REAL is non empty V33() set
NAT is non empty epsilon-transitive epsilon-connected ordinal V33() countable denumerable Element of bool REAL
bool REAL is non empty set
COMPLEX is non empty V33() set
omega is non empty epsilon-transitive epsilon-connected ordinal V33() countable denumerable set
bool omega is non empty set
bool NAT is non empty set
[:NAT,REAL:] is non empty Relation-like set
bool [:NAT,REAL:] is non empty set
K190() is non empty set
[:K190(),K190():] is non empty Relation-like set
[:[:K190(),K190():],K190():] is non empty Relation-like set
bool [:[:K190(),K190():],K190():] is non empty set
[:REAL,K190():] is non empty Relation-like set
[:[:REAL,K190():],K190():] is non empty Relation-like set
bool [:[:REAL,K190():],K190():] is non empty set
K196() is RLSStruct
the carrier of K196() is set
bool the carrier of K196() is non empty set
K200() is Element of bool the carrier of K196()
[:K200(),K200():] is Relation-like set
[:[:K200(),K200():],REAL:] is Relation-like set
bool [:[:K200(),K200():],REAL:] is non empty set
the_set_of_l1RealSequences is Element of bool the carrier of K196()
[:the_set_of_l1RealSequences,REAL:] is Relation-like set
bool [:the_set_of_l1RealSequences,REAL:] is non empty set
RAT is non empty V33() set
INT is non empty V33() set
[:REAL,REAL:] is non empty Relation-like set
bool [:REAL,REAL:] is non empty set
[:COMPLEX,COMPLEX:] is non empty Relation-like set
bool [:COMPLEX,COMPLEX:] is non empty set
[:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty Relation-like set
bool [:[:COMPLEX,COMPLEX:],COMPLEX:] is non empty set
[:[:REAL,REAL:],REAL:] is non empty Relation-like set
bool [:[:REAL,REAL:],REAL:] is non empty set
[:RAT,RAT:] is non empty Relation-like set
bool [:RAT,RAT:] is non empty set
[:[:RAT,RAT:],RAT:] is non empty Relation-like set
bool [:[:RAT,RAT:],RAT:] is non empty set
[:INT,INT:] is non empty Relation-like set
bool [:INT,INT:] is non empty set
[:[:INT,INT:],INT:] is non empty Relation-like set
bool [:[:INT,INT:],INT:] is non empty set
[:NAT,NAT:] is non empty Relation-like set
[:[:NAT,NAT:],NAT:] is non empty Relation-like set
bool [:[:NAT,NAT:],NAT:] is non empty set
1 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
[:1,1:] is non empty Relation-like set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is non empty Relation-like set
bool [:[:1,1:],1:] is non empty set
[:[:1,1:],REAL:] is non empty Relation-like set
bool [:[:1,1:],REAL:] is non empty set
2 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
[:2,2:] is non empty Relation-like set
[:[:2,2:],REAL:] is non empty Relation-like set
bool [:[:2,2:],REAL:] is non empty set
TOP-REAL 2 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() TopSpace-like V198() L20()
the carrier of (TOP-REAL 2) is non empty set
K646() is TopStruct
the carrier of K646() is set
K506() is V182() L19()
K651() is TopSpace-like TopStruct
{} is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V33() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding V140() V141() V142() Cardinal-yielding countable set
3 is non empty epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
- 1 is V11() real ext-real Element of REAL
Seg 1 is non empty V33() V40(1) countable Element of bool NAT
{1} is non empty set
Seg 2 is non empty V33() V40(2) countable Element of bool NAT
{1,2} is non empty set
0 is empty epsilon-transitive epsilon-connected ordinal T-Sequence-like c=-linear natural V11() real ext-real Relation-like non-empty empty-yielding NAT -defined RAT -valued Function-like one-to-one constant functional V33() FinSequence-like FinSubsequence-like FinSequence-membered complex-yielding V140() V141() V142() Cardinal-yielding countable Element of NAT
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
I is non empty set
J is non empty set
[:I,J:] is non empty Relation-like set
[:[:X,Y:],[:I,J:]:] is non empty Relation-like set
[:X,I:] is non empty Relation-like set
[:Y,J:] is non empty Relation-like set
[:[:X,I:],[:Y,J:]:] is non empty Relation-like set
[:[:[:X,Y:],[:I,J:]:],[:[:X,I:],[:Y,J:]:]:] is non empty Relation-like set
bool [:[:[:X,Y:],[:I,J:]:],[:[:X,I:],[:Y,J:]:]:] is non empty set
K is set
K is set
v is set
r is set
[v,r] is set
yy is set
x1 is set
[yy,x1] is set
[v,yy] is set
[r,x1] is set
[[v,yy],[r,x1]] is set
K is non empty Relation-like [:[:X,Y:],[:I,J:]:] -defined [:[:X,I:],[:Y,J:]:] -valued Function-like V26([:[:X,Y:],[:I,J:]:]) quasi_total Element of bool [:[:[:X,Y:],[:I,J:]:],[:[:X,I:],[:Y,J:]:]:]
K is set
v is set
r is set
yy is set
[K,v] is set
[r,yy] is set
K . ([K,v],[r,yy]) is set
[[K,v],[r,yy]] is set
K . [[K,v],[r,yy]] is set
[K,r] is set
[v,yy] is set
[[K,r],[v,yy]] is set
x1 is set
y1 is set
xx2 is set
yy2 is set
[x1,y1] is set
[xx2,yy2] is set
[x1,xx2] is set
[y1,yy2] is set
[[x1,xx2],[y1,yy2]] is set
K is set
v is set
K . K is set
K . v is set
r is set
yy is set
[r,yy] is set
x1 is set
y1 is set
[x1,y1] is set
xx2 is set
yy2 is set
[xx2,yy2] is set
I is set
v is set
[I,v] is set
x1 is set
y1 is set
[x1,y1] is set
v1 is set
Ix1 is set
[v1,Ix1] is set
[x1,xx2] is set
[y1,yy2] is set
[[x1,xx2],[y1,yy2]] is set
K . ([x1,y1],[xx2,yy2]) is set
[[x1,y1],[xx2,yy2]] is set
K . [[x1,y1],[xx2,yy2]] is set
K . ([x1,y1],[v1,Ix1]) is set
[[x1,y1],[v1,Ix1]] is set
K . [[x1,y1],[v1,Ix1]] is set
[x1,v1] is set
[y1,Ix1] is set
[[x1,v1],[y1,Ix1]] is set
K is set
v is set
r is set
[v,r] is set
yy is set
x1 is set
[yy,x1] is set
y1 is set
xx2 is set
[y1,xx2] is set
[yy,y1] is set
[x1,xx2] is set
[[yy,y1],[x1,xx2]] is set
K . ([yy,y1],[x1,xx2]) is set
K . [[yy,y1],[x1,xx2]] is set
yy2 is Element of [:[:X,Y:],[:I,J:]:]
K . yy2 is Element of [:[:X,I:],[:Y,J:]:]
rng K is non empty Relation-like [:X,I:] -defined [:Y,J:] -valued Element of bool [:[:X,I:],[:Y,J:]:]
bool [:[:X,I:],[:Y,J:]:] is non empty set
{1} is non empty countable Element of bool NAT
X is non empty set
Y is Relation-like Function-like set
dom Y is set
Y . 1 is set
product Y is functional with_common_domain product-like set
[:X,(product Y):] is Relation-like set
bool [:X,(product Y):] is non empty set
I is set
<*I*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
dom <*I*> is non empty countable Element of bool NAT
len <*I*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len <*I*>) is V33() V40( len <*I*>) countable Element of bool NAT
J is set
<*I*> . J is set
Y . J is set
I is Relation-like X -defined product Y -valued Function-like quasi_total Element of bool [:X,(product Y):]
rng Y is set
J is set
Y . J is set
J is set
K is set
I . J is Relation-like Function-like set
I . K is Relation-like Function-like set
<*J*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
<*K*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
J is set
K is Relation-like Function-like set
dom K is set
K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K . 1 is set
len K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
<*(K . 1)*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
I . (K . 1) is Relation-like Function-like set
rng I is functional with_common_domain Element of bool (product Y)
bool (product Y) is non empty set
{1,2} is non empty countable Element of bool NAT
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
I is Relation-like Function-like set
dom I is set
I . 1 is set
I . 2 is set
product I is functional with_common_domain product-like set
[:[:X,Y:],(product I):] is Relation-like set
bool [:[:X,Y:],(product I):] is non empty set
J is set
K is set
<*J,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
dom <*J,K*> is non empty countable Element of bool NAT
len <*J,K*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len <*J,K*>) is V33() V40( len <*J,K*>) countable Element of bool NAT
K is set
<*J,K*> . K is set
I . K is set
J is Relation-like [:X,Y:] -defined product I -valued Function-like quasi_total Element of bool [:[:X,Y:],(product I):]
rng I is set
K is set
I . K is set
K is set
K is set
J . K is Relation-like Function-like set
J . K is Relation-like Function-like set
v is set
r is set
[v,r] is set
yy is set
x1 is set
[yy,x1] is set
<*v,r*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
J . (v,r) is set
J . [v,r] is Relation-like Function-like set
J . (yy,x1) is set
J . [yy,x1] is Relation-like Function-like set
<*yy,x1*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
K is set
K is Relation-like Function-like set
dom K is set
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v . 1 is set
v . 2 is set
len v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
<*(v . 1),(v . 2)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[(v . 1),(v . 2)] is set
J . ((v . 1),(v . 2)) is set
J . [(v . 1),(v . 2)] is Relation-like Function-like set
x1 is Element of [:X,Y:]
J . x1 is Relation-like Function-like Element of product I
rng J is functional with_common_domain Element of bool (product I)
bool (product I) is non empty set
X is non empty set
<*X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
product <*X*> is functional with_common_domain product-like set
[:X,(product <*X*>):] is Relation-like set
bool [:X,(product <*X*>):] is non empty set
dom <*X*> is non empty countable Element of bool NAT
<*X*> . 1 is set
X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
Y is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I is set
dom (X ^ Y) is non empty countable Element of bool NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
X . J is set
(X ^ Y) . J is set
(X ^ Y) . I is set
dom Y is non empty countable Element of bool NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Y . K is set
(X ^ Y) . J is set
(X ^ Y) . I is set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty set
Y is non empty set
[:X,Y:] is non empty Relation-like set
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
product <*X,Y*> is functional with_common_domain product-like set
[:[:X,Y:],(product <*X,Y*>):] is Relation-like set
bool [:[:X,Y:],(product <*X,Y*>):] is non empty set
dom <*X,Y*> is non empty countable Element of bool NAT
<*X,Y*> . 1 is set
<*X,Y*> . 2 is set
X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product X is non empty functional with_common_domain product-like set
Y is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product Y is non empty functional with_common_domain product-like set
[:(product X),(product Y):] is non empty Relation-like set
X ^ Y is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (X ^ Y) is non empty functional with_common_domain product-like set
[:[:(product X),(product Y):],(product (X ^ Y)):] is non empty Relation-like set
bool [:[:(product X),(product Y):],(product (X ^ Y)):] is non empty set
I is set
J is set
dom X is non empty countable Element of bool NAT
K is Relation-like Function-like set
dom K is set
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len X) is V33() V40( len X) countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
v is Relation-like Function-like set
dom v is set
len Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len Y) is V33() V40( len Y) countable Element of bool NAT
K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
r is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K ^ r is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom (K ^ r) is countable Element of bool NAT
(len K) + (len r) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg ((len K) + (len r)) is V33() V40((len K) + (len r)) countable Element of bool NAT
len (X ^ Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len (X ^ Y)) is V33() V40( len (X ^ Y)) countable Element of bool NAT
dom (X ^ Y) is non empty countable Element of bool NAT
yy is set
(K ^ r) . yy is set
(X ^ Y) . yy is set
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K . x1 is set
X . x1 is set
(K ^ r) . x1 is set
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
r . y1 is set
Y . y1 is set
(K ^ r) . x1 is set
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
I is non empty Relation-like [:(product X),(product Y):] -defined product (X ^ Y) -valued Function-like V26([:(product X),(product Y):]) quasi_total Element of bool [:[:(product X),(product Y):],(product (X ^ Y)):]
J is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I . (J,K) is set
[J,K] is set
I . [J,K] is Relation-like Function-like set
J ^ K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K ^ v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
J is set
K is set
I . J is Relation-like Function-like set
I . K is Relation-like Function-like set
K is set
v is set
[K,v] is set
r is set
yy is set
[r,yy] is set
I . (K,v) is set
I . [K,v] is Relation-like Function-like set
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
x1 ^ y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I . (r,yy) is set
I . [r,yy] is Relation-like Function-like set
xx2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
yy2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
xx2 ^ yy2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom x1 is countable Element of bool NAT
dom X is non empty countable Element of bool NAT
dom xx2 is countable Element of bool NAT
(x1 ^ y1) | (dom x1) is Relation-like NAT -defined dom x1 -defined NAT -defined Function-like FinSubsequence-like set
J is set
dom (X ^ Y) is non empty countable Element of bool NAT
K is Relation-like Function-like set
dom K is set
len (X ^ Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len (X ^ Y)) is V33() V40( len (X ^ Y)) countable Element of bool NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K | (len X) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
K /^ (len X) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
rng K is set
(K | (len X)) ^ (K /^ (len X)) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(len X) + (len Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (K | (len X)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom (K | (len X)) is countable Element of bool NAT
Seg (len X) is V33() V40( len X) countable Element of bool NAT
yy is set
(K | (len X)) . yy is set
X . yy is set
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(K | (len X)) . x1 is set
K . x1 is set
X . x1 is set
(X ^ Y) . x1 is set
len (K /^ (len X)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(len K) - (len X) is V11() real ext-real Element of REAL
Seg (len (K /^ (len X))) is V33() V40( len (K /^ (len X))) countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
dom (K /^ (len X)) is countable Element of bool NAT
yy is set
(K /^ (len X)) . yy is set
Y . yy is set
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(K /^ (len X)) . x1 is set
(len X) + x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K . ((len X) + x1) is set
Y . x1 is set
(X ^ Y) . ((len X) + x1) is set
[(K | (len X)),(K /^ (len X))] is set
I . ((K | (len X)),(K /^ (len X))) is set
I . [(K | (len X)),(K /^ (len X))] is Relation-like Function-like set
yy is Element of [:(product X),(product Y):]
I . yy is Relation-like NAT -defined Function-like X ^ Y -compatible Element of product (X ^ Y)
rng I is non empty functional with_common_domain Element of bool (product (X ^ Y))
bool (product (X ^ Y)) is non empty set
I is set
X is non empty 1-sorted
the carrier of X is non empty set
Y is non empty 1-sorted
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
X is non empty addLoopStr
the carrier of X is non empty set
Y is non empty addLoopStr
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
I is set
J is set
K is Element of the carrier of X
K is Element of the carrier of Y
[K,K] is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K + v is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,v) is Element of the carrier of X
[K,v] is set
the addF of X . [K,v] is set
K + r is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,r) is Element of the carrier of Y
[K,r] is set
the addF of Y . [K,r] is set
[(K + v),(K + r)] is Element of [: the carrier of X, the carrier of Y:]
yy is Element of [: the carrier of X, the carrier of Y:]
x1 is Element of the carrier of X
xx2 is Element of the carrier of Y
[x1,xx2] is Element of [: the carrier of X, the carrier of Y:]
y1 is Element of the carrier of X
yy2 is Element of the carrier of Y
[y1,yy2] is Element of [: the carrier of X, the carrier of Y:]
x1 + y1 is Element of the carrier of X
the addF of X . (x1,y1) is Element of the carrier of X
[x1,y1] is set
the addF of X . [x1,y1] is set
xx2 + yy2 is Element of the carrier of Y
the addF of Y . (xx2,yy2) is Element of the carrier of Y
[xx2,yy2] is set
the addF of Y . [xx2,yy2] is set
[(x1 + y1),(xx2 + yy2)] is Element of [: the carrier of X, the carrier of Y:]
I is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
J is Element of the carrier of X
K is Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
K is Element of the carrier of X
v is Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
I . ([J,K],[K,v]) is Element of [: the carrier of X, the carrier of Y:]
[[J,K],[K,v]] is set
I . [[J,K],[K,v]] is set
r is Element of the carrier of X
x1 is Element of the carrier of Y
[r,x1] is Element of [: the carrier of X, the carrier of Y:]
yy is Element of the carrier of X
y1 is Element of the carrier of Y
[yy,y1] is Element of [: the carrier of X, the carrier of Y:]
r + yy is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (r,yy) is Element of the carrier of X
[r,yy] is set
the addF of X . [r,yy] is set
x1 + y1 is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (x1,y1) is Element of the carrier of Y
[x1,y1] is set
the addF of Y . [x1,y1] is set
[(r + yy),(x1 + y1)] is Element of [: the carrier of X, the carrier of Y:]
J + K is Element of the carrier of X
the addF of X . (J,K) is Element of the carrier of X
[J,K] is set
the addF of X . [J,K] is set
K + v is Element of the carrier of Y
the addF of Y . (K,v) is Element of the carrier of Y
[K,v] is set
the addF of Y . [K,v] is set
[(J + K),(K + v)] is Element of [: the carrier of X, the carrier of Y:]
I is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
J is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
K is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K is Element of [: the carrier of X, the carrier of Y:]
yy is Element of the carrier of X
x1 is Element of the carrier of Y
[yy,x1] is Element of [: the carrier of X, the carrier of Y:]
I . (K,K) is Element of [: the carrier of X, the carrier of Y:]
[K,K] is set
I . [K,K] is set
v + yy is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (v,yy) is Element of the carrier of X
[v,yy] is set
the addF of X . [v,yy] is set
r + x1 is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (r,x1) is Element of the carrier of Y
[r,x1] is set
the addF of Y . [r,x1] is set
[(v + yy),(r + x1)] is Element of [: the carrier of X, the carrier of Y:]
J . (K,K) is Element of [: the carrier of X, the carrier of Y:]
J . [K,K] is set
X is non empty RLSStruct
the carrier of X is non empty set
Y is non empty RLSStruct
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
K is set
K is set
r is Element of the carrier of X
yy is Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
v is V11() real ext-real Element of REAL
v * r is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (v,r) is set
[v,r] is set
the Mult of X . [v,r] is set
v * yy is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (v,yy) is set
[v,yy] is set
the Mult of Y . [v,yy] is set
[(v * r),(v * yy)] is Element of [: the carrier of X, the carrier of Y:]
y1 is V11() real ext-real Element of REAL
xx2 is Element of the carrier of X
yy2 is Element of the carrier of Y
[xx2,yy2] is Element of [: the carrier of X, the carrier of Y:]
y1 * xx2 is Element of the carrier of X
the Mult of X . (y1,xx2) is set
[y1,xx2] is set
the Mult of X . [y1,xx2] is set
y1 * yy2 is Element of the carrier of Y
the Mult of Y . (y1,yy2) is set
[y1,yy2] is set
the Mult of Y . [y1,yy2] is set
[(y1 * xx2),(y1 * yy2)] is Element of [: the carrier of X, the carrier of Y:]
K is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
K is V11() real ext-real Element of REAL
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K . (K,[v,r]) is Element of [: the carrier of X, the carrier of Y:]
[K,[v,r]] is set
K . [K,[v,r]] is set
yy is V11() real ext-real Element of REAL
x1 is Element of the carrier of X
y1 is Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
yy * x1 is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (yy,x1) is set
[yy,x1] is set
the Mult of X . [yy,x1] is set
yy * y1 is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (yy,y1) is set
[yy,y1] is set
the Mult of Y . [yy,y1] is set
[(yy * x1),(yy * y1)] is Element of [: the carrier of X, the carrier of Y:]
yy is V11() real ext-real Element of REAL
x1 is Element of the carrier of X
y1 is Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
yy * x1 is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (yy,x1) is set
[yy,x1] is set
the Mult of X . [yy,x1] is set
K * y1 is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,y1) is set
[K,y1] is set
the Mult of Y . [K,y1] is set
[(yy * x1),(K * y1)] is Element of [: the carrier of X, the carrier of Y:]
K * v is Element of the carrier of X
the Mult of X . (K,v) is set
[K,v] is set
the Mult of X . [K,v] is set
K * r is Element of the carrier of Y
the Mult of Y . (K,r) is set
[K,r] is set
the Mult of Y . [K,r] is set
[(K * v),(K * r)] is Element of [: the carrier of X, the carrier of Y:]
I is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
J is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
K is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K is V11() real ext-real Element of REAL
I . (K,K) is Element of [: the carrier of X, the carrier of Y:]
[K,K] is set
I . [K,K] is set
K * v is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,v) is set
[K,v] is set
the Mult of X . [K,v] is set
K * r is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,r) is set
[K,r] is set
the Mult of Y . [K,r] is set
[(K * v),(K * r)] is Element of [: the carrier of X, the carrier of Y:]
J . (K,K) is Element of [: the carrier of X, the carrier of Y:]
J . [K,K] is set
X is non empty addLoopStr
the carrier of X is non empty set
Y is non empty addLoopStr
the carrier of Y is non empty set
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
X is non empty addLoopStr
the carrier of X is non empty set
Y is non empty addLoopStr
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
X is non empty Abelian addLoopStr
Y is non empty Abelian addLoopStr
(X,Y) is non empty strict addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is Element of the carrier of (X,Y)
I + J is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
J + I is Element of the carrier of (X,Y)
the addF of (X,Y) . (J,I) is Element of the carrier of (X,Y)
[J,I] is set
the addF of (X,Y) . [J,I] is set
K is Element of the carrier of X
K is Element of the carrier of Y
[K,K] is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K + v is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,v) is Element of the carrier of X
[K,v] is set
the addF of X . [K,v] is set
K + r is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,r) is Element of the carrier of Y
[K,r] is set
the addF of Y . [K,r] is set
[(K + v),(K + r)] is Element of [: the carrier of X, the carrier of Y:]
X is non empty add-associative addLoopStr
Y is non empty add-associative addLoopStr
(X,Y) is non empty strict addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is Element of the carrier of (X,Y)
I + J is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
K is Element of the carrier of (X,Y)
(I + J) + K is Element of the carrier of (X,Y)
the addF of (X,Y) . ((I + J),K) is Element of the carrier of (X,Y)
[(I + J),K] is set
the addF of (X,Y) . [(I + J),K] is set
J + K is Element of the carrier of (X,Y)
the addF of (X,Y) . (J,K) is Element of the carrier of (X,Y)
[J,K] is set
the addF of (X,Y) . [J,K] is set
I + (J + K) is Element of the carrier of (X,Y)
the addF of (X,Y) . (I,(J + K)) is Element of the carrier of (X,Y)
[I,(J + K)] is set
the addF of (X,Y) . [I,(J + K)] is set
K is Element of the carrier of X
v is Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is Element of the carrier of X
yy is Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is Element of the carrier of X
y1 is Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
K + r is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,r) is Element of the carrier of X
[K,r] is set
the addF of X . [K,r] is set
(K + r) + x1 is Element of the carrier of X
the addF of X . ((K + r),x1) is Element of the carrier of X
[(K + r),x1] is set
the addF of X . [(K + r),x1] is set
r + x1 is Element of the carrier of X
the addF of X . (r,x1) is Element of the carrier of X
[r,x1] is set
the addF of X . [r,x1] is set
K + (r + x1) is Element of the carrier of X
the addF of X . (K,(r + x1)) is Element of the carrier of X
