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{i}) is non empty (R) (R) (R) (R) Element of bool the carrier of R
c is Element of I
i is set
c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c1 is Element of I
c1 is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
c1 * i is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (c1,i) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[c1,i] is non empty V18() set
{c1,i} is non empty finite set
{c1} is non empty trivial finite 1 -element set
{{c1,i},{c1}} is non empty finite V39() set
the multF of R . [c1,i] is set
r is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(c1 * i) + r is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
the addF of R . ((c1 * i),r) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(c1 * i),r] is non empty V18() set
{(c1 * i),r} is non empty finite set
{(c1 * i)} is non empty trivial finite 1 -element set
{{(c1 * i),r},{(c1 * i)}} is non empty finite V39() set
the addF of R . [(c1 * i),r] is set
D . r is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
l is Element of I
- (c1 * i) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
(- (c1 * i)) + c is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((- (c1 * i)),c) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(- (c1 * i)),c] is non empty V18() set
{(- (c1 * i)),c} is non empty finite set
{(- (c1 * i))} is non empty trivial finite 1 -element set
{{(- (c1 * i)),c},{(- (c1 * i))}} is non empty finite V39() set
the addF of R . [(- (c1 * i)),c] is set
(- (c1 * i)) + (c1 * i) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((- (c1 * i)),(c1 * i)) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(- (c1 * i)),(c1 * i)] is non empty V18() set
{(- (c1 * i)),(c1 * i)} is non empty finite set
{{(- (c1 * i)),(c1 * i)},{(- (c1 * i))}} is non empty finite V39() set
the addF of R . [(- (c1 * i)),(c1 * i)] is set
((- (c1 * i)) + (c1 * i)) + r is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . (((- (c1 * i)) + (c1 * i)),r) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[((- (c1 * i)) + (c1 * i)),r] is non empty V18() set
{((- (c1 * i)) + (c1 * i)),r} is non empty finite set
{((- (c1 * i)) + (c1 * i))} is non empty trivial finite 1 -element set
{{((- (c1 * i)) + (c1 * i)),r},{((- (c1 * i)) + (c1 * i))}} is non empty finite V39() set
the addF of R . [((- (c1 * i)) + (c1 * i)),r] is set
(0. R) + r is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the addF of R . ((0. R),r) is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
[(0. R),r] is non empty V18() set
{(0. R),r} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R),r},{(0. R)}} is non empty finite V39() set
the addF of R . [(0. R),r] is set
0. R is zero left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable right_complementable Element of the carrier of R
{(0. R)} is non empty trivial finite 1 -element (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty (R) (R) (R) Element of bool the carrier of R
D is Element of the carrier of R
{D} is non empty trivial finite 1 -element Element of bool the carrier of R
(R,{D}) is non empty (R) (R) (R) (R) Element of bool the carrier of R
R is non empty left_add-cancelable right_add-cancelable add-cancelable add-associative right_zeroed unital associative right-distributive left-distributive right_unital well-unital distributive left_unital left_zeroed () doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
I is non empty Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
D is non empty finite Element of bool the carrier of R
(R,D) is non empty (R) (R) (R) (R) Element of bool the carrier of R
bool I is non empty Element of bool (bool I)
bool I is non empty set
bool (bool I) is non empty set
e is set
e is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
Seg (len e) is finite len e -element V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
e /. D is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is Element of I
e9 * F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . (e9,F) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[e9,F] is non empty V18() set
{e9,F} is non empty finite set
{e9} is non empty trivial finite 1 -element set
{{e9,F},{e9}} is non empty finite V39() set
the multF of R . [e9,F] is set
f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(e9 * F) * f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((e9 * F),f) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(e9 * F),f] is non empty V18() set
{(e9 * F),f} is non empty finite set
{(e9 * F)} is non empty trivial finite 1 -element set
{{(e9 * F),f},{(e9 * F)}} is non empty finite V39() set
the multF of R . [(e9 * F),f] is set
i is Element of I
D is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom D is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
rng D is finite Element of bool I
e9 is finite Element of bool I
the Element of I is Element of I
0. R is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the ZeroF of R is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(0. R) * the Element of I is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((0. R), the Element of I) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(0. R), the Element of I] is non empty V18() set
{(0. R), the Element of I} is non empty finite set
{(0. R)} is non empty trivial finite 1 -element set
{{(0. R), the Element of I},{(0. R)}} is non empty finite V39() set
the multF of R . [(0. R), the Element of I] is set
((0. R) * the Element of I) * (0. R) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (((0. R) * the Element of I),(0. R)) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[((0. R) * the Element of I),(0. R)] is non empty V18() set
{((0. R) * the Element of I),(0. R)} is non empty finite set