[K,(r + x1)] is set
the addF of X . [K,(r + x1)] is set
v + yy is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (v,yy) is Element of the carrier of Y
[v,yy] is set
the addF of Y . [v,yy] is set
(v + yy) + y1 is Element of the carrier of Y
the addF of Y . ((v + yy),y1) is Element of the carrier of Y
[(v + yy),y1] is set
the addF of Y . [(v + yy),y1] is set
yy + y1 is Element of the carrier of Y
the addF of Y . (yy,y1) is Element of the carrier of Y
[yy,y1] is set
the addF of Y . [yy,y1] is set
v + (yy + y1) is Element of the carrier of Y
the addF of Y . (v,(yy + y1)) is Element of the carrier of Y
[v,(yy + y1)] is set
the addF of Y . [v,(yy + y1)] is set
[(K + r),(v + yy)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([(K + r),(v + yy)],[x1,y1]) is Element of [: the carrier of X, the carrier of Y:]
[[(K + r),(v + yy)],[x1,y1]] is set
(X,Y) . [[(K + r),(v + yy)],[x1,y1]] is set
[((K + r) + x1),((v + yy) + y1)] is Element of [: the carrier of X, the carrier of Y:]
[(r + x1),(yy + y1)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([K,v],[(r + x1),(yy + y1)]) is Element of [: the carrier of X, the carrier of Y:]
[[K,v],[(r + x1),(yy + y1)]] is set
(X,Y) . [[K,v],[(r + x1),(yy + y1)]] is set
X is non empty right_zeroed addLoopStr
Y is non empty right_zeroed addLoopStr
(X,Y) is non empty strict addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
I + (0. (X,Y)) is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,(0. (X,Y))) is Element of the carrier of (X,Y)
[I,(0. (X,Y))] is set
the addF of (X,Y) . [I,(0. (X,Y))] is set
J is Element of the carrier of X
K is Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
J + (0. X) is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,(0. X)) is Element of the carrier of X
[J,(0. X)] is set
the addF of X . [J,(0. X)] is set
K + (0. Y) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(0. Y)) is Element of the carrier of Y
[K,(0. Y)] is set
the addF of Y . [K,(0. Y)] is set
X is non empty right_complementable addLoopStr
Y is non empty right_complementable addLoopStr
(X,Y) is non empty strict addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. Y is V52(Y) right_complementable Element of the carrier of Y
the ZeroF of Y is right_complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is right_complementable Element of the carrier of X
K is right_complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
K is right_complementable Element of the carrier of X
J + K is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,K) is right_complementable Element of the carrier of X
[J,K] is set
the addF of X . [J,K] is set
v is right_complementable Element of the carrier of Y
K + v is right_complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,v) is right_complementable Element of the carrier of Y
[K,v] is set
the addF of Y . [K,v] is set
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is Element of the carrier of (X,Y)
I + r is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,r) is Element of the carrier of (X,Y)
[I,r] is set
the addF of (X,Y) . [I,r] is set
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
J is non empty addLoopStr
the carrier of J is non empty set
K is non empty addLoopStr
the carrier of K is non empty set
yy is non empty addLoopStr
x1 is non empty addLoopStr
(yy,x1) is non empty strict addLoopStr
the carrier of yy is non empty set
the carrier of x1 is non empty set
[: the carrier of yy, the carrier of x1:] is non empty Relation-like set
(yy,x1) is non empty Relation-like [:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:] -defined [: the carrier of yy, the carrier of x1:] -valued Function-like V26([:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:]) quasi_total Element of bool [:[:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:],[: the carrier of yy, the carrier of x1:]:]
[:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:] is non empty Relation-like set
[:[:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:],[: the carrier of yy, the carrier of x1:]:] is non empty Relation-like set
bool [:[:[: the carrier of yy, the carrier of x1:],[: the carrier of yy, the carrier of x1:]:],[: the carrier of yy, the carrier of x1:]:] is non empty set
(yy,x1) is Element of [: the carrier of yy, the carrier of x1:]
0. yy is V52(yy) Element of the carrier of yy
the ZeroF of yy is Element of the carrier of yy
0. x1 is V52(x1) Element of the carrier of x1
the ZeroF of x1 is Element of the carrier of x1
[(0. yy),(0. x1)] is Element of [: the carrier of yy, the carrier of x1:]
addLoopStr(# [: the carrier of yy, the carrier of x1:],(yy,x1),(yy,x1) #) is non empty strict addLoopStr
the carrier of (yy,x1) is non empty set
I is set
X is non empty addLoopStr
Y is non empty addLoopStr
(X,Y) is non empty strict addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
K is set
v is Element of the carrier of J
r is Element of the carrier of K
[v,r] is Element of [: the carrier of J, the carrier of K:]
[: the carrier of J, the carrier of K:] is non empty Relation-like set
(J,K) is non empty strict addLoopStr
(J,K) is non empty Relation-like [:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:] -defined [: the carrier of J, the carrier of K:] -valued Function-like V26([:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:]) quasi_total Element of bool [:[:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:],[: the carrier of J, the carrier of K:]:]
[:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:] is non empty Relation-like set
[:[:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:],[: the carrier of J, the carrier of K:]:] is non empty Relation-like set
bool [:[:[: the carrier of J, the carrier of K:],[: the carrier of J, the carrier of K:]:],[: the carrier of J, the carrier of K:]:] is non empty set
(J,K) is Element of [: the carrier of J, the carrier of K:]
0. J is V52(J) Element of the carrier of J
the ZeroF of J is Element of the carrier of J
0. K is V52(K) Element of the carrier of K
the ZeroF of K is Element of the carrier of K
[(0. J),(0. K)] is Element of [: the carrier of J, the carrier of K:]
addLoopStr(# [: the carrier of J, the carrier of K:],(J,K),(J,K) #) is non empty strict addLoopStr
the carrier of (J,K) is non empty set
y1 is Element of the carrier of (yy,x1)
yy2 is Element of the carrier of yy
v is Element of the carrier of x1
[yy2,v] is Element of [: the carrier of yy, the carrier of x1:]
xx2 is Element of the carrier of (yy,x1)
I is Element of the carrier of yy
x1 is Element of the carrier of x1
[I,x1] is Element of [: the carrier of yy, the carrier of x1:]
y1 + xx2 is Element of the carrier of (yy,x1)
the addF of (yy,x1) is non empty Relation-like [: the carrier of (yy,x1), the carrier of (yy,x1):] -defined the carrier of (yy,x1) -valued Function-like V26([: the carrier of (yy,x1), the carrier of (yy,x1):]) quasi_total Element of bool [:[: the carrier of (yy,x1), the carrier of (yy,x1):], the carrier of (yy,x1):]
[: the carrier of (yy,x1), the carrier of (yy,x1):] is non empty Relation-like set
[:[: the carrier of (yy,x1), the carrier of (yy,x1):], the carrier of (yy,x1):] is non empty Relation-like set
bool [:[: the carrier of (yy,x1), the carrier of (yy,x1):], the carrier of (yy,x1):] is non empty set
the addF of (yy,x1) . (y1,xx2) is Element of the carrier of (yy,x1)
[y1,xx2] is set
the addF of (yy,x1) . [y1,xx2] is set
yy2 + I is Element of the carrier of yy
the addF of yy is non empty Relation-like [: the carrier of yy, the carrier of yy:] -defined the carrier of yy -valued Function-like V26([: the carrier of yy, the carrier of yy:]) quasi_total Element of bool [:[: the carrier of yy, the carrier of yy:], the carrier of yy:]
[: the carrier of yy, the carrier of yy:] is non empty Relation-like set
[:[: the carrier of yy, the carrier of yy:], the carrier of yy:] is non empty Relation-like set
bool [:[: the carrier of yy, the carrier of yy:], the carrier of yy:] is non empty set
the addF of yy . (yy2,I) is Element of the carrier of yy
[yy2,I] is set
the addF of yy . [yy2,I] is set
v + x1 is Element of the carrier of x1
the addF of x1 is non empty Relation-like [: the carrier of x1, the carrier of x1:] -defined the carrier of x1 -valued Function-like V26([: the carrier of x1, the carrier of x1:]) quasi_total Element of bool [:[: the carrier of x1, the carrier of x1:], the carrier of x1:]
[: the carrier of x1, the carrier of x1:] is non empty Relation-like set
[:[: the carrier of x1, the carrier of x1:], the carrier of x1:] is non empty Relation-like set
bool [:[: the carrier of x1, the carrier of x1:], the carrier of x1:] is non empty set
the addF of x1 . (v,x1) is Element of the carrier of x1
[v,x1] is set
the addF of x1 . [v,x1] is set
[(yy2 + I),(v + x1)] is Element of [: the carrier of yy, the carrier of x1:]
y1 is non empty addLoopStr
v1 is non empty addLoopStr
(y1,v1) is non empty strict addLoopStr
the carrier of y1 is non empty set
the carrier of v1 is non empty set
[: the carrier of y1, the carrier of v1:] is non empty Relation-like set
(y1,v1) is non empty Relation-like [:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:] -defined [: the carrier of y1, the carrier of v1:] -valued Function-like V26([:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:]) quasi_total Element of bool [:[:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:],[: the carrier of y1, the carrier of v1:]:]
[:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:] is non empty Relation-like set
[:[:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:],[: the carrier of y1, the carrier of v1:]:] is non empty Relation-like set
bool [:[:[: the carrier of y1, the carrier of v1:],[: the carrier of y1, the carrier of v1:]:],[: the carrier of y1, the carrier of v1:]:] is non empty set
(y1,v1) is Element of [: the carrier of y1, the carrier of v1:]
0. y1 is V52(y1) Element of the carrier of y1
the ZeroF of y1 is Element of the carrier of y1
0. v1 is V52(v1) Element of the carrier of v1
the ZeroF of v1 is Element of the carrier of v1
[(0. y1),(0. v1)] is Element of [: the carrier of y1, the carrier of v1:]
addLoopStr(# [: the carrier of y1, the carrier of v1:],(y1,v1),(y1,v1) #) is non empty strict addLoopStr
0. (y1,v1) is V52((y1,v1)) Element of the carrier of (y1,v1)
the carrier of (y1,v1) is non empty set
the ZeroF of (y1,v1) is Element of the carrier of (y1,v1)
X is non empty right_complementable add-associative right_zeroed addLoopStr
Y is non empty right_complementable add-associative right_zeroed addLoopStr
(X,Y) is non empty strict right_complementable add-associative right_zeroed addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. Y is V52(Y) right_complementable Element of the carrier of Y
the ZeroF of Y is right_complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
the carrier of (X,Y) is non empty set
I is right_complementable Element of the carrier of (X,Y)
- I is right_complementable Element of the carrier of (X,Y)
J is right_complementable Element of the carrier of X
- J is right_complementable Element of the carrier of X
K is right_complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
- K is right_complementable Element of the carrier of Y
[(- J),(- K)] is Element of [: the carrier of X, the carrier of Y:]
K is right_complementable Element of the carrier of (X,Y)
I + K is right_complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,K) is right_complementable Element of the carrier of (X,Y)
[I,K] is set
the addF of (X,Y) . [I,K] is set
J + (- J) is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,(- J)) is right_complementable Element of the carrier of X
[J,(- J)] is set
the addF of X . [J,(- J)] is set
K + (- K) is right_complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(- K)) is right_complementable Element of the carrier of Y
[K,(- K)] is set
the addF of Y . [K,(- K)] is set
[(J + (- J)),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
[(0. X),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
0. (X,Y) is V52((X,Y)) right_complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is right_complementable Element of the carrier of (X,Y)
X is non empty strict left_complementable right_complementable complementable Abelian add-associative right_zeroed V103() addLoopStr
Y is non empty strict left_complementable right_complementable complementable Abelian add-associative right_zeroed V103() addLoopStr
(X,Y) is non empty strict left_complementable right_complementable complementable Abelian add-associative right_zeroed V103() addLoopStr
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
addLoopStr(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y) #) is non empty strict addLoopStr
X is non empty RLSStruct
the carrier of X is non empty set
Y is non empty RLSStruct
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
X is non empty Abelian RLSStruct
Y is non empty Abelian RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is Element of the carrier of (X,Y)
I + J is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
J + I is Element of the carrier of (X,Y)
the addF of (X,Y) . (J,I) is Element of the carrier of (X,Y)
[J,I] is set
the addF of (X,Y) . [J,I] is set
K is Element of the carrier of X
K is Element of the carrier of Y
[K,K] is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
K + v is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,v) is Element of the carrier of X
[K,v] is set
the addF of X . [K,v] is set
K + r is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,r) is Element of the carrier of Y
[K,r] is set
the addF of Y . [K,r] is set
[(K + v),(K + r)] is Element of [: the carrier of X, the carrier of Y:]
X is non empty add-associative RLSStruct
Y is non empty add-associative RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is Element of the carrier of (X,Y)
I + J is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
K is Element of the carrier of (X,Y)
(I + J) + K is Element of the carrier of (X,Y)
the addF of (X,Y) . ((I + J),K) is Element of the carrier of (X,Y)
[(I + J),K] is set
the addF of (X,Y) . [(I + J),K] is set
J + K is Element of the carrier of (X,Y)
the addF of (X,Y) . (J,K) is Element of the carrier of (X,Y)
[J,K] is set
the addF of (X,Y) . [J,K] is set
I + (J + K) is Element of the carrier of (X,Y)
the addF of (X,Y) . (I,(J + K)) is Element of the carrier of (X,Y)
[I,(J + K)] is set
the addF of (X,Y) . [I,(J + K)] is set
K is Element of the carrier of X
v is Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is Element of the carrier of X
yy is Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is Element of the carrier of X
y1 is Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
K + r is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,r) is Element of the carrier of X
[K,r] is set
the addF of X . [K,r] is set
(K + r) + x1 is Element of the carrier of X
the addF of X . ((K + r),x1) is Element of the carrier of X
[(K + r),x1] is set
the addF of X . [(K + r),x1] is set
r + x1 is Element of the carrier of X
the addF of X . (r,x1) is Element of the carrier of X
[r,x1] is set
the addF of X . [r,x1] is set
K + (r + x1) is Element of the carrier of X
the addF of X . (K,(r + x1)) is Element of the carrier of X
[K,(r + x1)] is set
the addF of X . [K,(r + x1)] is set
v + yy is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (v,yy) is Element of the carrier of Y
[v,yy] is set
the addF of Y . [v,yy] is set
(v + yy) + y1 is Element of the carrier of Y
the addF of Y . ((v + yy),y1) is Element of the carrier of Y
[(v + yy),y1] is set
the addF of Y . [(v + yy),y1] is set
yy + y1 is Element of the carrier of Y
the addF of Y . (yy,y1) is Element of the carrier of Y
[yy,y1] is set
the addF of Y . [yy,y1] is set
v + (yy + y1) is Element of the carrier of Y
the addF of Y . (v,(yy + y1)) is Element of the carrier of Y
[v,(yy + y1)] is set
the addF of Y . [v,(yy + y1)] is set
[(K + r),(v + yy)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([(K + r),(v + yy)],[x1,y1]) is Element of [: the carrier of X, the carrier of Y:]
[[(K + r),(v + yy)],[x1,y1]] is set
(X,Y) . [[(K + r),(v + yy)],[x1,y1]] is set
[(K + (r + x1)),(v + (yy + y1))] is Element of [: the carrier of X, the carrier of Y:]
[(r + x1),(yy + y1)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([K,v],[(r + x1),(yy + y1)]) is Element of [: the carrier of X, the carrier of Y:]
[[K,v],[(r + x1),(yy + y1)]] is set
(X,Y) . [[K,v],[(r + x1),(yy + y1)]] is set
X is non empty right_zeroed RLSStruct
Y is non empty right_zeroed RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
I + (0. (X,Y)) is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,(0. (X,Y))) is Element of the carrier of (X,Y)
[I,(0. (X,Y))] is set
the addF of (X,Y) . [I,(0. (X,Y))] is set
J is Element of the carrier of X
K is Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
J + (0. X) is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,(0. X)) is Element of the carrier of X
[J,(0. X)] is set
the addF of X . [J,(0. X)] is set
K + (0. Y) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(0. Y)) is Element of the carrier of Y
[K,(0. Y)] is set
the addF of Y . [K,(0. Y)] is set
X is non empty right_complementable RLSStruct
Y is non empty right_complementable RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. Y is V52(Y) right_complementable Element of the carrier of Y
the ZeroF of Y is right_complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is right_complementable Element of the carrier of X
K is right_complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
K is right_complementable Element of the carrier of X
J + K is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,K) is right_complementable Element of the carrier of X
[J,K] is set
the addF of X . [J,K] is set
v is right_complementable Element of the carrier of Y
K + v is right_complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,v) is right_complementable Element of the carrier of Y
[K,v] is set
the addF of Y . [K,v] is set
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is Element of the carrier of (X,Y)
I + r is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,r) is Element of the carrier of (X,Y)
[I,r] is set
the addF of (X,Y) . [I,r] is set
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
X is non empty RLSStruct
Y is non empty RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
I is Element of the carrier of (X,Y)
J is Element of the carrier of X
K is Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
K is V11() real ext-real set
K * I is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (K,I) is set
[K,I] is set
the Mult of (X,Y) . [K,I] is set
K * J is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,J) is set
[K,J] is set
the Mult of X . [K,J] is set
K * K is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,K) is set
[K,K] is set
the Mult of Y . [K,K] is set
[(K * J),(K * K)] is Element of [: the carrier of X, the carrier of Y:]
v is V11() real ext-real Element of REAL
(X,Y) . (v,I) is set
[v,I] is set
(X,Y) . [v,I] is set
I is set
J is set
K is Element of the carrier of X
K is Element of the carrier of Y
[K,K] is Element of [: the carrier of X, the carrier of Y:]
v is Element of the carrier of (X,Y)
yy is Element of the carrier of X
y1 is Element of the carrier of Y
[yy,y1] is Element of [: the carrier of X, the carrier of Y:]
r is Element of the carrier of (X,Y)
x1 is Element of the carrier of X
xx2 is Element of the carrier of Y
[x1,xx2] is Element of [: the carrier of X, the carrier of Y:]
v + r is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (v,r) is Element of the carrier of (X,Y)
[v,r] is set
the addF of (X,Y) . [v,r] is set
yy + x1 is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (yy,x1) is Element of the carrier of X
[yy,x1] is set
the addF of X . [yy,x1] is set
y1 + xx2 is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (y1,xx2) is Element of the carrier of Y
[y1,xx2] is set
the addF of Y . [y1,xx2] is set
[(yy + x1),(y1 + xx2)] is Element of [: the carrier of X, the carrier of Y:]
yy2 is Element of the carrier of (X,Y)
I is Element of the carrier of X
v is Element of the carrier of Y
[I,v] is Element of [: the carrier of X, the carrier of Y:]
x1 is V11() real ext-real set
x1 * yy2 is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (x1,yy2) is set
[x1,yy2] is set
the Mult of (X,Y) . [x1,yy2] is set
x1 * I is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (x1,I) is set
[x1,I] is set
the Mult of X . [x1,I] is set
x1 * v is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (x1,v) is set
[x1,v] is set
the Mult of Y . [x1,v] is set
[(x1 * I),(x1 * v)] is Element of [: the carrier of X, the carrier of Y:]
X is non empty right_complementable add-associative right_zeroed RLSStruct
Y is non empty right_complementable add-associative right_zeroed RLSStruct
(X,Y) is non empty right_complementable strict add-associative right_zeroed RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) right_complementable Element of the carrier of X
the ZeroF of X is right_complementable Element of the carrier of X
0. Y is V52(Y) right_complementable Element of the carrier of Y
the ZeroF of Y is right_complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is right_complementable Element of the carrier of (X,Y)
- I is right_complementable Element of the carrier of (X,Y)
J is right_complementable Element of the carrier of X
- J is right_complementable Element of the carrier of X
K is right_complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
- K is right_complementable Element of the carrier of Y
[(- J),(- K)] is Element of [: the carrier of X, the carrier of Y:]
K is right_complementable Element of the carrier of (X,Y)
I + K is right_complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,K) is right_complementable Element of the carrier of (X,Y)
[I,K] is set
the addF of (X,Y) . [I,K] is set
J + (- J) is right_complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,(- J)) is right_complementable Element of the carrier of X
[J,(- J)] is set
the addF of X . [J,(- J)] is set
K + (- K) is right_complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(- K)) is right_complementable Element of the carrier of Y
[K,(- K)] is set
the addF of Y . [K,(- K)] is set
[(J + (- J)),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
[(0. X),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
0. (X,Y) is V52((X,Y)) right_complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is right_complementable Element of the carrier of (X,Y)
X is non empty vector-distributive RLSStruct
Y is non empty vector-distributive RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is V11() real ext-real set
J is Element of the carrier of (X,Y)
K is Element of the carrier of (X,Y)
J + K is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (J,K) is Element of the carrier of (X,Y)
[J,K] is set
the addF of (X,Y) . [J,K] is set
I * (J + K) is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (I,(J + K)) is set
[I,(J + K)] is set
the Mult of (X,Y) . [I,(J + K)] is set
I * J is Element of the carrier of (X,Y)
the Mult of (X,Y) . (I,J) is set
[I,J] is set
the Mult of (X,Y) . [I,J] is set
I * K is Element of the carrier of (X,Y)
the Mult of (X,Y) . (I,K) is set
[I,K] is set
the Mult of (X,Y) . [I,K] is set
(I * J) + (I * K) is Element of the carrier of (X,Y)
the addF of (X,Y) . ((I * J),(I * K)) is Element of the carrier of (X,Y)
[(I * J),(I * K)] is set
the addF of (X,Y) . [(I * J),(I * K)] is set
v is Element of the carrier of X
r is Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
yy is Element of the carrier of X
x1 is Element of the carrier of Y
[yy,x1] is Element of [: the carrier of X, the carrier of Y:]
v + yy is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (v,yy) is Element of the carrier of X
[v,yy] is set
the addF of X . [v,yy] is set
K is V11() real ext-real Element of REAL
K * (v + yy) is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,(v + yy)) is set
[K,(v + yy)] is set
the Mult of X . [K,(v + yy)] is set
I * v is Element of the carrier of X
the Mult of X . (I,v) is set
[I,v] is set
the Mult of X . [I,v] is set
I * yy is Element of the carrier of X
the Mult of X . (I,yy) is set
[I,yy] is set
the Mult of X . [I,yy] is set
(I * v) + (I * yy) is Element of the carrier of X
the addF of X . ((I * v),(I * yy)) is Element of the carrier of X
[(I * v),(I * yy)] is set
the addF of X . [(I * v),(I * yy)] is set
r + x1 is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (r,x1) is Element of the carrier of Y
[r,x1] is set
the addF of Y . [r,x1] is set
K * (r + x1) is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,(r + x1)) is set
[K,(r + x1)] is set
the Mult of Y . [K,(r + x1)] is set
I * r is Element of the carrier of Y
the Mult of Y . (I,r) is set
[I,r] is set
the Mult of Y . [I,r] is set
I * x1 is Element of the carrier of Y
the Mult of Y . (I,x1) is set
[I,x1] is set
the Mult of Y . [I,x1] is set
(I * r) + (I * x1) is Element of the carrier of Y
the addF of Y . ((I * r),(I * x1)) is Element of the carrier of Y
[(I * r),(I * x1)] is set
the addF of Y . [(I * r),(I * x1)] is set
[(v + yy),(r + x1)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . (K,[(v + yy),(r + x1)]) is Element of [: the carrier of X, the carrier of Y:]
[K,[(v + yy),(r + x1)]] is set
(X,Y) . [K,[(v + yy),(r + x1)]] is set
[(K * (v + yy)),(K * (r + x1))] is Element of [: the carrier of X, the carrier of Y:]
K * v is Element of the carrier of X
the Mult of X . (K,v) is set
[K,v] is set
the Mult of X . [K,v] is set
K * r is Element of the carrier of Y
the Mult of Y . (K,r) is set
[K,r] is set
the Mult of Y . [K,r] is set
[(K * v),(K * r)] is Element of [: the carrier of X, the carrier of Y:]
K * yy is Element of the carrier of X
the Mult of X . (K,yy) is set
[K,yy] is set
the Mult of X . [K,yy] is set
K * x1 is Element of the carrier of Y
the Mult of Y . (K,x1) is set
[K,x1] is set
the Mult of Y . [K,x1] is set
[(K * yy),(K * x1)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([(K * v),(K * r)],[(K * yy),(K * x1)]) is Element of [: the carrier of X, the carrier of Y:]
[[(K * v),(K * r)],[(K * yy),(K * x1)]] is set
(X,Y) . [[(K * v),(K * r)],[(K * yy),(K * x1)]] is set
(X,Y) . (K,[v,r]) is Element of [: the carrier of X, the carrier of Y:]
[K,[v,r]] is set
(X,Y) . [K,[v,r]] is set
(X,Y) . (((X,Y) . (K,[v,r])),[(K * yy),(K * x1)]) is Element of [: the carrier of X, the carrier of Y:]
[((X,Y) . (K,[v,r])),[(K * yy),(K * x1)]] is set
(X,Y) . [((X,Y) . (K,[v,r])),[(K * yy),(K * x1)]] is set
X is non empty scalar-distributive RLSStruct
Y is non empty scalar-distributive RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is V11() real ext-real set
J is V11() real ext-real set
I + J is V11() real ext-real set
K is Element of the carrier of (X,Y)
(I + J) * K is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . ((I + J),K) is set
[(I + J),K] is set
the Mult of (X,Y) . [(I + J),K] is set
I * K is Element of the carrier of (X,Y)
the Mult of (X,Y) . (I,K) is set
[I,K] is set
the Mult of (X,Y) . [I,K] is set
J * K is Element of the carrier of (X,Y)
the Mult of (X,Y) . (J,K) is set
[J,K] is set
the Mult of (X,Y) . [J,K] is set
(I * K) + (J * K) is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . ((I * K),(J * K)) is Element of the carrier of (X,Y)
[(I * K),(J * K)] is set
the addF of (X,Y) . [(I * K),(J * K)] is set
r is Element of the carrier of X
yy is Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
K is V11() real ext-real Element of REAL
v is V11() real ext-real Element of REAL
K + v is V11() real ext-real Element of REAL
(K + v) * r is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((K + v),r) is set
[(K + v),r] is set
the Mult of X . [(K + v),r] is set
I * r is Element of the carrier of X
the Mult of X . (I,r) is set
[I,r] is set
the Mult of X . [I,r] is set
J * r is Element of the carrier of X
the Mult of X . (J,r) is set
[J,r] is set
the Mult of X . [J,r] is set
(I * r) + (J * r) is Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((I * r),(J * r)) is Element of the carrier of X
[(I * r),(J * r)] is set
the addF of X . [(I * r),(J * r)] is set
(K + v) * yy is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . ((K + v),yy) is set
[(K + v),yy] is set
the Mult of Y . [(K + v),yy] is set
I * yy is Element of the carrier of Y
the Mult of Y . (I,yy) is set
[I,yy] is set
the Mult of Y . [I,yy] is set
J * yy is Element of the carrier of Y
the Mult of Y . (J,yy) is set
[J,yy] is set
the Mult of Y . [J,yy] is set
(I * yy) + (J * yy) is Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . ((I * yy),(J * yy)) is Element of the carrier of Y
[(I * yy),(J * yy)] is set
the addF of Y . [(I * yy),(J * yy)] is set
[((K + v) * r),((K + v) * yy)] is Element of [: the carrier of X, the carrier of Y:]
K * r is Element of the carrier of X
the Mult of X . (K,r) is set
[K,r] is set
the Mult of X . [K,r] is set
K * yy is Element of the carrier of Y
the Mult of Y . (K,yy) is set
[K,yy] is set
the Mult of Y . [K,yy] is set
[(K * r),(K * yy)] is Element of [: the carrier of X, the carrier of Y:]
v * r is Element of the carrier of X
the Mult of X . (v,r) is set
[v,r] is set
the Mult of X . [v,r] is set
v * yy is Element of the carrier of Y
the Mult of Y . (v,yy) is set
[v,yy] is set
the Mult of Y . [v,yy] is set
[(v * r),(v * yy)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . ([(K * r),(K * yy)],[(v * r),(v * yy)]) is Element of [: the carrier of X, the carrier of Y:]
[[(K * r),(K * yy)],[(v * r),(v * yy)]] is set
(X,Y) . [[(K * r),(K * yy)],[(v * r),(v * yy)]] is set
(X,Y) . (K,[r,yy]) is Element of [: the carrier of X, the carrier of Y:]
[K,[r,yy]] is set
(X,Y) . [K,[r,yy]] is set
(X,Y) . (((X,Y) . (K,[r,yy])),[(v * r),(v * yy)]) is Element of [: the carrier of X, the carrier of Y:]
[((X,Y) . (K,[r,yy])),[(v * r),(v * yy)]] is set
(X,Y) . [((X,Y) . (K,[r,yy])),[(v * r),(v * yy)]] is set
X is non empty scalar-associative RLSStruct
Y is non empty scalar-associative RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is V11() real ext-real set
J is V11() real ext-real set
I * J is V11() real ext-real set
K is Element of the carrier of (X,Y)
(I * J) * K is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . ((I * J),K) is set
[(I * J),K] is set
the Mult of (X,Y) . [(I * J),K] is set
J * K is Element of the carrier of (X,Y)
the Mult of (X,Y) . (J,K) is set
[J,K] is set
the Mult of (X,Y) . [J,K] is set
I * (J * K) is Element of the carrier of (X,Y)
the Mult of (X,Y) . (I,(J * K)) is set
[I,(J * K)] is set
the Mult of (X,Y) . [I,(J * K)] is set
r is Element of the carrier of X
yy is Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
K is V11() real ext-real Element of REAL
v is V11() real ext-real Element of REAL
K * v is V11() real ext-real Element of REAL
(K * v) * r is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . ((K * v),r) is set
[(K * v),r] is set
the Mult of X . [(K * v),r] is set
J * r is Element of the carrier of X
the Mult of X . (J,r) is set
[J,r] is set
the Mult of X . [J,r] is set
I * (J * r) is Element of the carrier of X
the Mult of X . (I,(J * r)) is set
[I,(J * r)] is set
the Mult of X . [I,(J * r)] is set
(K * v) * yy is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . ((K * v),yy) is set
[(K * v),yy] is set
the Mult of Y . [(K * v),yy] is set
J * yy is Element of the carrier of Y
the Mult of Y . (J,yy) is set
[J,yy] is set
the Mult of Y . [J,yy] is set
I * (J * yy) is Element of the carrier of Y
the Mult of Y . (I,(J * yy)) is set
[I,(J * yy)] is set
the Mult of Y . [I,(J * yy)] is set
[((K * v) * r),((K * v) * yy)] is Element of [: the carrier of X, the carrier of Y:]
v * r is Element of the carrier of X
the Mult of X . (v,r) is set
[v,r] is set
the Mult of X . [v,r] is set
v * yy is Element of the carrier of Y
the Mult of Y . (v,yy) is set
[v,yy] is set
the Mult of Y . [v,yy] is set
[(v * r),(v * yy)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) . (K,[(v * r),(v * yy)]) is Element of [: the carrier of X, the carrier of Y:]
[K,[(v * r),(v * yy)]] is set
(X,Y) . [K,[(v * r),(v * yy)]] is set
X is non empty scalar-unital RLSStruct
Y is non empty scalar-unital RLSStruct
(X,Y) is non empty strict RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
1 * I is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (1,I) is set
[1,I] is set
the Mult of (X,Y) . [1,I] is set
J is Element of the carrier of X
K is Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
1 * J is Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (1,J) is set
[1,J] is set
the Mult of X . [1,J] is set
1 * K is Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (1,K) is set
[1,K] is set
the Mult of Y . [1,K] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
<*X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
Y is set
rng <*X*> is non empty set
dom <*X*> is non empty countable Element of bool NAT
I is set
<*X*> . I is set
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
I is set
rng <*X,Y*> is non empty set
dom <*X,Y*> is non empty countable Element of bool NAT
J is set
<*X,Y*> . J is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of X is non empty set
<*X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*X*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*X*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*X*>) is non empty functional with_common_domain product-like set
zeros <*X*> is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
addop <*X*> is Relation-like Function-like BinOps of carr <*X*>
[:(addop <*X*>):] is non empty Relation-like [:(product (carr <*X*>)),(product (carr <*X*>)):] -defined product (carr <*X*>) -valued Function-like V26([:(product (carr <*X*>)),(product (carr <*X*>)):]) quasi_total Element of bool [:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):]
[:(product (carr <*X*>)),(product (carr <*X*>)):] is non empty Relation-like set
[:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty Relation-like set
bool [:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty set
multop <*X*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*X*>
[:(multop <*X*>):] is non empty Relation-like [:REAL,(product (carr <*X*>)):] -defined product (carr <*X*>) -valued Function-like V26([:REAL,(product (carr <*X*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):]
[:REAL,(product (carr <*X*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty set
RLSStruct(# (product (carr <*X*>)),(zeros <*X*>),[:(addop <*X*>):],[:(multop <*X*>):] #) is non empty strict RLSStruct
the carrier of (product <*X*>) is non empty set
[: the carrier of X, the carrier of (product <*X*>):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (product <*X*>):] is non empty set
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. (product <*X*>) is V52( product <*X*>) left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the ZeroF of (product <*X*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
<* the carrier of X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
product <* the carrier of X*> is functional with_common_domain product-like set
[: the carrier of X,(product <* the carrier of X*>):] is Relation-like set
bool [: the carrier of X,(product <* the carrier of X*>):] is non empty set
I is Relation-like the carrier of X -defined product <* the carrier of X*> -valued Function-like quasi_total Element of bool [: the carrier of X,(product <* the carrier of X*>):]
len (carr <*X*>) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len <*X*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom <*X*> is non empty countable Element of bool NAT
<*X*> . 1 is set
(carr <*X*>) . 1 is set
J is non empty Relation-like the carrier of X -defined the carrier of (product <*X*>) -valued Function-like V26( the carrier of X) quasi_total Element of bool [: the carrier of X, the carrier of (product <*X*>):]
K is left_complementable right_complementable complementable Element of the carrier of X
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of X
K + K is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,K) is left_complementable right_complementable complementable Element of the carrier of X
[K,K] is set
the addF of X . [K,K] is set
J . (K + K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
(J . K) + (J . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) is non empty Relation-like [: the carrier of (product <*X*>), the carrier of (product <*X*>):] -defined the carrier of (product <*X*>) -valued Function-like V26([: the carrier of (product <*X*>), the carrier of (product <*X*>):]) quasi_total Element of bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):]
[: the carrier of (product <*X*>), the carrier of (product <*X*>):] is non empty Relation-like set
[:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty set
the addF of (product <*X*>) . ((J . K),(J . K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . K),(J . K)] is set
the addF of (product <*X*>) . [(J . K),(J . K)] is set
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*(K + K)*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*K*> . 1 is set
<*K*> . 1 is set
dom (carr <*X*>) is non empty countable Element of bool NAT
yy is Element of dom (carr <*X*>)
(carr <*X*>) . yy is non empty set
[:((carr <*X*>) . yy),((carr <*X*>) . yy):] is non empty Relation-like set
<*X*> . yy is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X*> . yy) is non empty set
[: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):] is non empty Relation-like set
(addop <*X*>) . yy is non empty Relation-like [:((carr <*X*>) . yy),((carr <*X*>) . yy):] -defined (carr <*X*>) . yy -valued Function-like V26([:((carr <*X*>) . yy),((carr <*X*>) . yy):]) quasi_total Element of bool [:[:((carr <*X*>) . yy),((carr <*X*>) . yy):],((carr <*X*>) . yy):]
[:[:((carr <*X*>) . yy),((carr <*X*>) . yy):],((carr <*X*>) . yy):] is non empty Relation-like set
bool [:[:((carr <*X*>) . yy),((carr <*X*>) . yy):],((carr <*X*>) . yy):] is non empty set
the addF of (<*X*> . yy) is non empty Relation-like [: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):] -defined the carrier of (<*X*> . yy) -valued Function-like V26([: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):]) quasi_total Element of bool [:[: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):], the carrier of (<*X*> . yy):]
[:[: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):], the carrier of (<*X*> . yy):] is non empty Relation-like set
bool [:[: the carrier of (<*X*> . yy), the carrier of (<*X*> . yy):], the carrier of (<*X*> . yy):] is non empty set
v is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
r is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
[:(addop <*X*>):] . (v,r) is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
[v,r] is set
[:(addop <*X*>):] . [v,r] is Relation-like Function-like set
([:(addop <*X*>):] . (v,r)) . yy is Element of (carr <*X*>) . yy
v . yy is Element of (carr <*X*>) . yy
r . yy is Element of (carr <*X*>) . yy
((addop <*X*>) . yy) . ((v . yy),(r . yy)) is Element of (carr <*X*>) . yy
[(v . yy),(r . yy)] is set
((addop <*X*>) . yy) . [(v . yy),(r . yy)] is set
x1 is Relation-like Function-like set
dom x1 is set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is left_complementable right_complementable complementable Element of the carrier of X
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
K is V11() real ext-real Element of REAL
K * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,K) is set
[K,K] is set
the Mult of X . [K,K] is set
J . (K * K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
K * (J . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the Mult of (product <*X*>) is non empty Relation-like [:REAL, the carrier of (product <*X*>):] -defined the carrier of (product <*X*>) -valued Function-like V26([:REAL, the carrier of (product <*X*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):]
[:REAL, the carrier of (product <*X*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty set
the Mult of (product <*X*>) . (K,(J . K)) is set
[K,(J . K)] is set
the Mult of (product <*X*>) . [K,(J . K)] is set
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*(K * K)*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*K*> . 1 is set
dom (carr <*X*>) is non empty countable Element of bool NAT
v is Element of dom (carr <*X*>)
(carr <*X*>) . v is non empty set
[:REAL,((carr <*X*>) . v):] is non empty Relation-like set
<*X*> . v is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X*> . v) is non empty set
[:REAL, the carrier of (<*X*> . v):] is non empty Relation-like set
(multop <*X*>) . v is non empty Relation-like [:REAL,((carr <*X*>) . v):] -defined (carr <*X*>) . v -valued Function-like V26([:REAL,((carr <*X*>) . v):]) quasi_total Element of bool [:[:REAL,((carr <*X*>) . v):],((carr <*X*>) . v):]
[:[:REAL,((carr <*X*>) . v):],((carr <*X*>) . v):] is non empty Relation-like set
bool [:[:REAL,((carr <*X*>) . v):],((carr <*X*>) . v):] is non empty set
the Mult of (<*X*> . v) is non empty Relation-like [:REAL, the carrier of (<*X*> . v):] -defined the carrier of (<*X*> . v) -valued Function-like V26([:REAL, the carrier of (<*X*> . v):]) quasi_total Element of bool [:[:REAL, the carrier of (<*X*> . v):], the carrier of (<*X*> . v):]
[:[:REAL, the carrier of (<*X*> . v):], the carrier of (<*X*> . v):] is non empty Relation-like set
bool [:[:REAL, the carrier of (<*X*> . v):], the carrier of (<*X*> . v):] is non empty set
r is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
[:(multop <*X*>):] . (K,r) is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
[K,r] is set
[:(multop <*X*>):] . [K,r] is Relation-like Function-like set
([:(multop <*X*>):] . (K,r)) . v is Element of (carr <*X*>) . v
r . v is Element of (carr <*X*>) . v
((multop <*X*>) . v) . (K,(r . v)) is Element of (carr <*X*>) . v
[K,(r . v)] is set
((multop <*X*>) . v) . [K,(r . v)] is set
yy is Relation-like Function-like set
dom yy is set
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
J . (0. X) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
(0. X) + (0. X) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . ((0. X),(0. X)) is left_complementable right_complementable complementable Element of the carrier of X
[(0. X),(0. X)] is set
the addF of X . [(0. X),(0. X)] is set
J . ((0. X) + (0. X)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
(J . (0. X)) + (J . (0. X)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) is non empty Relation-like [: the carrier of (product <*X*>), the carrier of (product <*X*>):] -defined the carrier of (product <*X*>) -valued Function-like V26([: the carrier of (product <*X*>), the carrier of (product <*X*>):]) quasi_total Element of bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):]
[: the carrier of (product <*X*>), the carrier of (product <*X*>):] is non empty Relation-like set
[:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty set
the addF of (product <*X*>) . ((J . (0. X)),(J . (0. X))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . (0. X)),(J . (0. X))] is set
the addF of (product <*X*>) . [(J . (0. X)),(J . (0. X))] is set
(J . (0. X)) - (J . (0. X)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
- (J . (0. X)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
(J . (0. X)) + (- (J . (0. X))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) . ((J . (0. X)),(- (J . (0. X)))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . (0. X)),(- (J . (0. X)))] is set
the addF of (product <*X*>) . [(J . (0. X)),(- (J . (0. X)))] is set
(J . (0. X)) + ((J . (0. X)) - (J . (0. X))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) . ((J . (0. X)),((J . (0. X)) - (J . (0. X)))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . (0. X)),((J . (0. X)) - (J . (0. X)))] is set
the addF of (product <*X*>) . [(J . (0. X)),((J . (0. X)) - (J . (0. X)))] is set
(J . (0. X)) + (0. (product <*X*>)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) . ((J . (0. X)),(0. (product <*X*>))) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . (0. X)),(0. (product <*X*>))] is set
the addF of (product <*X*>) . [(J . (0. X)),(0. (product <*X*>))] is set
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I is set
rng (X ^ Y) is non empty set
dom (X ^ Y) is non empty countable Element of bool NAT
J is set
(X ^ Y) . J is set
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
(X ^ Y) . K is set
X . K is set
rng X is non empty set
dom Y is non empty countable Element of bool NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(X ^ Y) . K is set
Y . K is set
rng Y is non empty set
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
(X,Y) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*X,Y*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*X,Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*X,Y*>) is non empty functional with_common_domain product-like set
zeros <*X,Y*> is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
addop <*X,Y*> is Relation-like Function-like BinOps of carr <*X,Y*>
[:(addop <*X,Y*>):] is non empty Relation-like [:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
multop <*X,Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*X,Y*>
[:(multop <*X,Y*>):] is non empty Relation-like [:REAL,(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:REAL,(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:REAL,(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
RLSStruct(# (product (carr <*X,Y*>)),(zeros <*X,Y*>),[:(addop <*X,Y*>):],[:(multop <*X,Y*>):] #) is non empty strict RLSStruct
the carrier of (product <*X,Y*>) is non empty set
[: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
0. (product <*X,Y*>) is V52( product <*X,Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the ZeroF of (product <*X,Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
<* the carrier of X, the carrier of Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
product <* the carrier of X, the carrier of Y*> is functional with_common_domain product-like set
[:[: the carrier of X, the carrier of Y:],(product <* the carrier of X, the carrier of Y*>):] is Relation-like set
bool [:[: the carrier of X, the carrier of Y:],(product <* the carrier of X, the carrier of Y*>):] is non empty set
K is Relation-like [: the carrier of X, the carrier of Y:] -defined product <* the carrier of X, the carrier of Y*> -valued Function-like quasi_total Element of bool [:[: the carrier of X, the carrier of Y:],(product <* the carrier of X, the carrier of Y*>):]
len (carr <*X,Y*>) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len <*X,Y*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom (carr <*X,Y*>) is non empty countable Element of bool NAT
dom <*X,Y*> is non empty countable Element of bool NAT
<*X,Y*> . 1 is set
<*X,Y*> . 2 is set
(carr <*X,Y*>) . 1 is set
(carr <*X,Y*>) . 2 is set
K is non empty Relation-like the carrier of (X,Y) -defined the carrier of (product <*X,Y*>) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):]
v is left_complementable right_complementable complementable Element of the carrier of X
r is left_complementable right_complementable complementable Element of the carrier of Y
K . (v,r) is set
[v,r] is set
K . [v,r] is set
<*v,r*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
r is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v + r is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (v,r) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[v,r] is set
the addF of (X,Y) . [v,r] is set
K . (v + r) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K . r is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . v) + (K . r) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . ((K . v),(K . r)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . v),(K . r)] is set
the addF of (product <*X,Y*>) . [(K . v),(K . r)] is set
yy is left_complementable right_complementable complementable Element of the carrier of X
x1 is left_complementable right_complementable complementable Element of the carrier of Y
[yy,x1] is Element of [: the carrier of X, the carrier of Y:]
y1 is left_complementable right_complementable complementable Element of the carrier of X
xx2 is left_complementable right_complementable complementable Element of the carrier of Y
[y1,xx2] is Element of [: the carrier of X, the carrier of Y:]
K . (yy,x1) is set
[yy,x1] is set
K . [yy,x1] is set
K . (y1,xx2) is set
[y1,xx2] is set
K . [y1,xx2] is set
<*yy,x1*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*y1,xx2*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
yy + y1 is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (yy,y1) is left_complementable right_complementable complementable Element of the carrier of X
[yy,y1] is set
the addF of X . [yy,y1] is set
x1 + xx2 is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (x1,xx2) is left_complementable right_complementable complementable Element of the carrier of Y
[x1,xx2] is set
the addF of Y . [x1,xx2] is set
K . ((yy + y1),(x1 + xx2)) is set
[(yy + y1),(x1 + xx2)] is set
K . [(yy + y1),(x1 + xx2)] is set
<*(yy + y1),(x1 + xx2)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*yy,x1*> . 1 is set
<*y1,xx2*> . 1 is set
<*yy,x1*> . 2 is set
<*y1,xx2*> . 2 is set
v is Element of dom (carr <*X,Y*>)
(carr <*X,Y*>) . v is non empty set
[:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):] is non empty Relation-like set
<*X,Y*> . v is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X,Y*> . v) is non empty set
[: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):] is non empty Relation-like set