{((0. R) * the Element of I)} is non empty trivial finite 1 -element set
{{((0. R) * the Element of I),(0. R)},{((0. R) * the Element of I)}} is non empty finite V39() set
the multF of R . [((0. R) * the Element of I),(0. R)] is set
<*(((0. R) * the Element of I) * (0. R))*> is non empty trivial Relation-like NAT -defined the carrier of R -valued Function-like finite 1 -element FinSequence-like FinSubsequence-like FinSequence of the carrier of R
f is set
dom <*(((0. R) * the Element of I) * (0. R))*> is non empty trivial finite 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
len <*(((0. R) * the Element of I) * (0. R))*> is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len <*(((0. R) * the Element of I) * (0. R))*>) is non empty finite len <*(((0. R) * the Element of I) * (0. R))*> -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
{1} is non empty trivial finite V39() 1 -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
<*(((0. R) * the Element of I) * (0. R))*> /. f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
k is Element of I
F * k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . (F,k) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[F,k] is non empty V18() set
{F,k} is non empty finite set
{F} is non empty trivial finite 1 -element set
{{F,k},{F}} is non empty finite V39() set
the multF of R . [F,k] is set
i is zero left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(F * k) * i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R . ((F * k),i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(F * k),i] is non empty V18() set
{(F * k),i} is non empty finite set
{(F * k)} is non empty trivial finite 1 -element set
{{(F * k),i},{(F * k)}} is non empty finite V39() set
the multF of R . [(F * k),i] is set
<*> the carrier of R is empty trivial proper Relation-like non-empty empty-yielding NAT -defined the carrier of R -valued Function-like one-to-one functional V29() V33() V34() finite finite-yielding V39() cardinal {} -element FinSequence-like FinSubsequence-like FinSequence-membered V106() complex ext-real non positive non negative V148() V149() V150() V151() V152() V153() V154() V155() V158() V159() V160() V161() V162() V163() V164() V194() V195() V196() V197() FinSequence of the carrier of R
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
f is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum f is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
len f is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Seg (len f) is non empty finite len f -element V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
i is Element of I
f /. F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
F is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom F is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
rng F is finite Element of bool I
i is finite Element of bool I
dom f is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
[:D,(bool I):] is non empty Relation-like set
bool [:D,(bool I):] is non empty set
e is non empty Relation-like D -defined bool I -valued Function-like total V21(D, bool I) finite Element of bool [:D,(bool I):]
e is set
rng e is non empty finite Element of bool (bool I)
bool (bool I) is non empty set
dom e is non empty finite Element of bool D
bool D is non empty finite V39() set
D is set
e . D is set
e9 is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum e9 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
f is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom f is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom e9 is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
rng f is finite Element of bool I
e is Element of bool (bool I)
union e is Element of bool I
e9 is set
f is set
e . f is set
F is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum F is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
i is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom i is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
dom F is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
rng i is finite Element of bool I
D is Element of bool I
i is set
i is set
e . i is set
k is non empty Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,I)
Sum k is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
dom k is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
F is non empty finite Element of bool the carrier of R
i is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom i is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
rng i is finite Element of bool I
i is Relation-like NAT -defined I -valued Function-like finite FinSequence-like FinSubsequence-like FinSequence of I
dom i is finite V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of bool NAT
rng i is finite Element of bool I
i is set
i /. i is Element of I
i . i is set
k /. i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c * (i /. i) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(c * (i /. i)) * c1 is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
the multF of R is non empty Relation-like [: the carrier of R, the carrier of R:] -defined the carrier of R -valued Function-like total V21([: the carrier of R, the carrier of R:], the carrier of R) Element of bool [:[: the carrier of R, the carrier of R:], the carrier of R:]
[: the carrier of R, the carrier of R:] is non empty Relation-like set
[:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty Relation-like set
bool [:[: the carrier of R, the carrier of R:], the carrier of R:] is non empty set
the multF of R . ((c * (i /. i)),c1) is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
[(c * (i /. i)),c1] is non empty V18() set
{(c * (i /. i)),c1} is non empty finite set
{(c * (i /. i))} is non empty trivial finite 1 -element set