(addop <*X,Y*>) . v is non empty Relation-like [:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):] -defined (carr <*X,Y*>) . v -valued Function-like V26([:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):]) quasi_total Element of bool [:[:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):],((carr <*X,Y*>) . v):]
[:[:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):],((carr <*X,Y*>) . v):] is non empty Relation-like set
bool [:[:((carr <*X,Y*>) . v),((carr <*X,Y*>) . v):],((carr <*X,Y*>) . v):] is non empty set
the addF of (<*X,Y*> . v) is non empty Relation-like [: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):] -defined the carrier of (<*X,Y*> . v) -valued Function-like V26([: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):]) quasi_total Element of bool [:[: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):], the carrier of (<*X,Y*> . v):]
[:[: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):], the carrier of (<*X,Y*> . v):] is non empty Relation-like set
bool [:[: the carrier of (<*X,Y*> . v), the carrier of (<*X,Y*> . v):], the carrier of (<*X,Y*> . v):] is non empty set
yy2 is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
I is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
[:(addop <*X,Y*>):] . (yy2,I) is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
[yy2,I] is set
[:(addop <*X,Y*>):] . [yy2,I] is Relation-like Function-like set
([:(addop <*X,Y*>):] . (yy2,I)) . v is Element of (carr <*X,Y*>) . v
yy2 . v is Element of (carr <*X,Y*>) . v
I . v is Element of (carr <*X,Y*>) . v
((addop <*X,Y*>) . v) . ((yy2 . v),(I . v)) is Element of (carr <*X,Y*>) . v
[(yy2 . v),(I . v)] is set
((addop <*X,Y*>) . v) . [(yy2 . v),(I . v)] is set
x1 is Element of dom (carr <*X,Y*>)
(carr <*X,Y*>) . x1 is non empty set
[:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):] is non empty Relation-like set
<*X,Y*> . x1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X,Y*> . x1) is non empty set
[: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):] is non empty Relation-like set
(addop <*X,Y*>) . x1 is non empty Relation-like [:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):] -defined (carr <*X,Y*>) . x1 -valued Function-like V26([:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):]) quasi_total Element of bool [:[:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):],((carr <*X,Y*>) . x1):]
[:[:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):],((carr <*X,Y*>) . x1):] is non empty Relation-like set
bool [:[:((carr <*X,Y*>) . x1),((carr <*X,Y*>) . x1):],((carr <*X,Y*>) . x1):] is non empty set
the addF of (<*X,Y*> . x1) is non empty Relation-like [: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):] -defined the carrier of (<*X,Y*> . x1) -valued Function-like V26([: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):]) quasi_total Element of bool [:[: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):], the carrier of (<*X,Y*> . x1):]
[:[: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):], the carrier of (<*X,Y*> . x1):] is non empty Relation-like set
bool [:[: the carrier of (<*X,Y*> . x1), the carrier of (<*X,Y*> . x1):], the carrier of (<*X,Y*> . x1):] is non empty set
([:(addop <*X,Y*>):] . (yy2,I)) . x1 is Element of (carr <*X,Y*>) . x1
yy2 . x1 is Element of (carr <*X,Y*>) . x1
I . x1 is Element of (carr <*X,Y*>) . x1
((addop <*X,Y*>) . x1) . ((yy2 . x1),(I . x1)) is Element of (carr <*X,Y*>) . x1
[(yy2 . x1),(I . x1)] is set
((addop <*X,Y*>) . x1) . [(yy2 . x1),(I . x1)] is set
y1 is Relation-like Function-like set
dom y1 is set
v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len v1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r is V11() real ext-real Element of REAL
r * v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (r,v) is set
[r,v] is set
the Mult of (X,Y) . [r,v] is set
K . (r * v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r * (K . v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the Mult of (product <*X,Y*>) is non empty Relation-like [:REAL, the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([:REAL, the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[:REAL, the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the Mult of (product <*X,Y*>) . (r,(K . v)) is set
[r,(K . v)] is set
the Mult of (product <*X,Y*>) . [r,(K . v)] is set
yy is left_complementable right_complementable complementable Element of the carrier of X
x1 is left_complementable right_complementable complementable Element of the carrier of Y
[yy,x1] is Element of [: the carrier of X, the carrier of Y:]
K . (yy,x1) is set
[yy,x1] is set
K . [yy,x1] is set
<*yy,x1*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
r * yy is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (r,yy) is set
[r,yy] is set
the Mult of X . [r,yy] is set
r * x1 is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (r,x1) is set
[r,x1] is set
the Mult of Y . [r,x1] is set
K . ((r * yy),(r * x1)) is set
[(r * yy),(r * x1)] is set
K . [(r * yy),(r * x1)] is set
<*(r * yy),(r * x1)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*yy,x1*> . 1 is set
<*yy,x1*> . 2 is set
y1 is Element of dom (carr <*X,Y*>)
(carr <*X,Y*>) . y1 is non empty set
[:REAL,((carr <*X,Y*>) . y1):] is non empty Relation-like set
<*X,Y*> . y1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X,Y*> . y1) is non empty set
[:REAL, the carrier of (<*X,Y*> . y1):] is non empty Relation-like set
(multop <*X,Y*>) . y1 is non empty Relation-like [:REAL,((carr <*X,Y*>) . y1):] -defined (carr <*X,Y*>) . y1 -valued Function-like V26([:REAL,((carr <*X,Y*>) . y1):]) quasi_total Element of bool [:[:REAL,((carr <*X,Y*>) . y1):],((carr <*X,Y*>) . y1):]
[:[:REAL,((carr <*X,Y*>) . y1):],((carr <*X,Y*>) . y1):] is non empty Relation-like set
bool [:[:REAL,((carr <*X,Y*>) . y1):],((carr <*X,Y*>) . y1):] is non empty set
the Mult of (<*X,Y*> . y1) is non empty Relation-like [:REAL, the carrier of (<*X,Y*> . y1):] -defined the carrier of (<*X,Y*> . y1) -valued Function-like V26([:REAL, the carrier of (<*X,Y*> . y1):]) quasi_total Element of bool [:[:REAL, the carrier of (<*X,Y*> . y1):], the carrier of (<*X,Y*> . y1):]
[:[:REAL, the carrier of (<*X,Y*> . y1):], the carrier of (<*X,Y*> . y1):] is non empty Relation-like set
bool [:[:REAL, the carrier of (<*X,Y*> . y1):], the carrier of (<*X,Y*> . y1):] is non empty set
xx2 is Element of dom (carr <*X,Y*>)
(carr <*X,Y*>) . xx2 is non empty set
[:REAL,((carr <*X,Y*>) . xx2):] is non empty Relation-like set
<*X,Y*> . xx2 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (<*X,Y*> . xx2) is non empty set
[:REAL, the carrier of (<*X,Y*> . xx2):] is non empty Relation-like set
(multop <*X,Y*>) . xx2 is non empty Relation-like [:REAL,((carr <*X,Y*>) . xx2):] -defined (carr <*X,Y*>) . xx2 -valued Function-like V26([:REAL,((carr <*X,Y*>) . xx2):]) quasi_total Element of bool [:[:REAL,((carr <*X,Y*>) . xx2):],((carr <*X,Y*>) . xx2):]
[:[:REAL,((carr <*X,Y*>) . xx2):],((carr <*X,Y*>) . xx2):] is non empty Relation-like set
bool [:[:REAL,((carr <*X,Y*>) . xx2):],((carr <*X,Y*>) . xx2):] is non empty set
the Mult of (<*X,Y*> . xx2) is non empty Relation-like [:REAL, the carrier of (<*X,Y*> . xx2):] -defined the carrier of (<*X,Y*> . xx2) -valued Function-like V26([:REAL, the carrier of (<*X,Y*> . xx2):]) quasi_total Element of bool [:[:REAL, the carrier of (<*X,Y*> . xx2):], the carrier of (<*X,Y*> . xx2):]
[:[:REAL, the carrier of (<*X,Y*> . xx2):], the carrier of (<*X,Y*> . xx2):] is non empty Relation-like set
bool [:[:REAL, the carrier of (<*X,Y*> . xx2):], the carrier of (<*X,Y*> . xx2):] is non empty set
yy2 is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
[:(multop <*X,Y*>):] . (r,yy2) is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
[r,yy2] is set
[:(multop <*X,Y*>):] . [r,yy2] is Relation-like Function-like set
([:(multop <*X,Y*>):] . (r,yy2)) . y1 is Element of (carr <*X,Y*>) . y1
yy2 . y1 is Element of (carr <*X,Y*>) . y1
((multop <*X,Y*>) . y1) . (r,(yy2 . y1)) is Element of (carr <*X,Y*>) . y1
[r,(yy2 . y1)] is set
((multop <*X,Y*>) . y1) . [r,(yy2 . y1)] is set
([:(multop <*X,Y*>):] . (r,yy2)) . xx2 is Element of (carr <*X,Y*>) . xx2
yy2 . xx2 is Element of (carr <*X,Y*>) . xx2
((multop <*X,Y*>) . xx2) . (r,(yy2 . xx2)) is Element of (carr <*X,Y*>) . xx2
[r,(yy2 . xx2)] is set
((multop <*X,Y*>) . xx2) . [r,(yy2 . xx2)] is set
I is Relation-like Function-like set
dom I is set
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(0. (X,Y)) + (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . ((0. (X,Y)),(0. (X,Y))) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[(0. (X,Y)),(0. (X,Y))] is set
the addF of (X,Y) . [(0. (X,Y)),(0. (X,Y))] is set
K . ((0. (X,Y)) + (0. (X,Y))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . (0. (X,Y))) + (K . (0. (X,Y))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . ((K . (0. (X,Y))),(K . (0. (X,Y)))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . (0. (X,Y))),(K . (0. (X,Y)))] is set
the addF of (product <*X,Y*>) . [(K . (0. (X,Y))),(K . (0. (X,Y)))] is set
(K . (0. (X,Y))) - (K . (0. (X,Y))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- (K . (0. (X,Y))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . (0. (X,Y))) + (- (K . (0. (X,Y)))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((K . (0. (X,Y))),(- (K . (0. (X,Y))))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . (0. (X,Y))),(- (K . (0. (X,Y))))] is set
the addF of (product <*X,Y*>) . [(K . (0. (X,Y))),(- (K . (0. (X,Y))))] is set
(K . (0. (X,Y))) + ((K . (0. (X,Y))) - (K . (0. (X,Y)))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((K . (0. (X,Y))),((K . (0. (X,Y))) - (K . (0. (X,Y))))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . (0. (X,Y))),((K . (0. (X,Y))) - (K . (0. (X,Y))))] is set
the addF of (product <*X,Y*>) . [(K . (0. (X,Y))),((K . (0. (X,Y))) - (K . (0. (X,Y))))] is set
(K . (0. (X,Y))) + (0. (product <*X,Y*>)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((K . (0. (X,Y))),(0. (product <*X,Y*>))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . (0. (X,Y))),(0. (product <*X,Y*>))] is set
the addF of (product <*X,Y*>) . [(K . (0. (X,Y))),(0. (product <*X,Y*>))] is set
X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product X is non empty functional with_common_domain product-like set
Y is set
dom X is non empty countable Element of bool NAT
I is Relation-like Function-like set
dom I is set
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len X) is V33() V40( len X) countable Element of bool NAT
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product X is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr X) is non empty functional with_common_domain product-like set
zeros X is Relation-like NAT -defined Function-like carr X -compatible Element of product (carr X)
addop X is Relation-like Function-like BinOps of carr X
[:(addop X):] is non empty Relation-like [:(product (carr X)),(product (carr X)):] -defined product (carr X) -valued Function-like V26([:(product (carr X)),(product (carr X)):]) quasi_total Element of bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):]
[:(product (carr X)),(product (carr X)):] is non empty Relation-like set
[:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty set
multop X is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr X
[:(multop X):] is non empty Relation-like [:REAL,(product (carr X)):] -defined product (carr X) -valued Function-like V26([:REAL,(product (carr X)):]) quasi_total Element of bool [:[:REAL,(product (carr X)):],(product (carr X)):]
[:REAL,(product (carr X)):] is non empty Relation-like set
[:[:REAL,(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:REAL,(product (carr X)):],(product (carr X)):] is non empty set
RLSStruct(# (product (carr X)),(zeros X),[:(addop X):],[:(multop X):] #) is non empty strict RLSStruct
Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product Y is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr Y is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr Y) is non empty functional with_common_domain product-like set
zeros Y is Relation-like NAT -defined Function-like carr Y -compatible Element of product (carr Y)
addop Y is Relation-like Function-like BinOps of carr Y
[:(addop Y):] is non empty Relation-like [:(product (carr Y)),(product (carr Y)):] -defined product (carr Y) -valued Function-like V26([:(product (carr Y)),(product (carr Y)):]) quasi_total Element of bool [:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):]
[:(product (carr Y)),(product (carr Y)):] is non empty Relation-like set
[:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):] is non empty Relation-like set
bool [:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):] is non empty set
multop Y is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr Y
[:(multop Y):] is non empty Relation-like [:REAL,(product (carr Y)):] -defined product (carr Y) -valued Function-like V26([:REAL,(product (carr Y)):]) quasi_total Element of bool [:[:REAL,(product (carr Y)):],(product (carr Y)):]
[:REAL,(product (carr Y)):] is non empty Relation-like set
[:[:REAL,(product (carr Y)):],(product (carr Y)):] is non empty Relation-like set
bool [:[:REAL,(product (carr Y)):],(product (carr Y)):] is non empty set
RLSStruct(# (product (carr Y)),(zeros Y),[:(addop Y):],[:(multop Y):] #) is non empty strict RLSStruct
((product X),(product Y)) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (product X) is non empty set
the carrier of (product Y) is non empty set
[: the carrier of (product X), the carrier of (product Y):] is non empty Relation-like set
((product X),(product Y)) is Element of [: the carrier of (product X), the carrier of (product Y):]
0. (product X) is V52( product X) left_complementable right_complementable complementable Element of the carrier of (product X)
the ZeroF of (product X) is left_complementable right_complementable complementable Element of the carrier of (product X)
0. (product Y) is V52( product Y) left_complementable right_complementable complementable Element of the carrier of (product Y)
the ZeroF of (product Y) is left_complementable right_complementable complementable Element of the carrier of (product Y)
[(0. (product X)),(0. (product Y))] is Element of [: the carrier of (product X), the carrier of (product Y):]
((product X),(product Y)) is non empty Relation-like [:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
((product X),(product Y)) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:REAL,[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
RLSStruct(# [: the carrier of (product X), the carrier of (product Y):],((product X),(product Y)),((product X),(product Y)),((product X),(product Y)) #) is non empty strict RLSStruct
the carrier of ((product X),(product Y)) is non empty set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product (X ^ Y) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr (X ^ Y) is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr (X ^ Y)) is non empty functional with_common_domain product-like set
zeros (X ^ Y) is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
addop (X ^ Y) is Relation-like Function-like BinOps of carr (X ^ Y)
[:(addop (X ^ Y)):] is non empty Relation-like [:(product (carr (X ^ Y))),(product (carr (X ^ Y))):] -defined product (carr (X ^ Y)) -valued Function-like V26([:(product (carr (X ^ Y))),(product (carr (X ^ Y))):]) quasi_total Element of bool [:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):]
[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):] is non empty Relation-like set
[:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty Relation-like set
bool [:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty set
multop (X ^ Y) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr (X ^ Y)
[:(multop (X ^ Y)):] is non empty Relation-like [:REAL,(product (carr (X ^ Y))):] -defined product (carr (X ^ Y)) -valued Function-like V26([:REAL,(product (carr (X ^ Y))):]) quasi_total Element of bool [:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):]
[:REAL,(product (carr (X ^ Y))):] is non empty Relation-like set
[:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty Relation-like set
bool [:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty set
RLSStruct(# (product (carr (X ^ Y))),(zeros (X ^ Y)),[:(addop (X ^ Y)):],[:(multop (X ^ Y)):] #) is non empty strict RLSStruct
the carrier of (product (X ^ Y)) is non empty set
[: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty set
0. ((product X),(product Y)) is V52(((product X),(product Y))) left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the ZeroF of ((product X),(product Y)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
0. (product (X ^ Y)) is V52( product (X ^ Y)) left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the ZeroF of (product (X ^ Y)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
I is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
J is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (carr (X ^ Y)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (X ^ Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
product I is non empty functional with_common_domain product-like set
product J is non empty functional with_common_domain product-like set
[:(product I),(product J):] is non empty Relation-like set
I ^ J is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (I ^ J) is non empty functional with_common_domain product-like set
[:[:(product I),(product J):],(product (I ^ J)):] is non empty Relation-like set
bool [:[:(product I),(product J):],(product (I ^ J)):] is non empty set
K is non empty Relation-like [:(product I),(product J):] -defined product (I ^ J) -valued Function-like V26([:(product I),(product J):]) quasi_total Element of bool [:[:(product I),(product J):],(product (I ^ J)):]
(len X) + (len Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (I ^ J) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom (carr (X ^ Y)) is non empty countable Element of bool NAT
dom (I ^ J) is non empty countable Element of bool NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(carr (X ^ Y)) . r is set
(I ^ J) . r is set
dom (X ^ Y) is non empty countable Element of bool NAT
yy is Element of dom (X ^ Y)
(X ^ Y) . yy is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of ((X ^ Y) . yy) is non empty set
dom X is non empty countable Element of bool NAT
dom I is non empty countable Element of bool NAT
x1 is Element of dom X
X . x1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (X . x1) is non empty set
I . r is set
dom Y is non empty countable Element of bool NAT
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom J is non empty countable Element of bool NAT
y1 is Element of dom Y
Y . y1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (Y . y1) is non empty set
J . x1 is set
dom X is non empty countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
r is non empty Relation-like the carrier of ((product X),(product Y)) -defined the carrier of (product (X ^ Y)) -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total Element of bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):]
yy is left_complementable right_complementable complementable Element of the carrier of (product X)
x1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
r . (yy,x1) is set
[yy,x1] is set
r . [yy,x1] is set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
xx2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 ^ xx2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
yy is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
yy + x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the addF of ((product X),(product Y)) is non empty Relation-like [: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the addF of ((product X),(product Y)) . (yy,x1) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
[yy,x1] is set
the addF of ((product X),(product Y)) . [yy,x1] is set
r . (yy + x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
r . yy is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
r . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(r . yy) + (r . x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) is non empty Relation-like [: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the addF of (product (X ^ Y)) . ((r . yy),(r . x1)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(r . yy),(r . x1)] is set
the addF of (product (X ^ Y)) . [(r . yy),(r . x1)] is set
y1 is left_complementable right_complementable complementable Element of the carrier of (product X)
xx2 is left_complementable right_complementable complementable Element of the carrier of (product Y)
[y1,xx2] is Element of [: the carrier of (product X), the carrier of (product Y):]
yy2 is left_complementable right_complementable complementable Element of the carrier of (product X)
I is left_complementable right_complementable complementable Element of the carrier of (product Y)
[yy2,I] is Element of [: the carrier of (product X), the carrier of (product Y):]
y1 + yy2 is left_complementable right_complementable complementable Element of the carrier of (product X)
the addF of (product X) is non empty Relation-like [: the carrier of (product X), the carrier of (product X):] -defined the carrier of (product X) -valued Function-like V26([: the carrier of (product X), the carrier of (product X):]) quasi_total Element of bool [:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):]
[: the carrier of (product X), the carrier of (product X):] is non empty Relation-like set
[:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):] is non empty set
the addF of (product X) . (y1,yy2) is left_complementable right_complementable complementable Element of the carrier of (product X)
[y1,yy2] is set
the addF of (product X) . [y1,yy2] is set
xx2 + I is left_complementable right_complementable complementable Element of the carrier of (product Y)
the addF of (product Y) is non empty Relation-like [: the carrier of (product Y), the carrier of (product Y):] -defined the carrier of (product Y) -valued Function-like V26([: the carrier of (product Y), the carrier of (product Y):]) quasi_total Element of bool [:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):]
[: the carrier of (product Y), the carrier of (product Y):] is non empty Relation-like set
[:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):] is non empty Relation-like set
bool [:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):] is non empty set
the addF of (product Y) . (xx2,I) is left_complementable right_complementable complementable Element of the carrier of (product Y)
[xx2,I] is set
the addF of (product Y) . [xx2,I] is set
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom v is countable Element of bool NAT
dom I is non empty countable Element of bool NAT
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom x1 is countable Element of bool NAT
Ix1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom Ix1 is countable Element of bool NAT
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom y1 is countable Element of bool NAT
dom J is non empty countable Element of bool NAT
v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom v1 is countable Element of bool NAT
Iy1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom Iy1 is countable Element of bool NAT
r . (y1,xx2) is set
[y1,xx2] is set
r . [y1,xx2] is set
r . (yy2,I) is set
[yy2,I] is set
r . [yy2,I] is set
v ^ y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
x1 ^ v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
r . ((y1 + yy2),(xx2 + I)) is set
[(y1 + yy2),(xx2 + I)] is set
r . [(y1 + yy2),(xx2 + I)] is set
Ix1 ^ Iy1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom (Ix1 ^ Iy1) is countable Element of bool NAT
Iy is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom Iy is countable Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
Iy . k is set
(Ix1 ^ Iy1) . k is set
k1 is Element of dom (carr (X ^ Y))
(carr (X ^ Y)) . k1 is non empty set
(addop (X ^ Y)) . k1 is non empty Relation-like [:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):] -defined (carr (X ^ Y)) . k1 -valued Function-like V26([:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):]) quasi_total Element of bool [:[:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):],((carr (X ^ Y)) . k1):]
[:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):] is non empty Relation-like set
[:[:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):],((carr (X ^ Y)) . k1):] is non empty Relation-like set
bool [:[:((carr (X ^ Y)) . k1),((carr (X ^ Y)) . k1):],((carr (X ^ Y)) . k1):] is non empty set
Iv is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
Iv . k1 is Element of (carr (X ^ Y)) . k1
Ix is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
Ix . k1 is Element of (carr (X ^ Y)) . k1
((addop (X ^ Y)) . k1) . ((Iv . k1),(Ix . k1)) is Element of (carr (X ^ Y)) . k1
[(Iv . k1),(Ix . k1)] is set
((addop (X ^ Y)) . k1) . [(Iv . k1),(Ix . k1)] is set
(X ^ Y) . k1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the addF of ((X ^ Y) . k1) is non empty Relation-like [: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):] -defined the carrier of ((X ^ Y) . k1) -valued Function-like V26([: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):]) quasi_total Element of bool [:[: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):], the carrier of ((X ^ Y) . k1):]
the carrier of ((X ^ Y) . k1) is non empty set
[: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):] is non empty Relation-like set
[:[: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):], the carrier of ((X ^ Y) . k1):] is non empty Relation-like set
bool [:[: the carrier of ((X ^ Y) . k1), the carrier of ((X ^ Y) . k1):], the carrier of ((X ^ Y) . k1):] is non empty set
the addF of ((X ^ Y) . k1) . ((Iv . k1),(Ix . k1)) is set
the addF of ((X ^ Y) . k1) . [(Iv . k1),(Ix . k1)] is set
dom X is non empty countable Element of bool NAT
X . k is set
v . k1 is set
x1 . k1 is set
Ix1 . k is set
dom (carr X) is non empty countable Element of bool NAT
n is Element of dom (carr X)
(addop X) . n is non empty Relation-like [:((carr X) . n),((carr X) . n):] -defined (carr X) . n -valued Function-like V26([:((carr X) . n),((carr X) . n):]) quasi_total Element of bool [:[:((carr X) . n),((carr X) . n):],((carr X) . n):]
(carr X) . n is non empty set
[:((carr X) . n),((carr X) . n):] is non empty Relation-like set
[:[:((carr X) . n),((carr X) . n):],((carr X) . n):] is non empty Relation-like set
bool [:[:((carr X) . n),((carr X) . n):],((carr X) . n):] is non empty set
v . n is set
x1 . n is set
((addop X) . n) . ((v . n),(x1 . n)) is set
[(v . n),(x1 . n)] is set
((addop X) . n) . [(v . n),(x1 . n)] is set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len I) + n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len I) + n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len Ix1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom Y is non empty countable Element of bool NAT
Y . n is set
y1 . n is set
v1 . n is set
Iy1 . n is set
dom (carr Y) is non empty countable Element of bool NAT
n1 is Element of dom (carr Y)
(addop Y) . n1 is non empty Relation-like [:((carr Y) . n1),((carr Y) . n1):] -defined (carr Y) . n1 -valued Function-like V26([:((carr Y) . n1),((carr Y) . n1):]) quasi_total Element of bool [:[:((carr Y) . n1),((carr Y) . n1):],((carr Y) . n1):]
(carr Y) . n1 is non empty set
[:((carr Y) . n1),((carr Y) . n1):] is non empty Relation-like set
[:[:((carr Y) . n1),((carr Y) . n1):],((carr Y) . n1):] is non empty Relation-like set
bool [:[:((carr Y) . n1),((carr Y) . n1):],((carr Y) . n1):] is non empty set
y1 . n1 is set
v1 . n1 is set
((addop Y) . n1) . ((y1 . n1),(v1 . n1)) is set
[(y1 . n1),(v1 . n1)] is set
((addop Y) . n1) . [(y1 . n1),(v1 . n1)] is set
yy is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
r . yy is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
x1 is V11() real ext-real Element of REAL
x1 * yy is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the Mult of ((product X),(product Y)) is non empty Relation-like [:REAL, the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([:REAL, the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[:REAL, the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the Mult of ((product X),(product Y)) . (x1,yy) is set
[x1,yy] is set
the Mult of ((product X),(product Y)) . [x1,yy] is set
r . (x1 * yy) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
x1 * (r . yy) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the Mult of (product (X ^ Y)) is non empty Relation-like [:REAL, the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([:REAL, the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[:REAL, the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the Mult of (product (X ^ Y)) . (x1,(r . yy)) is set
[x1,(r . yy)] is set
the Mult of (product (X ^ Y)) . [x1,(r . yy)] is set
y1 is left_complementable right_complementable complementable Element of the carrier of (product X)
xx2 is left_complementable right_complementable complementable Element of the carrier of (product Y)
[y1,xx2] is Element of [: the carrier of (product X), the carrier of (product Y):]
x1 * y1 is left_complementable right_complementable complementable Element of the carrier of (product X)
the Mult of (product X) is non empty Relation-like [:REAL, the carrier of (product X):] -defined the carrier of (product X) -valued Function-like V26([:REAL, the carrier of (product X):]) quasi_total Element of bool [:[:REAL, the carrier of (product X):], the carrier of (product X):]
[:REAL, the carrier of (product X):] is non empty Relation-like set
[:[:REAL, the carrier of (product X):], the carrier of (product X):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product X):], the carrier of (product X):] is non empty set
the Mult of (product X) . (x1,y1) is set
[x1,y1] is set
the Mult of (product X) . [x1,y1] is set
x1 * xx2 is left_complementable right_complementable complementable Element of the carrier of (product Y)
the Mult of (product Y) is non empty Relation-like [:REAL, the carrier of (product Y):] -defined the carrier of (product Y) -valued Function-like V26([:REAL, the carrier of (product Y):]) quasi_total Element of bool [:[:REAL, the carrier of (product Y):], the carrier of (product Y):]
[:REAL, the carrier of (product Y):] is non empty Relation-like set
[:[:REAL, the carrier of (product Y):], the carrier of (product Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product Y):], the carrier of (product Y):] is non empty set
the Mult of (product Y) . (x1,xx2) is set
[x1,xx2] is set
the Mult of (product Y) . [x1,xx2] is set
yy2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom yy2 is countable Element of bool NAT
dom I is non empty countable Element of bool NAT
I is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom I is countable Element of bool NAT
dom J is non empty countable Element of bool NAT
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom v is countable Element of bool NAT
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom x1 is countable Element of bool NAT
r . (y1,xx2) is set
[y1,xx2] is set
r . [y1,xx2] is set
yy2 ^ I is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
r . ((x1 * y1),(x1 * xx2)) is set
[(x1 * y1),(x1 * xx2)] is set
r . [(x1 * y1),(x1 * xx2)] is set
v ^ x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom v1 is countable Element of bool NAT
dom (v ^ x1) is countable Element of bool NAT
Ix1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
v1 . Ix1 is set
(v ^ x1) . Ix1 is set
Iy1 is Element of dom (carr (X ^ Y))
(carr (X ^ Y)) . Iy1 is non empty set
(multop (X ^ Y)) . Iy1 is non empty Relation-like [:REAL,((carr (X ^ Y)) . Iy1):] -defined (carr (X ^ Y)) . Iy1 -valued Function-like V26([:REAL,((carr (X ^ Y)) . Iy1):]) quasi_total Element of bool [:[:REAL,((carr (X ^ Y)) . Iy1):],((carr (X ^ Y)) . Iy1):]
[:REAL,((carr (X ^ Y)) . Iy1):] is non empty Relation-like set
[:[:REAL,((carr (X ^ Y)) . Iy1):],((carr (X ^ Y)) . Iy1):] is non empty Relation-like set
bool [:[:REAL,((carr (X ^ Y)) . Iy1):],((carr (X ^ Y)) . Iy1):] is non empty set
y1 is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
y1 . Iy1 is Element of (carr (X ^ Y)) . Iy1
((multop (X ^ Y)) . Iy1) . (x1,(y1 . Iy1)) is Element of (carr (X ^ Y)) . Iy1
[x1,(y1 . Iy1)] is set
((multop (X ^ Y)) . Iy1) . [x1,(y1 . Iy1)] is set
(X ^ Y) . Iy1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the Mult of ((X ^ Y) . Iy1) is non empty Relation-like [:REAL, the carrier of ((X ^ Y) . Iy1):] -defined the carrier of ((X ^ Y) . Iy1) -valued Function-like V26([:REAL, the carrier of ((X ^ Y) . Iy1):]) quasi_total Element of bool [:[:REAL, the carrier of ((X ^ Y) . Iy1):], the carrier of ((X ^ Y) . Iy1):]
the carrier of ((X ^ Y) . Iy1) is non empty set
[:REAL, the carrier of ((X ^ Y) . Iy1):] is non empty Relation-like set
[:[:REAL, the carrier of ((X ^ Y) . Iy1):], the carrier of ((X ^ Y) . Iy1):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((X ^ Y) . Iy1):], the carrier of ((X ^ Y) . Iy1):] is non empty set
the Mult of ((X ^ Y) . Iy1) . (x1,(y1 . Iy1)) is set
the Mult of ((X ^ Y) . Iy1) . [x1,(y1 . Iy1)] is set
dom X is non empty countable Element of bool NAT
X . Ix1 is set
yy2 . Iy1 is set
v . Ix1 is set
dom (carr X) is non empty countable Element of bool NAT
Iv is Element of dom (carr X)
(multop X) . Iv is non empty Relation-like [:REAL,((carr X) . Iv):] -defined (carr X) . Iv -valued Function-like V26([:REAL,((carr X) . Iv):]) quasi_total Element of bool [:[:REAL,((carr X) . Iv):],((carr X) . Iv):]
(carr X) . Iv is non empty set
[:REAL,((carr X) . Iv):] is non empty Relation-like set
[:[:REAL,((carr X) . Iv):],((carr X) . Iv):] is non empty Relation-like set
bool [:[:REAL,((carr X) . Iv):],((carr X) . Iv):] is non empty set
yy2 . Iv is set
((multop X) . Iv) . (x1,(yy2 . Iv)) is set
[x1,(yy2 . Iv)] is set
((multop X) . Iv) . [x1,(yy2 . Iv)] is set
Iv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len I) + Iv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Iv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len I) + Iv is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len yy2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom Y is non empty countable Element of bool NAT
Y . Iv is set
I . Iv is set
x1 . Iv is set
dom (carr Y) is non empty countable Element of bool NAT
Ix is Element of dom (carr Y)
(multop Y) . Ix is non empty Relation-like [:REAL,((carr Y) . Ix):] -defined (carr Y) . Ix -valued Function-like V26([:REAL,((carr Y) . Ix):]) quasi_total Element of bool [:[:REAL,((carr Y) . Ix):],((carr Y) . Ix):]
(carr Y) . Ix is non empty set
[:REAL,((carr Y) . Ix):] is non empty Relation-like set
[:[:REAL,((carr Y) . Ix):],((carr Y) . Ix):] is non empty Relation-like set
bool [:[:REAL,((carr Y) . Ix):],((carr Y) . Ix):] is non empty set
I . Ix is set
((multop Y) . Ix) . (x1,(I . Ix)) is set
[x1,(I . Ix)] is set
((multop Y) . Ix) . [x1,(I . Ix)] is set
r . (0. ((product X),(product Y))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(0. ((product X),(product Y))) + (0. ((product X),(product Y))) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the addF of ((product X),(product Y)) is non empty Relation-like [: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the addF of ((product X),(product Y)) . ((0. ((product X),(product Y))),(0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
[(0. ((product X),(product Y))),(0. ((product X),(product Y)))] is set
the addF of ((product X),(product Y)) . [(0. ((product X),(product Y))),(0. ((product X),(product Y)))] is set
r . ((0. ((product X),(product Y))) + (0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(r . (0. ((product X),(product Y)))) + (r . (0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) is non empty Relation-like [: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the addF of (product (X ^ Y)) . ((r . (0. ((product X),(product Y)))),(r . (0. ((product X),(product Y))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(r . (0. ((product X),(product Y)))),(r . (0. ((product X),(product Y))))] is set
the addF of (product (X ^ Y)) . [(r . (0. ((product X),(product Y)))),(r . (0. ((product X),(product Y))))] is set
(r . (0. ((product X),(product Y)))) - (r . (0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
- (r . (0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(r . (0. ((product X),(product Y)))) + (- (r . (0. ((product X),(product Y))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) . ((r . (0. ((product X),(product Y)))),(- (r . (0. ((product X),(product Y)))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(r . (0. ((product X),(product Y)))),(- (r . (0. ((product X),(product Y)))))] is set
the addF of (product (X ^ Y)) . [(r . (0. ((product X),(product Y)))),(- (r . (0. ((product X),(product Y)))))] is set
(r . (0. ((product X),(product Y)))) + ((r . (0. ((product X),(product Y)))) - (r . (0. ((product X),(product Y))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) . ((r . (0. ((product X),(product Y)))),((r . (0. ((product X),(product Y)))) - (r . (0. ((product X),(product Y)))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(r . (0. ((product X),(product Y)))),((r . (0. ((product X),(product Y)))) - (r . (0. ((product X),(product Y)))))] is set
the addF of (product (X ^ Y)) . [(r . (0. ((product X),(product Y)))),((r . (0. ((product X),(product Y)))) - (r . (0. ((product X),(product Y)))))] is set
(r . (0. ((product X),(product Y)))) + (0. (product (X ^ Y))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) . ((r . (0. ((product X),(product Y)))),(0. (product (X ^ Y)))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(r . (0. ((product X),(product Y)))),(0. (product (X ^ Y)))] is set
the addF of (product (X ^ Y)) . [(r . (0. ((product X),(product Y)))),(0. (product (X ^ Y)))] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*X,Y*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*X,Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*X,Y*>) is non empty functional with_common_domain product-like set
zeros <*X,Y*> is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
addop <*X,Y*> is Relation-like Function-like BinOps of carr <*X,Y*>
[:(addop <*X,Y*>):] is non empty Relation-like [:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
multop <*X,Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*X,Y*>
[:(multop <*X,Y*>):] is non empty Relation-like [:REAL,(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:REAL,(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:REAL,(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
RLSStruct(# (product (carr <*X,Y*>)),(zeros <*X,Y*>),[:(addop <*X,Y*>):],[:(multop <*X,Y*>):] #) is non empty strict RLSStruct
the carrier of (product <*X,Y*>) is non empty set
the carrier of X is non empty set
the carrier of Y is non empty set
0. (product <*X,Y*>) is V52( product <*X,Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the ZeroF of (product <*X,Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
<*(0. X),(0. Y)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
(X,Y) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
[: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I is non empty Relation-like the carrier of (X,Y) -defined the carrier of (product <*X,Y*>) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):]
I . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J is set
rng I is non empty Element of bool the carrier of (product <*X,Y*>)
bool the carrier of (product <*X,Y*>) is non empty set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
I . (r,yy) is set
[r,yy] is set
I . [r,yy] is set
<*r,yy*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,K] is Element of [: the carrier of X, the carrier of Y:]
I . [K,K] is set
I . (K,K) is set
[K,K] is set
I . [K,K] is set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J + K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . (J,K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[J,K] is set
the addF of (product <*X,Y*>) . [J,K] is set
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of X
K + v is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,v) is left_complementable right_complementable complementable Element of the carrier of X
[K,v] is set
the addF of X . [K,v] is set
r is left_complementable right_complementable complementable Element of the carrier of Y
<*K,r*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
yy is left_complementable right_complementable complementable Element of the carrier of Y
<*v,yy*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
r + yy is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (r,yy) is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is set
the addF of Y . [r,yy] is set
<*(K + v),(r + yy)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,r] is Element of [: the carrier of X, the carrier of Y:]
[v,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
y1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
x1 + y1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (x1,y1) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[x1,y1] is set
the addF of (X,Y) . [x1,y1] is set
[(K + v),(r + yy)] is Element of [: the carrier of X, the carrier of Y:]
I . ((K + v),(r + yy)) is set
[(K + v),(r + yy)] is set
I . [(K + v),(r + yy)] is set
I . (K,r) is set
[K,r] is set
I . [K,r] is set
I . (v,yy) is set
[v,yy] is set
I . [v,yy] is set
I . ((0. X),(0. Y)) is set
[(0. X),(0. Y)] is set
I . [(0. X),(0. Y)] is set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
- K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
- K is left_complementable right_complementable complementable Element of the carrier of Y
<*(- K),(- K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J + v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . (J,v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[J,v] is set
the addF of (product <*X,Y*>) . [J,v] is set
K + (- K) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,(- K)) is left_complementable right_complementable complementable Element of the carrier of X
[K,(- K)] is set
the addF of X . [K,(- K)] is set
K + (- K) is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(- K)) is left_complementable right_complementable complementable Element of the carrier of Y
[K,(- K)] is set
the addF of Y . [K,(- K)] is set
<*(K + (- K)),(K + (- K))*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*(0. X),(K + (- K))*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is V11() real ext-real set
v * J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the Mult of (product <*X,Y*>) is non empty Relation-like [:REAL, the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([:REAL, the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[:REAL, the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the Mult of (product <*X,Y*>) . (v,J) is set
[v,J] is set
the Mult of (product <*X,Y*>) . [v,J] is set
v * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (v,K) is set
[v,K] is set
the Mult of X . [v,K] is set
v * K is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (v,K) is set
[v,K] is set
the Mult of Y . [v,K] is set
<*(v * K),(v * K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,K] is Element of [: the carrier of X, the carrier of Y:]
I . (K,K) is set
[K,K] is set
I . [K,K] is set
yy is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v * yy is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (v,yy) is set
[v,yy] is set
the Mult of (X,Y) . [v,yy] is set
I . (v * yy) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r is V11() real ext-real Element of REAL
r * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (r,K) is set
[r,K] is set
the Mult of X . [r,K] is set
r * K is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y . (r,K) is set
[r,K] is set
the Mult of Y . [r,K] is set
I . ((r * K),(r * K)) is set
[(r * K),(r * K)] is set
I . [(r * K),(r * K)] is set
<*(r * K),(r * K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
X is non empty NORMSTR
the carrier of X is non empty set
Y is non empty NORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
REAL 2 is non empty functional FinSequence-membered M10( REAL )
K361(2,REAL) is functional FinSequence-membered M10( REAL )
I is set
J is set
K is Element of the carrier of X
||.K.|| is V11() real ext-real Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . K is V11() real ext-real Element of REAL
K is Element of the carrier of Y
||.K.|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.K.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
v is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.v.| is V11() real ext-real non negative Element of REAL
sqr v is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr v) is V11() real ext-real Element of REAL
sqrt (Sum (sqr v)) is V11() real ext-real Element of REAL
I is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
J is Element of the carrier of X
K is Element of the carrier of Y
I . (J,K) is V11() real ext-real Element of REAL
[J,K] is set
I . [J,K] is V11() real ext-real set
||.J.|| is V11() real ext-real Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . J is V11() real ext-real Element of REAL
||.K.|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.J.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
K is Element of the carrier of X
v is Element of the carrier of Y
r is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
||.K.|| is V11() real ext-real Element of REAL
the normF of X . K is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real Element of REAL
the normF of Y . v is V11() real ext-real Element of REAL
<*||.K.||,||.v.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
|.r.| is V11() real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr r) is V11() real ext-real Element of REAL
sqrt (Sum (sqr r)) is V11() real ext-real Element of REAL
I is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
J is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
K is Element of the carrier of X
||.K.|| is V11() real ext-real Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . K is V11() real ext-real Element of REAL
K is Element of the carrier of Y
||.K.|| is V11() real ext-real Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.K.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
I . (K,K) is V11() real ext-real Element of REAL
[K,K] is set
I . [K,K] is V11() real ext-real set
J . (K,K) is V11() real ext-real Element of REAL
J . [K,K] is V11() real ext-real set
v is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.v.| is V11() real ext-real non negative Element of REAL
sqr v is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr v) is V11() real ext-real Element of REAL
sqrt (Sum (sqr v)) is V11() real ext-real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.r.| is V11() real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr r) is V11() real ext-real Element of REAL
sqrt (Sum (sqr r)) is V11() real ext-real Element of REAL
X is non empty NORMSTR
the carrier of X is non empty set
Y is non empty NORMSTR
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) Element of the carrier of X
the ZeroF of X is Element of the carrier of X
0. Y is V52(Y) Element of the carrier of Y
the ZeroF of Y is Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty strict NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
I is Element of the carrier of (X,Y)
J is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
||.J.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . J is V11() real ext-real Element of REAL
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.J.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (J,K) is V11() real ext-real Element of REAL
[J,K] is set
(X,Y) . [J,K] is V11() real ext-real set
K is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.K.| is V11() real ext-real non negative Element of REAL
sqr K is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr K) is V11() real ext-real Element of REAL
sqrt (Sum (sqr K)) is V11() real ext-real Element of REAL
0. (X,Y) is V52((X,Y)) Element of the carrier of (X,Y)
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
0* 2 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
K362(REAL,2,0) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of K361(2,REAL)
||.I.|| is V11() real ext-real Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . I is V11() real ext-real Element of REAL
||.(0. (X,Y)).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (0. (X,Y)) is V11() real ext-real Element of REAL
I is Element of the carrier of (X,Y)
J is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
||.J.|| is V11() real ext-real non negative Element of REAL
the normF of X . J is V11() real ext-real Element of REAL
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of Y . K is V11() real ext-real Element of REAL
<*||.J.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (J,K) is V11() real ext-real Element of REAL
[J,K] is set
(X,Y) . [J,K] is V11() real ext-real set
K is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.K.| is V11() real ext-real non negative Element of REAL
sqr K is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr K) is V11() real ext-real Element of REAL
sqrt (Sum (sqr K)) is V11() real ext-real Element of REAL