{{(c * (i /. i)),c1},{(c * (i /. i))}} is non empty finite V39() set
the multF of R . [(c * (i /. i)),c1] is set
i is Relation-like NAT -defined the carrier of R -valued Function-like finite FinSequence-like FinSubsequence-like (R,F)
Sum i is left_add-cancelable right_add-cancelable add-cancelable Element of the carrier of R
(R,F) is non empty (R) (R) (R) (R) Element of bool the carrier of R
(R,(R,F)) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty set
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
I is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
rng I is non empty Element of bool the carrier of R
D is non empty Element of bool the carrier of R
(R,D) is non empty (R) (R) (R) Element of bool the carrier of R
e is non empty finite Element of bool the carrier of R
(R,e) is non empty (R) (R) (R) (R) Element of bool the carrier of R
dom I is non empty V158() V159() V160() V161() V162() V163() V192() V194() Element of bool NAT
e is set
D is set
I . D is set
[:e,NAT:] is non empty non trivial Relation-like RAT -valued INT -valued non finite V148() V149() V150() V151() set
bool [:e,NAT:] is non empty non trivial non finite set
e is non empty Relation-like e -defined NAT -valued Function-like total V21(e, NAT ) finite V148() V149() V150() V151() Element of bool [:e,NAT:]
rng e is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
e9 is non empty finite V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of bool NAT
max e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Segm (f + 1) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
I | (Segm (f + 1)) is Relation-like NAT -defined Segm (f + 1) -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
rng (I | (Segm (f + 1))) is Element of bool the carrier of R
dom e is non empty finite Element of bool e
bool e is non empty finite V39() set
i is set
e . i is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
I . (e . i) is Element of the carrier of R
i is non empty Element of bool the carrier of R
(R,i) is non empty (R) (R) (R) Element of bool the carrier of R
I . (f + 1) is Element of the carrier of R
I | (f + 1) is Relation-like NAT -defined f + 1 -defined NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
rng (I | (f + 1)) is finite Element of bool the carrier of R
(R,(rng (I | (f + 1)))) is non empty (R) (R) (R) Element of bool the carrier of R
R is non empty set
I is non empty set
[:R,I:] is non empty Relation-like set
bool [:R,I:] is non empty set
bool R is non empty set
D is non empty Relation-like R -defined I -valued Function-like total V21(R,I) Element of bool [:R,I:]
e is non empty Element of bool R
D | e is Relation-like R -defined e -defined R -defined I -valued Function-like Element of bool [:R,I:]
dom D is non empty Element of bool R
(dom D) /\ e is Element of bool R
R is non empty doubleLoopStr
the carrier of R is non empty set
[:NAT, the carrier of R:] is non empty non trivial Relation-like non finite set
bool [:NAT, the carrier of R:] is non empty non trivial non finite set
bool the carrier of R is non empty Element of bool (bool the carrier of R)
bool the carrier of R is non empty set
bool (bool the carrier of R) is non empty set
[:NAT,(bool the carrier of R):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool the carrier of R):] is non empty non trivial non finite set
I is non empty Relation-like NAT -defined bool the carrier of R -valued Function-like total V21( NAT , bool the carrier of R) Element of bool [:NAT,(bool the carrier of R):]
I . 0 is Element of bool the carrier of R
D is set
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is set
I . e is Element of bool the carrier of R
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
e + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
I . e is Element of bool the carrier of R
I . D is Element of bool the carrier of R
(I . e) \ (I . D) is Element of bool the carrier of R
e9 is set
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D is non empty Relation-like NAT -defined the carrier of R -valued Function-like total V21( NAT , the carrier of R) Element of bool [:NAT, the carrier of R:]
e is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
D . (e + 1) is Element of the carrier of R
D | (e + 1) is Relation-like NAT -defined e + 1 -defined NAT -defined the carrier of R -valued Function-like finite Element of bool [:NAT, the carrier of R:]
rng (D | (e + 1)) is finite Element of bool the carrier of R
(R,(rng (D | (e + 1)))) is non empty (R) (R) (R) Element of bool the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Segm (D + 1) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
D | (Segm (D + 1)) is Relation-like NAT -defined Segm (D + 1) -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
rng (D | (Segm (D + 1))) is Element of bool the carrier of R
I . D is Element of bool the carrier of R
(D + 1) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Segm ((D + 1) + 1) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
D | (Segm ((D + 1) + 1)) is Relation-like NAT -defined Segm ((D + 1) + 1) -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
rng (D | (Segm ((D + 1) + 1))) is Element of bool the carrier of R
I . (D + 1) is Element of bool the carrier of R
e9 is set
dom (D | (Segm ((D + 1) + 1))) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
f is set
(D | (Segm ((D + 1) + 1))) . f is set
dom D is non empty V158() V159() V160() V161() V162() V163() V192() V194() Element of bool NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
D . F is Element of the carrier of R
D . F is Element of the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
k + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
I . i is Element of bool the carrier of R
I . k is Element of bool the carrier of R