||.I.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . I is V11() real ext-real Element of REAL
<*0,0*> is non empty Relation-like NAT -defined NAT -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() V142() countable FinSequence of NAT
K . 1 is V11() real ext-real set
K . 2 is V11() real ext-real set
I is Element of the carrier of (X,Y)
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
J is Element of the carrier of (X,Y)
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of X . K is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of Y . v is V11() real ext-real Element of REAL
<*||.K.||,||.v.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (K,v) is V11() real ext-real Element of REAL
[K,v] is set
(X,Y) . [K,v] is V11() real ext-real set
x1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.x1.| is V11() real ext-real non negative Element of REAL
sqr x1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr x1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr x1)) is V11() real ext-real Element of REAL
||.r.|| is V11() real ext-real non negative Element of REAL
the normF of X . r is V11() real ext-real Element of REAL
||.yy.|| is V11() real ext-real non negative Element of REAL
the normF of Y . yy is V11() real ext-real Element of REAL
<*||.r.||,||.yy.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (r,yy) is V11() real ext-real Element of REAL
[r,yy] is set
(X,Y) . [r,yy] is V11() real ext-real set
y1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.y1.| is V11() real ext-real non negative Element of REAL
sqr y1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr y1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr y1)) is V11() real ext-real Element of REAL
K is V11() real ext-real Element of REAL
K * I is Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (K,I) is set
[K,I] is set
the Mult of (X,Y) . [K,I] is set
||.(K * I).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (K * I) is V11() real ext-real Element of REAL
abs K is V11() real ext-real Element of REAL
||.I.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . I is V11() real ext-real Element of REAL
(abs K) * ||.I.|| is V11() real ext-real Element of REAL
K * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,K) is set
[K,K] is set
the Mult of X . [K,K] is set
||.(K * K).|| is V11() real ext-real non negative Element of REAL
the normF of X . (K * K) is V11() real ext-real Element of REAL
K * v is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,v) is set
[K,v] is set
the Mult of Y . [K,v] is set
||.(K * v).|| is V11() real ext-real non negative Element of REAL
the normF of Y . (K * v) is V11() real ext-real Element of REAL
<*||.(K * K).||,||.(K * v).||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . ((K * K),(K * v)) is V11() real ext-real Element of REAL
[(K * K),(K * v)] is set
(X,Y) . [(K * K),(K * v)] is V11() real ext-real set
xx2 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.xx2.| is V11() real ext-real non negative Element of REAL
sqr xx2 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr xx2) is V11() real ext-real Element of REAL
sqrt (Sum (sqr xx2)) is V11() real ext-real Element of REAL
(abs K) * ||.K.|| is V11() real ext-real Element of REAL
(abs K) * ||.v.|| is V11() real ext-real Element of REAL
I is V11() real ext-real set
v is V11() real ext-real set
|[I,v]| is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like left_complementable right_complementable complementable complex-yielding V140() V141() countable Element of the carrier of (TOP-REAL 2)
yy2 is V11() real ext-real set
yy2 * |[I,v]| is V40(2) FinSequence-like left_complementable right_complementable complementable V141() Element of the carrier of (TOP-REAL 2)
the Mult of (TOP-REAL 2) is non empty Relation-like [:REAL, the carrier of (TOP-REAL 2):] -defined the carrier of (TOP-REAL 2) -valued Function-like V26([:REAL, the carrier of (TOP-REAL 2):]) quasi_total Element of bool [:[:REAL, the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):]
[:REAL, the carrier of (TOP-REAL 2):] is non empty Relation-like set
[:[:REAL, the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):] is non empty Relation-like set
bool [:[:REAL, the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):] is non empty set
the Mult of (TOP-REAL 2) . (yy2,|[I,v]|) is set
[yy2,|[I,v]|] is set
the Mult of (TOP-REAL 2) . [yy2,|[I,v]|] is set
(abs K) * x1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
abs (abs K) is V11() real ext-real Element of REAL
(abs (abs K)) * |.x1.| is V11() real ext-real Element of REAL
(abs K) * |.x1.| is V11() real ext-real Element of REAL
I + J is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
||.(I + J).|| is V11() real ext-real Element of REAL
the normF of (X,Y) . (I + J) is V11() real ext-real Element of REAL
||.J.|| is V11() real ext-real Element of REAL
the normF of (X,Y) . J is V11() real ext-real Element of REAL
||.I.|| + ||.J.|| is V11() real ext-real Element of REAL
K + r is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,r) is left_complementable right_complementable complementable Element of the carrier of X
[K,r] is set
the addF of X . [K,r] is set
||.(K + r).|| is V11() real ext-real non negative Element of REAL
the normF of X . (K + r) is V11() real ext-real Element of REAL
v + yy is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (v,yy) is left_complementable right_complementable complementable Element of the carrier of Y
[v,yy] is set
the addF of Y . [v,yy] is set
||.(v + yy).|| is V11() real ext-real non negative Element of REAL
the normF of Y . (v + yy) is V11() real ext-real Element of REAL
<*||.(K + r).||,||.(v + yy).||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . ((K + r),(v + yy)) is V11() real ext-real Element of REAL
[(K + r),(v + yy)] is set
(X,Y) . [(K + r),(v + yy)] is V11() real ext-real set
xx2 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.xx2.| is V11() real ext-real non negative Element of REAL
sqr xx2 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr xx2) is V11() real ext-real Element of REAL
sqrt (Sum (sqr xx2)) is V11() real ext-real Element of REAL
||.K.|| + ||.r.|| is V11() real ext-real non negative Element of REAL
||.v.|| + ||.yy.|| is V11() real ext-real non negative Element of REAL
<*(||.K.|| + ||.r.||),(||.v.|| + ||.yy.||)*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
len xx2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
I is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
len I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len xx2) is V33() V40( len xx2) countable Element of bool NAT
xx2 . v is V11() real ext-real set
I . v is V11() real ext-real set
xx2 . v is V11() real ext-real set
I . v is V11() real ext-real set
|.I.| is V11() real ext-real non negative Element of REAL
sqr I is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr I) is V11() real ext-real Element of REAL
sqrt (Sum (sqr I)) is V11() real ext-real Element of REAL
|[||.K.||,||.v.||]| is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like left_complementable right_complementable complementable complex-yielding V140() V141() countable Element of the carrier of (TOP-REAL 2)
|[||.r.||,||.yy.||]| is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like left_complementable right_complementable complementable complex-yielding V140() V141() countable Element of the carrier of (TOP-REAL 2)
|[||.K.||,||.v.||]| + |[||.r.||,||.yy.||]| is V40(2) FinSequence-like left_complementable right_complementable complementable V141() Element of the carrier of (TOP-REAL 2)
the addF of (TOP-REAL 2) is non empty Relation-like [: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):] -defined the carrier of (TOP-REAL 2) -valued Function-like V26([: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):]) quasi_total Element of bool [:[: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):]
[: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):] is non empty Relation-like set
[:[: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):] is non empty Relation-like set
bool [:[: the carrier of (TOP-REAL 2), the carrier of (TOP-REAL 2):], the carrier of (TOP-REAL 2):] is non empty set
the addF of (TOP-REAL 2) . (|[||.K.||,||.v.||]|,|[||.r.||,||.yy.||]|) is V40(2) FinSequence-like left_complementable right_complementable complementable V141() Element of the carrier of (TOP-REAL 2)
[|[||.K.||,||.v.||]|,|[||.r.||,||.yy.||]|] is set
the addF of (TOP-REAL 2) . [|[||.K.||,||.v.||]|,|[||.r.||,||.yy.||]|] is set
x1 + y1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.x1.| + |.y1.| is V11() real ext-real non negative Element of REAL
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
the ZeroF of (X,Y) is Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
RLSStruct(# the carrier of (X,Y), the ZeroF of (X,Y), the addF of (X,Y), the Mult of (X,Y) #) is non empty strict RLSStruct
I is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
J is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
(I,J) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of I is non empty set
the carrier of J is non empty set
[: the carrier of I, the carrier of J:] is non empty Relation-like set
(I,J) is Element of [: the carrier of I, the carrier of J:]
0. I is V52(I) left_complementable right_complementable complementable Element of the carrier of I
the ZeroF of I is left_complementable right_complementable complementable Element of the carrier of I
0. J is V52(J) left_complementable right_complementable complementable Element of the carrier of J
the ZeroF of J is left_complementable right_complementable complementable Element of the carrier of J
[(0. I),(0. J)] is Element of [: the carrier of I, the carrier of J:]
(I,J) is non empty Relation-like [:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:] -defined [: the carrier of I, the carrier of J:] -valued Function-like V26([:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:]) quasi_total Element of bool [:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:]
[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
[:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
bool [:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty set
(I,J) is non empty Relation-like [:REAL,[: the carrier of I, the carrier of J:]:] -defined [: the carrier of I, the carrier of J:] -valued Function-like V26([:REAL,[: the carrier of I, the carrier of J:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:]
[:REAL,[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty set
RLSStruct(# [: the carrier of I, the carrier of J:],(I,J),(I,J),(I,J) #) is non empty strict RLSStruct
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
<*X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
Y is set
rng <*X*> is non empty set
dom <*X*> is non empty countable Element of bool NAT
I is set
<*X*> . I is set
len <*X*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
I is set
rng <*X,Y*> is non empty set
dom <*X,Y*> is non empty countable Element of bool NAT
J is set
<*X,Y*> . J is set
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product <*X,Y*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*X,Y*>) is non empty set
[: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty set
0. (product <*X,Y*>) is V52( product <*X,Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the ZeroF of (product <*X,Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
J is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
(I,J) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of I is non empty set
the carrier of J is non empty set
[: the carrier of I, the carrier of J:] is non empty Relation-like set
(I,J) is Element of [: the carrier of I, the carrier of J:]
0. I is V52(I) left_complementable right_complementable complementable Element of the carrier of I
the ZeroF of I is left_complementable right_complementable complementable Element of the carrier of I
0. J is V52(J) left_complementable right_complementable complementable Element of the carrier of J
the ZeroF of J is left_complementable right_complementable complementable Element of the carrier of J
[(0. I),(0. J)] is Element of [: the carrier of I, the carrier of J:]
(I,J) is non empty Relation-like [:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:] -defined [: the carrier of I, the carrier of J:] -valued Function-like V26([:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:]) quasi_total Element of bool [:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:]
[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
[:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
bool [:[:[: the carrier of I, the carrier of J:],[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty set
(I,J) is non empty Relation-like [:REAL,[: the carrier of I, the carrier of J:]:] -defined [: the carrier of I, the carrier of J:] -valued Function-like V26([:REAL,[: the carrier of I, the carrier of J:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:]
[:REAL,[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of I, the carrier of J:]:],[: the carrier of I, the carrier of J:]:] is non empty set
RLSStruct(# [: the carrier of I, the carrier of J:],(I,J),(I,J),(I,J) #) is non empty strict RLSStruct
the carrier of (I,J) is non empty set
<*I,J*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*I,J*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*I,J*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*I,J*>) is non empty functional with_common_domain product-like set
zeros <*I,J*> is Relation-like NAT -defined Function-like carr <*I,J*> -compatible Element of product (carr <*I,J*>)
addop <*I,J*> is Relation-like Function-like BinOps of carr <*I,J*>
[:(addop <*I,J*>):] is non empty Relation-like [:(product (carr <*I,J*>)),(product (carr <*I,J*>)):] -defined product (carr <*I,J*>) -valued Function-like V26([:(product (carr <*I,J*>)),(product (carr <*I,J*>)):]) quasi_total Element of bool [:[:(product (carr <*I,J*>)),(product (carr <*I,J*>)):],(product (carr <*I,J*>)):]
[:(product (carr <*I,J*>)),(product (carr <*I,J*>)):] is non empty Relation-like set
[:[:(product (carr <*I,J*>)),(product (carr <*I,J*>)):],(product (carr <*I,J*>)):] is non empty Relation-like set
bool [:[:(product (carr <*I,J*>)),(product (carr <*I,J*>)):],(product (carr <*I,J*>)):] is non empty set
multop <*I,J*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*I,J*>
[:(multop <*I,J*>):] is non empty Relation-like [:REAL,(product (carr <*I,J*>)):] -defined product (carr <*I,J*>) -valued Function-like V26([:REAL,(product (carr <*I,J*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*I,J*>)):],(product (carr <*I,J*>)):]
[:REAL,(product (carr <*I,J*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*I,J*>)):],(product (carr <*I,J*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*I,J*>)):],(product (carr <*I,J*>)):] is non empty set
RLSStruct(# (product (carr <*I,J*>)),(zeros <*I,J*>),[:(addop <*I,J*>):],[:(multop <*I,J*>):] #) is non empty strict RLSStruct
the carrier of (product <*I,J*>) is non empty set
[: the carrier of (I,J), the carrier of (product <*I,J*>):] is non empty Relation-like set
bool [: the carrier of (I,J), the carrier of (product <*I,J*>):] is non empty set
0. (product <*I,J*>) is V52( product <*I,J*>) left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
the ZeroF of (product <*I,J*>) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
0. (I,J) is V52((I,J)) left_complementable right_complementable complementable Element of the carrier of (I,J)
the ZeroF of (I,J) is left_complementable right_complementable complementable Element of the carrier of (I,J)
K is non empty Relation-like the carrier of (I,J) -defined the carrier of (product <*I,J*>) -valued Function-like V26( the carrier of (I,J)) quasi_total Element of bool [: the carrier of (I,J), the carrier of (product <*I,J*>):]
K . (0. (I,J)) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
carr <*X,Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*X,Y*>) is non empty functional with_common_domain product-like set
zeros <*X,Y*> is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
addop <*X,Y*> is Relation-like Function-like BinOps of carr <*X,Y*>
[:(addop <*X,Y*>):] is non empty Relation-like [:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*X,Y*>)),(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
multop <*X,Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*X,Y*>
[:(multop <*X,Y*>):] is non empty Relation-like [:REAL,(product (carr <*X,Y*>)):] -defined product (carr <*X,Y*>) -valued Function-like V26([:REAL,(product (carr <*X,Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):]
[:REAL,(product (carr <*X,Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*X,Y*>)):],(product (carr <*X,Y*>)):] is non empty set
productnorm <*X,Y*> is non empty Relation-like product (carr <*X,Y*>) -defined REAL -valued Function-like V26( product (carr <*X,Y*>)) quasi_total complex-yielding V140() V141() Element of bool [:(product (carr <*X,Y*>)),REAL:]
[:(product (carr <*X,Y*>)),REAL:] is non empty Relation-like set
bool [:(product (carr <*X,Y*>)),REAL:] is non empty set
NORMSTR(# (product (carr <*X,Y*>)),(zeros <*X,Y*>),[:(addop <*X,Y*>):],[:(multop <*X,Y*>):],(productnorm <*X,Y*>) #) is strict NORMSTR
K is non empty Relation-like the carrier of (X,Y) -defined the carrier of (product <*X,Y*>) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):]
K . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
v is left_complementable right_complementable complementable Element of the carrier of X
r is left_complementable right_complementable complementable Element of the carrier of Y
K . (v,r) is set
[v,r] is set
K . [v,r] is set
<*v,r*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
r is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v + r is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (v,r) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[v,r] is set
the addF of (X,Y) . [v,r] is set
K . (v + r) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K . r is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . v) + (K . r) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . ((K . v),(K . r)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . v),(K . r)] is set
the addF of (product <*X,Y*>) . [(K . v),(K . r)] is set
yy is left_complementable right_complementable complementable Element of the carrier of (I,J)
x1 is left_complementable right_complementable complementable Element of the carrier of (I,J)
yy + x1 is left_complementable right_complementable complementable Element of the carrier of (I,J)
the addF of (I,J) is non empty Relation-like [: the carrier of (I,J), the carrier of (I,J):] -defined the carrier of (I,J) -valued Function-like V26([: the carrier of (I,J), the carrier of (I,J):]) quasi_total Element of bool [:[: the carrier of (I,J), the carrier of (I,J):], the carrier of (I,J):]
[: the carrier of (I,J), the carrier of (I,J):] is non empty Relation-like set
[:[: the carrier of (I,J), the carrier of (I,J):], the carrier of (I,J):] is non empty Relation-like set
bool [:[: the carrier of (I,J), the carrier of (I,J):], the carrier of (I,J):] is non empty set
the addF of (I,J) . (yy,x1) is left_complementable right_complementable complementable Element of the carrier of (I,J)
[yy,x1] is set
the addF of (I,J) . [yy,x1] is set
K . (yy + x1) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
K . yy is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
K . x1 is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
(K . yy) + (K . x1) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
the addF of (product <*I,J*>) is non empty Relation-like [: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):] -defined the carrier of (product <*I,J*>) -valued Function-like V26([: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):]) quasi_total Element of bool [:[: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):]
[: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):] is non empty Relation-like set
[:[: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*I,J*>), the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):] is non empty set
the addF of (product <*I,J*>) . ((K . yy),(K . x1)) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
[(K . yy),(K . x1)] is set
the addF of (product <*I,J*>) . [(K . yy),(K . x1)] is set
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r is V11() real ext-real Element of REAL
r * v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (r,v) is set
[r,v] is set
the Mult of (X,Y) . [r,v] is set
K . (r * v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r * (K . v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the Mult of (product <*X,Y*>) is non empty Relation-like [:REAL, the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([:REAL, the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[:REAL, the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the Mult of (product <*X,Y*>) . (r,(K . v)) is set
[r,(K . v)] is set
the Mult of (product <*X,Y*>) . [r,(K . v)] is set
yy is left_complementable right_complementable complementable Element of the carrier of (I,J)
r * yy is left_complementable right_complementable complementable Element of the carrier of (I,J)
the Mult of (I,J) is non empty Relation-like [:REAL, the carrier of (I,J):] -defined the carrier of (I,J) -valued Function-like V26([:REAL, the carrier of (I,J):]) quasi_total Element of bool [:[:REAL, the carrier of (I,J):], the carrier of (I,J):]
[:REAL, the carrier of (I,J):] is non empty Relation-like set
[:[:REAL, the carrier of (I,J):], the carrier of (I,J):] is non empty Relation-like set
bool [:[:REAL, the carrier of (I,J):], the carrier of (I,J):] is non empty set
the Mult of (I,J) . (r,yy) is set
[r,yy] is set
the Mult of (I,J) . [r,yy] is set
K . (r * yy) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
K . yy is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
r * (K . yy) is left_complementable right_complementable complementable Element of the carrier of (product <*I,J*>)
the Mult of (product <*I,J*>) is non empty Relation-like [:REAL, the carrier of (product <*I,J*>):] -defined the carrier of (product <*I,J*>) -valued Function-like V26([:REAL, the carrier of (product <*I,J*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):]
[:REAL, the carrier of (product <*I,J*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*I,J*>):], the carrier of (product <*I,J*>):] is non empty set
the Mult of (product <*I,J*>) . (r,(K . yy)) is set
[r,(K . yy)] is set
the Mult of (product <*I,J*>) . [r,(K . yy)] is set
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
||.(K . v).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) is non empty Relation-like the carrier of (product <*X,Y*>) -defined REAL -valued Function-like V26( the carrier of (product <*X,Y*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*X,Y*>),REAL:]
[: the carrier of (product <*X,Y*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*X,Y*>),REAL:] is non empty set
the normF of (product <*X,Y*>) . (K . v) is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . v is V11() real ext-real Element of REAL
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
||.r.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . r is V11() real ext-real Element of REAL
||.yy.|| is V11() real ext-real non negative Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . yy is V11() real ext-real Element of REAL
<*||.r.||,||.yy.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (r,yy) is V11() real ext-real Element of REAL
[r,yy] is set
(X,Y) . [r,yy] is V11() real ext-real set
x1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.x1.| is V11() real ext-real non negative Element of REAL
sqr x1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr x1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr x1)) is V11() real ext-real Element of REAL
K . (r,yy) is set
K . [r,yy] is set
<*r,yy*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*r,yy*> . 1 is set
<*r,yy*> . 2 is set
dom <*X,Y*> is non empty countable Element of bool NAT
y1 is Relation-like NAT -defined Function-like carr <*X,Y*> -compatible Element of product (carr <*X,Y*>)
normsequence (<*X,Y*>,y1) is Relation-like NAT -defined REAL -valued Function-like V33() V40( len <*X,Y*>) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL (len <*X,Y*>)
len <*X,Y*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
REAL (len <*X,Y*>) is non empty functional FinSequence-membered M10( REAL )
K361((len <*X,Y*>),REAL) is functional FinSequence-membered M10( REAL )
xx2 is Element of dom <*X,Y*>
(normsequence (<*X,Y*>,y1)) . xx2 is V11() real ext-real set
<*X,Y*> . xx2 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of (<*X,Y*> . xx2) is non empty Relation-like the carrier of (<*X,Y*> . xx2) -defined REAL -valued Function-like V26( the carrier of (<*X,Y*> . xx2)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (<*X,Y*> . xx2),REAL:]
the carrier of (<*X,Y*> . xx2) is non empty set
[: the carrier of (<*X,Y*> . xx2),REAL:] is non empty Relation-like set
bool [: the carrier of (<*X,Y*> . xx2),REAL:] is non empty set
y1 . xx2 is set
the normF of (<*X,Y*> . xx2) . (y1 . xx2) is V11() real ext-real set
yy2 is Element of dom <*X,Y*>
(normsequence (<*X,Y*>,y1)) . yy2 is V11() real ext-real set
<*X,Y*> . yy2 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of (<*X,Y*> . yy2) is non empty Relation-like the carrier of (<*X,Y*> . yy2) -defined REAL -valued Function-like V26( the carrier of (<*X,Y*> . yy2)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (<*X,Y*> . yy2),REAL:]
the carrier of (<*X,Y*> . yy2) is non empty set
[: the carrier of (<*X,Y*> . yy2),REAL:] is non empty Relation-like set
bool [: the carrier of (<*X,Y*> . yy2),REAL:] is non empty set
y1 . yy2 is set
the normF of (<*X,Y*> . yy2) . (y1 . yy2) is V11() real ext-real set
len (normsequence (<*X,Y*>,y1)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
||.(K . v).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) is non empty Relation-like the carrier of (product <*X,Y*>) -defined REAL -valued Function-like V26( the carrier of (product <*X,Y*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*X,Y*>),REAL:]
[: the carrier of (product <*X,Y*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*X,Y*>),REAL:] is non empty set
the normF of (product <*X,Y*>) . (K . v) is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . v is V11() real ext-real Element of REAL
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the carrier of X is non empty set
<*X*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product <*X*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*X*>) is non empty set
[: the carrier of X, the carrier of (product <*X*>):] is non empty Relation-like set
bool [: the carrier of X, the carrier of (product <*X*>):] is non empty set
0. (product <*X*>) is V52( product <*X*>) left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the ZeroF of (product <*X*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of Y is non empty set
<*Y*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*Y*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*Y*>) is non empty functional with_common_domain product-like set
zeros <*Y*> is Relation-like NAT -defined Function-like carr <*Y*> -compatible Element of product (carr <*Y*>)
addop <*Y*> is Relation-like Function-like BinOps of carr <*Y*>
[:(addop <*Y*>):] is non empty Relation-like [:(product (carr <*Y*>)),(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:(product (carr <*Y*>)),(product (carr <*Y*>)):]) quasi_total Element of bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:(product (carr <*Y*>)),(product (carr <*Y*>)):] is non empty Relation-like set
[:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
multop <*Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*Y*>
[:(multop <*Y*>):] is non empty Relation-like [:REAL,(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:REAL,(product (carr <*Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:REAL,(product (carr <*Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
RLSStruct(# (product (carr <*Y*>)),(zeros <*Y*>),[:(addop <*Y*>):],[:(multop <*Y*>):] #) is non empty strict RLSStruct
the carrier of (product <*Y*>) is non empty set
[: the carrier of Y, the carrier of (product <*Y*>):] is non empty Relation-like set
bool [: the carrier of Y, the carrier of (product <*Y*>):] is non empty set
0. (product <*Y*>) is V52( product <*Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the ZeroF of (product <*Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
I is non empty Relation-like the carrier of Y -defined the carrier of (product <*Y*>) -valued Function-like V26( the carrier of Y) quasi_total Element of bool [: the carrier of Y, the carrier of (product <*Y*>):]
I . (0. Y) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
carr <*X*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*X*>) is non empty functional with_common_domain product-like set
zeros <*X*> is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
addop <*X*> is Relation-like Function-like BinOps of carr <*X*>
[:(addop <*X*>):] is non empty Relation-like [:(product (carr <*X*>)),(product (carr <*X*>)):] -defined product (carr <*X*>) -valued Function-like V26([:(product (carr <*X*>)),(product (carr <*X*>)):]) quasi_total Element of bool [:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):]
[:(product (carr <*X*>)),(product (carr <*X*>)):] is non empty Relation-like set
[:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty Relation-like set
bool [:[:(product (carr <*X*>)),(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty set
multop <*X*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*X*>
[:(multop <*X*>):] is non empty Relation-like [:REAL,(product (carr <*X*>)):] -defined product (carr <*X*>) -valued Function-like V26([:REAL,(product (carr <*X*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):]
[:REAL,(product (carr <*X*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*X*>)):],(product (carr <*X*>)):] is non empty set
productnorm <*X*> is non empty Relation-like product (carr <*X*>) -defined REAL -valued Function-like V26( product (carr <*X*>)) quasi_total complex-yielding V140() V141() Element of bool [:(product (carr <*X*>)),REAL:]
[:(product (carr <*X*>)),REAL:] is non empty Relation-like set
bool [:(product (carr <*X*>)),REAL:] is non empty set
NORMSTR(# (product (carr <*X*>)),(zeros <*X*>),[:(addop <*X*>):],[:(multop <*X*>):],(productnorm <*X*>) #) is strict NORMSTR
J is non empty Relation-like the carrier of X -defined the carrier of (product <*X*>) -valued Function-like V26( the carrier of X) quasi_total Element of bool [: the carrier of X, the carrier of (product <*X*>):]
J . (0. X) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
K is left_complementable right_complementable complementable Element of the carrier of X
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of X
K + K is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,K) is left_complementable right_complementable complementable Element of the carrier of X
[K,K] is set
the addF of X . [K,K] is set
J . (K + K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
(J . K) + (J . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the addF of (product <*X*>) is non empty Relation-like [: the carrier of (product <*X*>), the carrier of (product <*X*>):] -defined the carrier of (product <*X*>) -valued Function-like V26([: the carrier of (product <*X*>), the carrier of (product <*X*>):]) quasi_total Element of bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):]
[: the carrier of (product <*X*>), the carrier of (product <*X*>):] is non empty Relation-like set
[:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X*>), the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty set
the addF of (product <*X*>) . ((J . K),(J . K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
[(J . K),(J . K)] is set
the addF of (product <*X*>) . [(J . K),(J . K)] is set
v is left_complementable right_complementable complementable Element of the carrier of Y
I . v is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
r is left_complementable right_complementable complementable Element of the carrier of Y
I . r is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
(I . v) + (I . r) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the addF of (product <*Y*>) is non empty Relation-like [: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([: the carrier of (product <*Y*>), the carrier of (product <*Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the addF of (product <*Y*>) . ((I . v),(I . r)) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[(I . v),(I . r)] is set
the addF of (product <*Y*>) . [(I . v),(I . r)] is set
K is left_complementable right_complementable complementable Element of the carrier of X
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
K is V11() real ext-real Element of REAL
K * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,K) is set
[K,K] is set
the Mult of X . [K,K] is set
J . (K * K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
K * (J . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
the Mult of (product <*X*>) is non empty Relation-like [:REAL, the carrier of (product <*X*>):] -defined the carrier of (product <*X*>) -valued Function-like V26([:REAL, the carrier of (product <*X*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):]
[:REAL, the carrier of (product <*X*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X*>):], the carrier of (product <*X*>):] is non empty set
the Mult of (product <*X*>) . (K,(J . K)) is set
[K,(J . K)] is set
the Mult of (product <*X*>) . [K,(J . K)] is set
v is left_complementable right_complementable complementable Element of the carrier of Y
I . v is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
K * (I . v) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the Mult of (product <*Y*>) is non empty Relation-like [:REAL, the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([:REAL, the carrier of (product <*Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[:REAL, the carrier of (product <*Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the Mult of (product <*Y*>) . (K,(I . v)) is set
[K,(I . v)] is set
the Mult of (product <*Y*>) . [K,(I . v)] is set
K is left_complementable right_complementable complementable Element of the carrier of X
J . K is left_complementable right_complementable complementable Element of the carrier of (product <*X*>)
||.(J . K).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X*>) is non empty Relation-like the carrier of (product <*X*>) -defined REAL -valued Function-like V26( the carrier of (product <*X*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*X*>),REAL:]
[: the carrier of (product <*X*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*X*>),REAL:] is non empty set
the normF of (product <*X*>) . (J . K) is V11() real ext-real Element of REAL
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . K is V11() real ext-real Element of REAL
<*||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
len <*||.K.||*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
REAL 1 is non empty functional FinSequence-membered M10( REAL )
K361(1,REAL) is functional FinSequence-membered M10( REAL )
K is Relation-like NAT -defined REAL -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 1
|.K.| is V11() real ext-real non negative Element of REAL
sqr K is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr K) is V11() real ext-real Element of REAL
sqrt (Sum (sqr K)) is V11() real ext-real Element of REAL
||.K.|| ^2 is V11() real ext-real Element of REAL
<*(||.K.|| ^2)*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum <*(||.K.|| ^2)*> is V11() real ext-real Element of REAL
sqrt (Sum <*(||.K.|| ^2)*>) is V11() real ext-real Element of REAL
sqrt (||.K.|| ^2) is V11() real ext-real Element of REAL
<*K*> is non empty Relation-like NAT -defined the carrier of X -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of X
<*K*> . 1 is set
dom <*X*> is non empty countable Element of bool NAT
v is Relation-like NAT -defined Function-like carr <*X*> -compatible Element of product (carr <*X*>)
normsequence (<*X*>,v) is Relation-like NAT -defined REAL -valued Function-like V33() V40( len <*X*>) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL (len <*X*>)
len <*X*> is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
REAL (len <*X*>) is non empty functional FinSequence-membered M10( REAL )
K361((len <*X*>),REAL) is functional FinSequence-membered M10( REAL )
r is Element of dom <*X*>
(normsequence (<*X*>,v)) . r is V11() real ext-real set
<*X*> . r is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of (<*X*> . r) is non empty Relation-like the carrier of (<*X*> . r) -defined REAL -valued Function-like V26( the carrier of (<*X*> . r)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (<*X*> . r),REAL:]
the carrier of (<*X*> . r) is non empty set
[: the carrier of (<*X*> . r),REAL:] is non empty Relation-like set
bool [: the carrier of (<*X*> . r),REAL:] is non empty set
v . r is set
the normF of (<*X*> . r) . (v . r) is V11() real ext-real set
len (normsequence (<*X*>,v)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
rng (X ^ Y) is non empty set
I is set
dom (X ^ Y) is non empty countable Element of bool NAT
J is set
(X ^ Y) . J is set
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
X . K is set
rng X is non empty set
dom Y is non empty countable Element of bool NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Y . K is set
rng Y is non empty set
K is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom X is non empty countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
X is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Y is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
X ^ Y is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
sqr (X ^ Y) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
sqr X is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
sqr Y is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(sqr X) ^ (sqr Y) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
((product X),(product Y)) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product X) is non empty set
the carrier of (product Y) is non empty set
[: the carrier of (product X), the carrier of (product Y):] is non empty Relation-like set
((product X),(product Y)) is Element of [: the carrier of (product X), the carrier of (product Y):]
0. (product X) is V52( product X) left_complementable right_complementable complementable Element of the carrier of (product X)
the ZeroF of (product X) is left_complementable right_complementable complementable Element of the carrier of (product X)
0. (product Y) is V52( product Y) left_complementable right_complementable complementable Element of the carrier of (product Y)
the ZeroF of (product Y) is left_complementable right_complementable complementable Element of the carrier of (product Y)
[(0. (product X)),(0. (product Y))] is Element of [: the carrier of (product X), the carrier of (product Y):]
((product X),(product Y)) is non empty Relation-like [:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
((product X),(product Y)) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:REAL,[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
((product X),(product Y)) is non empty Relation-like [: the carrier of (product X), the carrier of (product Y):] -defined REAL -valued Function-like V26([: the carrier of (product X), the carrier of (product Y):]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of (product X), the carrier of (product Y):],REAL:]
[:[: the carrier of (product X), the carrier of (product Y):],REAL:] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of (product Y):],REAL:] is non empty set
NORMSTR(# [: the carrier of (product X), the carrier of (product Y):],((product X),(product Y)),((product X),(product Y)),((product X),(product Y)),((product X),(product Y)) #) is strict NORMSTR
the carrier of ((product X),(product Y)) is non empty set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product (X ^ Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product (X ^ Y)) is non empty set
[: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty set
0. ((product X),(product Y)) is V52(((product X),(product Y))) left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the ZeroF of ((product X),(product Y)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
0. (product (X ^ Y)) is V52( product (X ^ Y)) left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the ZeroF of (product (X ^ Y)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
I is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product I is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr I is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr I) is non empty functional with_common_domain product-like set
zeros I is Relation-like NAT -defined Function-like carr I -compatible Element of product (carr I)
addop I is Relation-like Function-like BinOps of carr I
[:(addop I):] is non empty Relation-like [:(product (carr I)),(product (carr I)):] -defined product (carr I) -valued Function-like V26([:(product (carr I)),(product (carr I)):]) quasi_total Element of bool [:[:(product (carr I)),(product (carr I)):],(product (carr I)):]
[:(product (carr I)),(product (carr I)):] is non empty Relation-like set
[:[:(product (carr I)),(product (carr I)):],(product (carr I)):] is non empty Relation-like set
bool [:[:(product (carr I)),(product (carr I)):],(product (carr I)):] is non empty set
multop I is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr I
[:(multop I):] is non empty Relation-like [:REAL,(product (carr I)):] -defined product (carr I) -valued Function-like V26([:REAL,(product (carr I)):]) quasi_total Element of bool [:[:REAL,(product (carr I)):],(product (carr I)):]
[:REAL,(product (carr I)):] is non empty Relation-like set
[:[:REAL,(product (carr I)):],(product (carr I)):] is non empty Relation-like set
bool [:[:REAL,(product (carr I)):],(product (carr I)):] is non empty set
RLSStruct(# (product (carr I)),(zeros I),[:(addop I):],[:(multop I):] #) is non empty strict RLSStruct
J is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product J is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr J is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr J) is non empty functional with_common_domain product-like set
zeros J is Relation-like NAT -defined Function-like carr J -compatible Element of product (carr J)
addop J is Relation-like Function-like BinOps of carr J
[:(addop J):] is non empty Relation-like [:(product (carr J)),(product (carr J)):] -defined product (carr J) -valued Function-like V26([:(product (carr J)),(product (carr J)):]) quasi_total Element of bool [:[:(product (carr J)),(product (carr J)):],(product (carr J)):]
[:(product (carr J)),(product (carr J)):] is non empty Relation-like set
[:[:(product (carr J)),(product (carr J)):],(product (carr J)):] is non empty Relation-like set
bool [:[:(product (carr J)),(product (carr J)):],(product (carr J)):] is non empty set
multop J is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr J
[:(multop J):] is non empty Relation-like [:REAL,(product (carr J)):] -defined product (carr J) -valued Function-like V26([:REAL,(product (carr J)):]) quasi_total Element of bool [:[:REAL,(product (carr J)):],(product (carr J)):]
[:REAL,(product (carr J)):] is non empty Relation-like set
[:[:REAL,(product (carr J)):],(product (carr J)):] is non empty Relation-like set
bool [:[:REAL,(product (carr J)):],(product (carr J)):] is non empty set
RLSStruct(# (product (carr J)),(zeros J),[:(addop J):],[:(multop J):] #) is non empty strict RLSStruct
carr X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
carr Y is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr X) is non empty functional with_common_domain product-like set
zeros X is Relation-like NAT -defined Function-like carr X -compatible Element of product (carr X)
addop X is Relation-like Function-like BinOps of carr X
[:(addop X):] is non empty Relation-like [:(product (carr X)),(product (carr X)):] -defined product (carr X) -valued Function-like V26([:(product (carr X)),(product (carr X)):]) quasi_total Element of bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):]
[:(product (carr X)),(product (carr X)):] is non empty Relation-like set
[:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty set
multop X is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr X
[:(multop X):] is non empty Relation-like [:REAL,(product (carr X)):] -defined product (carr X) -valued Function-like V26([:REAL,(product (carr X)):]) quasi_total Element of bool [:[:REAL,(product (carr X)):],(product (carr X)):]
[:REAL,(product (carr X)):] is non empty Relation-like set
[:[:REAL,(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:REAL,(product (carr X)):],(product (carr X)):] is non empty set
productnorm X is non empty Relation-like product (carr X) -defined REAL -valued Function-like V26( product (carr X)) quasi_total complex-yielding V140() V141() Element of bool [:(product (carr X)),REAL:]
[:(product (carr X)),REAL:] is non empty Relation-like set
bool [:(product (carr X)),REAL:] is non empty set
NORMSTR(# (product (carr X)),(zeros X),[:(addop X):],[:(multop X):],(productnorm X) #) is strict NORMSTR
product (carr Y) is non empty functional with_common_domain product-like set
zeros Y is Relation-like NAT -defined Function-like carr Y -compatible Element of product (carr Y)
addop Y is Relation-like Function-like BinOps of carr Y
[:(addop Y):] is non empty Relation-like [:(product (carr Y)),(product (carr Y)):] -defined product (carr Y) -valued Function-like V26([:(product (carr Y)),(product (carr Y)):]) quasi_total Element of bool [:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):]
[:(product (carr Y)),(product (carr Y)):] is non empty Relation-like set
[:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):] is non empty Relation-like set
bool [:[:(product (carr Y)),(product (carr Y)):],(product (carr Y)):] is non empty set
multop Y is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr Y
[:(multop Y):] is non empty Relation-like [:REAL,(product (carr Y)):] -defined product (carr Y) -valued Function-like V26([:REAL,(product (carr Y)):]) quasi_total Element of bool [:[:REAL,(product (carr Y)):],(product (carr Y)):]
[:REAL,(product (carr Y)):] is non empty Relation-like set
[:[:REAL,(product (carr Y)):],(product (carr Y)):] is non empty Relation-like set
bool [:[:REAL,(product (carr Y)):],(product (carr Y)):] is non empty set
productnorm Y is non empty Relation-like product (carr Y) -defined REAL -valued Function-like V26( product (carr Y)) quasi_total complex-yielding V140() V141() Element of bool [:(product (carr Y)),REAL:]
[:(product (carr Y)),REAL:] is non empty Relation-like set
bool [:(product (carr Y)),REAL:] is non empty set
NORMSTR(# (product (carr Y)),(zeros Y),[:(addop Y):],[:(multop Y):],(productnorm Y) #) is strict NORMSTR
((product I),(product J)) is non empty Relation-like [:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:] -defined [: the carrier of (product I), the carrier of (product J):] -valued Function-like V26([:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:]) quasi_total Element of bool [:[:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:]
the carrier of (product I) is non empty set
the carrier of (product J) is non empty set
[: the carrier of (product I), the carrier of (product J):] is non empty Relation-like set
[:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:] is non empty Relation-like set
[:[:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product I), the carrier of (product J):],[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:] is non empty set
yy2 is left_complementable right_complementable complementable Element of the carrier of (product X)
I is left_complementable right_complementable complementable Element of the carrier of (product X)
yy2 + I is left_complementable right_complementable complementable Element of the carrier of (product X)
the addF of (product X) is non empty Relation-like [: the carrier of (product X), the carrier of (product X):] -defined the carrier of (product X) -valued Function-like V26([: the carrier of (product X), the carrier of (product X):]) quasi_total Element of bool [:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):]
[: the carrier of (product X), the carrier of (product X):] is non empty Relation-like set
[:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of (product X):], the carrier of (product X):] is non empty set
the addF of (product X) . (yy2,I) is left_complementable right_complementable complementable Element of the carrier of (product X)
[yy2,I] is set
the addF of (product X) . [yy2,I] is set
v is left_complementable right_complementable complementable Element of the carrier of (product Y)
[yy2,v] is Element of [: the carrier of (product X), the carrier of (product Y):]
x1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
[I,x1] is Element of [: the carrier of (product X), the carrier of (product Y):]
((product I),(product J)) . ([yy2,v],[I,x1]) is set
[[yy2,v],[I,x1]] is set
((product I),(product J)) . [[yy2,v],[I,x1]] is set
v + x1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