(I . i) \ (I . k) is Element of bool the carrier of R
I . e is Element of bool the carrier of R
(R,(I . e)) is non empty (R) (R) (R) Element of bool the carrier of R
0 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
Segm (0 + 1) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
D | (Segm (0 + 1)) is Relation-like NAT -defined Segm (0 + 1) -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
rng (D | (Segm (0 + 1))) is Element of bool the carrier of R
D is set
Segm 1 is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
D | (Segm 1) is Relation-like NAT -defined Segm 1 -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
dom (D | (Segm 1)) is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
e9 is set
(D | (Segm 1)) . e9 is set
D . e9 is set
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
I . f is Element of bool the carrier of R
I . F is Element of bool the carrier of R
(I . f) \ (I . F) is Element of bool the carrier of R
e is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative set
Segm e is V158() V159() V160() V161() V162() V163() V194() Element of bool NAT
D | (Segm e) is Relation-like NAT -defined Segm e -defined NAT -defined the carrier of R -valued Function-like Element of bool [:NAT, the carrier of R:]
rng (D | (Segm e)) is Element of bool the carrier of R
(R,(rng (D | (Segm e)))) is non empty (R) (R) (R) Element of bool the carrier of R
D is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
e9 + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
I . D is Element of bool the carrier of R
I . e9 is Element of bool the carrier of R
(I . D) \ (I . e9) is Element of bool the carrier of R
R is non empty doubleLoopStr
the carrier of R is non empty set
bool the carrier of R is non empty Element of bool (bool the carrier of R)
bool the carrier of R is non empty set
bool (bool the carrier of R) is non empty set
[:NAT,(bool the carrier of R):] is non empty non trivial Relation-like non finite set
bool [:NAT,(bool the carrier of R):] is non empty non trivial non finite set
I is non empty (R) (R) (R) Element of bool the carrier of R
bool I is non empty set
{ b1 where b1 is Element of bool the carrier of R : b1 is non empty finite Element of bool I } is set
e is set
e is Element of the carrier of R
{e} is non empty trivial finite 1 -element Element of bool the carrier of R
D is set
e9 is Element of bool the carrier of R
D is non empty Element of bool (bool the carrier of R)
f is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F is Element of D
(R,F) is non empty (R) (R) (R) Element of bool the carrier of R
I \ (R,F) is Element of bool the carrier of R
i is Element of bool the carrier of R
(R,i) is non empty (R) (R) (R) Element of bool the carrier of R
(R,I) is non empty (R) (R) (R) Element of bool the carrier of R
i is set
k is non empty finite Element of bool I
{i} is non empty trivial finite 1 -element set
k \/ {i} is non empty finite set
c is set
c1 is Element of D
c is Element of the carrier of R
{c} is non empty trivial finite 1 -element Element of bool the carrier of R
F \/ {c} is non empty Element of bool the carrier of R
[:NAT,D:] is non empty non trivial Relation-like non finite set
bool [:NAT,D:] is non empty non trivial non finite set
e9 is Element of D
f is non empty Relation-like NAT -defined D -valued Function-like total V21( NAT ,D) Element of bool [:NAT,D:]
f . 0 is Element of D
F is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
f . F is Element of D
i is Element of bool the carrier of R
(R,i) is non empty (R) (R) (R) Element of bool the carrier of R
k is Element of bool the carrier of R
F is non empty Relation-like NAT -defined bool the carrier of R -valued Function-like total V21( NAT , bool the carrier of R) Element of bool [:NAT,(bool the carrier of R):]
k is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F . i is Element of bool the carrier of R
F . k is Element of bool the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative set
(i + 1) + i is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F . i is Element of bool the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . (i + 1) is Element of bool the carrier of R
f . i is Element of D
c is Element of bool the carrier of R
(R,c) is non empty (R) (R) (R) Element of bool the carrier of R
f . (i + 1) is Element of D
c1 is Element of bool the carrier of R
(R,c1) is non empty (R) (R) (R) Element of bool the carrier of R
r is Element of bool the carrier of R
I \ (R,c) is Element of bool the carrier of R
r is Element of the carrier of R
{r} is non empty trivial finite 1 -element Element of bool the carrier of R
c \/ {r} is non empty Element of bool the carrier of R
l is Element of bool the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
(i + 1) + i is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . ((i + 1) + i) is Element of bool the carrier of R
i + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
(i + 1) + (i + 1) is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . ((i + 1) + (i + 1)) is Element of bool the carrier of R
((i + 1) + i) + 1 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . (((i + 1) + i) + 1) is Element of bool the carrier of R
(i + 1) + 0 is non empty V29() V33() V34() finite cardinal V106() complex ext-real positive non negative V147() V158() V159() V160() V161() V162() V163() V192() V193() V194() V195() V196() Element of NAT
F . ((i + 1) + 0) is Element of bool the carrier of R
i is V29() V33() V34() finite cardinal V106() complex ext-real non negative V147() V158() V159() V160() V161() V162() V163() V194() V195() V196() Element of NAT
F . i is Element of bool the carrier of R
f . i is Element of D
k is Element of bool the carrier of R
(R,k) is non empty (R) (R) (R) Element of bool the carrier of R