the addF of (product Y) is non empty Relation-like [: the carrier of (product Y), the carrier of (product Y):] -defined the carrier of (product Y) -valued Function-like V26([: the carrier of (product Y), the carrier of (product Y):]) quasi_total Element of bool [:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):]
[: the carrier of (product Y), the carrier of (product Y):] is non empty Relation-like set
[:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):] is non empty Relation-like set
bool [:[: the carrier of (product Y), the carrier of (product Y):], the carrier of (product Y):] is non empty set
the addF of (product Y) . (v,x1) is left_complementable right_complementable complementable Element of the carrier of (product Y)
[v,x1] is set
the addF of (product Y) . [v,x1] is set
[(yy2 + I),(v + x1)] is Element of [: the carrier of (product X), the carrier of (product Y):]
y1 is left_complementable right_complementable complementable Element of the carrier of (product I)
v1 is left_complementable right_complementable complementable Element of the carrier of (product I)
y1 + v1 is left_complementable right_complementable complementable Element of the carrier of (product I)
the addF of (product I) is non empty Relation-like [: the carrier of (product I), the carrier of (product I):] -defined the carrier of (product I) -valued Function-like V26([: the carrier of (product I), the carrier of (product I):]) quasi_total Element of bool [:[: the carrier of (product I), the carrier of (product I):], the carrier of (product I):]
[: the carrier of (product I), the carrier of (product I):] is non empty Relation-like set
[:[: the carrier of (product I), the carrier of (product I):], the carrier of (product I):] is non empty Relation-like set
bool [:[: the carrier of (product I), the carrier of (product I):], the carrier of (product I):] is non empty set
the addF of (product I) . (y1,v1) is left_complementable right_complementable complementable Element of the carrier of (product I)
[y1,v1] is set
the addF of (product I) . [y1,v1] is set
Ix1 is left_complementable right_complementable complementable Element of the carrier of (product J)
Iy1 is left_complementable right_complementable complementable Element of the carrier of (product J)
Ix1 + Iy1 is left_complementable right_complementable complementable Element of the carrier of (product J)
the addF of (product J) is non empty Relation-like [: the carrier of (product J), the carrier of (product J):] -defined the carrier of (product J) -valued Function-like V26([: the carrier of (product J), the carrier of (product J):]) quasi_total Element of bool [:[: the carrier of (product J), the carrier of (product J):], the carrier of (product J):]
[: the carrier of (product J), the carrier of (product J):] is non empty Relation-like set
[:[: the carrier of (product J), the carrier of (product J):], the carrier of (product J):] is non empty Relation-like set
bool [:[: the carrier of (product J), the carrier of (product J):], the carrier of (product J):] is non empty set
the addF of (product J) . (Ix1,Iy1) is left_complementable right_complementable complementable Element of the carrier of (product J)
[Ix1,Iy1] is set
the addF of (product J) . [Ix1,Iy1] is set
((product I),(product J)) is non empty Relation-like [:REAL,[: the carrier of (product I), the carrier of (product J):]:] -defined [: the carrier of (product I), the carrier of (product J):] -valued Function-like V26([:REAL,[: the carrier of (product I), the carrier of (product J):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:]
[:REAL,[: the carrier of (product I), the carrier of (product J):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product I), the carrier of (product J):]:],[: the carrier of (product I), the carrier of (product J):]:] is non empty set
yy2 is V11() real ext-real Element of REAL
I is left_complementable right_complementable complementable Element of the carrier of (product X)
yy2 * I is left_complementable right_complementable complementable Element of the carrier of (product X)
the Mult of (product X) is non empty Relation-like [:REAL, the carrier of (product X):] -defined the carrier of (product X) -valued Function-like V26([:REAL, the carrier of (product X):]) quasi_total Element of bool [:[:REAL, the carrier of (product X):], the carrier of (product X):]
[:REAL, the carrier of (product X):] is non empty Relation-like set
[:[:REAL, the carrier of (product X):], the carrier of (product X):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product X):], the carrier of (product X):] is non empty set
the Mult of (product X) . (yy2,I) is set
[yy2,I] is set
the Mult of (product X) . [yy2,I] is set
v is left_complementable right_complementable complementable Element of the carrier of (product Y)
[I,v] is Element of [: the carrier of (product X), the carrier of (product Y):]
((product I),(product J)) . (yy2,[I,v]) is set
[yy2,[I,v]] is set
((product I),(product J)) . [yy2,[I,v]] is set
yy2 * v is left_complementable right_complementable complementable Element of the carrier of (product Y)
the Mult of (product Y) is non empty Relation-like [:REAL, the carrier of (product Y):] -defined the carrier of (product Y) -valued Function-like V26([:REAL, the carrier of (product Y):]) quasi_total Element of bool [:[:REAL, the carrier of (product Y):], the carrier of (product Y):]
[:REAL, the carrier of (product Y):] is non empty Relation-like set
[:[:REAL, the carrier of (product Y):], the carrier of (product Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product Y):], the carrier of (product Y):] is non empty set
the Mult of (product Y) . (yy2,v) is set
[yy2,v] is set
the Mult of (product Y) . [yy2,v] is set
[(yy2 * I),(yy2 * v)] is Element of [: the carrier of (product X), the carrier of (product Y):]
x1 is left_complementable right_complementable complementable Element of the carrier of (product I)
yy2 * x1 is left_complementable right_complementable complementable Element of the carrier of (product I)
the Mult of (product I) is non empty Relation-like [:REAL, the carrier of (product I):] -defined the carrier of (product I) -valued Function-like V26([:REAL, the carrier of (product I):]) quasi_total Element of bool [:[:REAL, the carrier of (product I):], the carrier of (product I):]
[:REAL, the carrier of (product I):] is non empty Relation-like set
[:[:REAL, the carrier of (product I):], the carrier of (product I):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product I):], the carrier of (product I):] is non empty set
the Mult of (product I) . (yy2,x1) is set
[yy2,x1] is set
the Mult of (product I) . [yy2,x1] is set
y1 is left_complementable right_complementable complementable Element of the carrier of (product J)
yy2 * y1 is left_complementable right_complementable complementable Element of the carrier of (product J)
the Mult of (product J) is non empty Relation-like [:REAL, the carrier of (product J):] -defined the carrier of (product J) -valued Function-like V26([:REAL, the carrier of (product J):]) quasi_total Element of bool [:[:REAL, the carrier of (product J):], the carrier of (product J):]
[:REAL, the carrier of (product J):] is non empty Relation-like set
[:[:REAL, the carrier of (product J):], the carrier of (product J):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product J):], the carrier of (product J):] is non empty set
the Mult of (product J) . (yy2,y1) is set
[yy2,y1] is set
the Mult of (product J) . [yy2,y1] is set
carr (X ^ Y) is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len (carr (X ^ Y)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (X ^ Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
I ^ J is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
carr (I ^ J) is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len (carr (I ^ J)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len (I ^ J) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
yy is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len yy is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len X is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
x1 is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len Y is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
y1 is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len y1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len I is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
xx2 is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
len xx2 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len J is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
((product I),(product J)) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
((product I),(product J)) is Element of [: the carrier of (product I), the carrier of (product J):]
0. (product I) is V52( product I) left_complementable right_complementable complementable Element of the carrier of (product I)
the ZeroF of (product I) is left_complementable right_complementable complementable Element of the carrier of (product I)
0. (product J) is V52( product J) left_complementable right_complementable complementable Element of the carrier of (product J)
the ZeroF of (product J) is left_complementable right_complementable complementable Element of the carrier of (product J)
[(0. (product I)),(0. (product J))] is Element of [: the carrier of (product I), the carrier of (product J):]
RLSStruct(# [: the carrier of (product I), the carrier of (product J):],((product I),(product J)),((product I),(product J)),((product I),(product J)) #) is non empty strict RLSStruct
the carrier of ((product I),(product J)) is non empty set
product (I ^ J) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
product (carr (I ^ J)) is non empty functional with_common_domain product-like set
zeros (I ^ J) is Relation-like NAT -defined Function-like carr (I ^ J) -compatible Element of product (carr (I ^ J))
addop (I ^ J) is Relation-like Function-like BinOps of carr (I ^ J)
[:(addop (I ^ J)):] is non empty Relation-like [:(product (carr (I ^ J))),(product (carr (I ^ J))):] -defined product (carr (I ^ J)) -valued Function-like V26([:(product (carr (I ^ J))),(product (carr (I ^ J))):]) quasi_total Element of bool [:[:(product (carr (I ^ J))),(product (carr (I ^ J))):],(product (carr (I ^ J))):]
[:(product (carr (I ^ J))),(product (carr (I ^ J))):] is non empty Relation-like set
[:[:(product (carr (I ^ J))),(product (carr (I ^ J))):],(product (carr (I ^ J))):] is non empty Relation-like set
bool [:[:(product (carr (I ^ J))),(product (carr (I ^ J))):],(product (carr (I ^ J))):] is non empty set
multop (I ^ J) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr (I ^ J)
[:(multop (I ^ J)):] is non empty Relation-like [:REAL,(product (carr (I ^ J))):] -defined product (carr (I ^ J)) -valued Function-like V26([:REAL,(product (carr (I ^ J))):]) quasi_total Element of bool [:[:REAL,(product (carr (I ^ J))):],(product (carr (I ^ J))):]
[:REAL,(product (carr (I ^ J))):] is non empty Relation-like set
[:[:REAL,(product (carr (I ^ J))):],(product (carr (I ^ J))):] is non empty Relation-like set
bool [:[:REAL,(product (carr (I ^ J))):],(product (carr (I ^ J))):] is non empty set
RLSStruct(# (product (carr (I ^ J))),(zeros (I ^ J)),[:(addop (I ^ J)):],[:(multop (I ^ J)):] #) is non empty strict RLSStruct
the carrier of (product (I ^ J)) is non empty set
[: the carrier of ((product I),(product J)), the carrier of (product (I ^ J)):] is non empty Relation-like set
bool [: the carrier of ((product I),(product J)), the carrier of (product (I ^ J)):] is non empty set
0. (product (I ^ J)) is V52( product (I ^ J)) left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
the ZeroF of (product (I ^ J)) is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
0. ((product I),(product J)) is V52(((product I),(product J))) left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
the ZeroF of ((product I),(product J)) is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
yy2 is non empty Relation-like the carrier of ((product I),(product J)) -defined the carrier of (product (I ^ J)) -valued Function-like V26( the carrier of ((product I),(product J))) quasi_total Element of bool [: the carrier of ((product I),(product J)), the carrier of (product (I ^ J)):]
yy2 . (0. ((product I),(product J))) is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
product (carr (X ^ Y)) is non empty functional with_common_domain product-like set
zeros (X ^ Y) is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
addop (X ^ Y) is Relation-like Function-like BinOps of carr (X ^ Y)
[:(addop (X ^ Y)):] is non empty Relation-like [:(product (carr (X ^ Y))),(product (carr (X ^ Y))):] -defined product (carr (X ^ Y)) -valued Function-like V26([:(product (carr (X ^ Y))),(product (carr (X ^ Y))):]) quasi_total Element of bool [:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):]
[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):] is non empty Relation-like set
[:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty Relation-like set
bool [:[:(product (carr (X ^ Y))),(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty set
multop (X ^ Y) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr (X ^ Y)
[:(multop (X ^ Y)):] is non empty Relation-like [:REAL,(product (carr (X ^ Y))):] -defined product (carr (X ^ Y)) -valued Function-like V26([:REAL,(product (carr (X ^ Y))):]) quasi_total Element of bool [:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):]
[:REAL,(product (carr (X ^ Y))):] is non empty Relation-like set
[:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty Relation-like set
bool [:[:REAL,(product (carr (X ^ Y))):],(product (carr (X ^ Y))):] is non empty set
productnorm (X ^ Y) is non empty Relation-like product (carr (X ^ Y)) -defined REAL -valued Function-like V26( product (carr (X ^ Y))) quasi_total complex-yielding V140() V141() Element of bool [:(product (carr (X ^ Y))),REAL:]
[:(product (carr (X ^ Y))),REAL:] is non empty Relation-like set
bool [:(product (carr (X ^ Y))),REAL:] is non empty set
NORMSTR(# (product (carr (X ^ Y))),(zeros (X ^ Y)),[:(addop (X ^ Y)):],[:(multop (X ^ Y)):],(productnorm (X ^ Y)) #) is strict NORMSTR
I is non empty Relation-like the carrier of ((product X),(product Y)) -defined the carrier of (product (X ^ Y)) -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total Element of bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):]
I . (0. ((product X),(product Y))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
v is left_complementable right_complementable complementable Element of the carrier of (product X)
x1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
I . (v,x1) is set
[v,x1] is set
I . [v,x1] is set
v is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I . (v,x1) is set
[v,x1] is set
I . [v,x1] is set
v ^ x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 ^ v1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v + x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the addF of ((product X),(product Y)) is non empty Relation-like [: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the addF of ((product X),(product Y)) . (v,x1) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
[v,x1] is set
the addF of ((product X),(product Y)) . [v,x1] is set
I . (v + x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
I . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
I . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(I . v) + (I . x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) is non empty Relation-like [: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the addF of (product (X ^ Y)) . ((I . v),(I . x1)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(I . v),(I . x1)] is set
the addF of (product (X ^ Y)) . [(I . v),(I . x1)] is set
y1 is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
v1 is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
y1 + v1 is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
the addF of ((product I),(product J)) is non empty Relation-like [: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):] -defined the carrier of ((product I),(product J)) -valued Function-like V26([: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):]) quasi_total Element of bool [:[: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):]
[: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):] is non empty Relation-like set
[:[: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):] is non empty Relation-like set
bool [:[: the carrier of ((product I),(product J)), the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):] is non empty set
the addF of ((product I),(product J)) . (y1,v1) is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
[y1,v1] is set
the addF of ((product I),(product J)) . [y1,v1] is set
yy2 . y1 is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
yy2 . v1 is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
(yy2 . y1) + (yy2 . v1) is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
the addF of (product (I ^ J)) is non empty Relation-like [: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):] -defined the carrier of (product (I ^ J)) -valued Function-like V26([: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):]) quasi_total Element of bool [:[: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):]
[: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):] is non empty Relation-like set
[:[: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):] is non empty Relation-like set
bool [:[: the carrier of (product (I ^ J)), the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):] is non empty set
the addF of (product (I ^ J)) . ((yy2 . y1),(yy2 . v1)) is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
[(yy2 . y1),(yy2 . v1)] is set
the addF of (product (I ^ J)) . [(yy2 . y1),(yy2 . v1)] is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
I . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
x1 is V11() real ext-real Element of REAL
x1 * v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the Mult of ((product X),(product Y)) is non empty Relation-like [:REAL, the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([:REAL, the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[:REAL, the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the Mult of ((product X),(product Y)) . (x1,v) is set
[x1,v] is set
the Mult of ((product X),(product Y)) . [x1,v] is set
I . (x1 * v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
x1 * (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the Mult of (product (X ^ Y)) is non empty Relation-like [:REAL, the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([:REAL, the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[:REAL, the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the Mult of (product (X ^ Y)) . (x1,(I . v)) is set
[x1,(I . v)] is set
the Mult of (product (X ^ Y)) . [x1,(I . v)] is set
y1 is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
x1 * y1 is left_complementable right_complementable complementable Element of the carrier of ((product I),(product J))
the Mult of ((product I),(product J)) is non empty Relation-like [:REAL, the carrier of ((product I),(product J)):] -defined the carrier of ((product I),(product J)) -valued Function-like V26([:REAL, the carrier of ((product I),(product J)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):]
[:REAL, the carrier of ((product I),(product J)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product I),(product J)):], the carrier of ((product I),(product J)):] is non empty set
the Mult of ((product I),(product J)) . (x1,y1) is set
[x1,y1] is set
the Mult of ((product I),(product J)) . [x1,y1] is set
yy2 . y1 is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
x1 * (yy2 . y1) is left_complementable right_complementable complementable Element of the carrier of (product (I ^ J))
the Mult of (product (I ^ J)) is non empty Relation-like [:REAL, the carrier of (product (I ^ J)):] -defined the carrier of (product (I ^ J)) -valued Function-like V26([:REAL, the carrier of (product (I ^ J)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):]
[:REAL, the carrier of (product (I ^ J)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (I ^ J)):], the carrier of (product (I ^ J)):] is non empty set
the Mult of (product (I ^ J)) . (x1,(yy2 . y1)) is set
[x1,(yy2 . y1)] is set
the Mult of (product (I ^ J)) . [x1,(yy2 . y1)] is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
I . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
||.(I . v).|| is V11() real ext-real non negative Element of REAL
the normF of (product (X ^ Y)) is non empty Relation-like the carrier of (product (X ^ Y)) -defined REAL -valued Function-like V26( the carrier of (product (X ^ Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product (X ^ Y)),REAL:]
[: the carrier of (product (X ^ Y)),REAL:] is non empty Relation-like set
bool [: the carrier of (product (X ^ Y)),REAL:] is non empty set
the normF of (product (X ^ Y)) . (I . v) is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of ((product X),(product Y)) is non empty Relation-like the carrier of ((product X),(product Y)) -defined REAL -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((product X),(product Y)),REAL:]
[: the carrier of ((product X),(product Y)),REAL:] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)),REAL:] is non empty set
the normF of ((product X),(product Y)) . v is V11() real ext-real Element of REAL
x1 is left_complementable right_complementable complementable Element of the carrier of (product X)
y1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
[x1,y1] is Element of [: the carrier of (product X), the carrier of (product Y):]
||.x1.|| is V11() real ext-real non negative Element of REAL
the normF of (product X) is non empty Relation-like the carrier of (product X) -defined REAL -valued Function-like V26( the carrier of (product X)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product X),REAL:]
[: the carrier of (product X),REAL:] is non empty Relation-like set
bool [: the carrier of (product X),REAL:] is non empty set
the normF of (product X) . x1 is V11() real ext-real Element of REAL
||.y1.|| is V11() real ext-real non negative Element of REAL
the normF of (product Y) is non empty Relation-like the carrier of (product Y) -defined REAL -valued Function-like V26( the carrier of (product Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product Y),REAL:]
[: the carrier of (product Y),REAL:] is non empty Relation-like set
bool [: the carrier of (product Y),REAL:] is non empty set
the normF of (product Y) . y1 is V11() real ext-real Element of REAL
<*||.x1.||,||.y1.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
((product X),(product Y)) . (x1,y1) is V11() real ext-real Element of REAL
[x1,y1] is set
((product X),(product Y)) . [x1,y1] is V11() real ext-real set
v1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.v1.| is V11() real ext-real non negative Element of REAL
sqr v1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr v1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr v1)) is V11() real ext-real Element of REAL
Ix1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom Ix1 is countable Element of bool NAT
dom (carr X) is non empty countable Element of bool NAT
Iy1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
dom Iy1 is countable Element of bool NAT
dom (carr Y) is non empty countable Element of bool NAT
I . (x1,y1) is set
I . [x1,y1] is set
Ix1 ^ Iy1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
Iv is Relation-like NAT -defined Function-like carr (X ^ Y) -compatible Element of product (carr (X ^ Y))
normsequence ((X ^ Y),Iv) is Relation-like NAT -defined REAL -valued Function-like V33() V40( len (X ^ Y)) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL (len (X ^ Y))
REAL (len (X ^ Y)) is non empty functional FinSequence-membered M10( REAL )
K361((len (X ^ Y)),REAL) is functional FinSequence-membered M10( REAL )
|.(normsequence ((X ^ Y),Iv)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence ((X ^ Y),Iv)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr (normsequence ((X ^ Y),Iv))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence ((X ^ Y),Iv)))) is V11() real ext-real Element of REAL
sqr (normsequence ((X ^ Y),Iv)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of K361((len (X ^ Y)),REAL)
Sum (sqr (normsequence ((X ^ Y),Iv))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence ((X ^ Y),Iv)))) is V11() real ext-real Element of REAL
len (normsequence ((X ^ Y),Iv)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Ix is Relation-like NAT -defined Function-like carr X -compatible Element of product (carr X)
normsequence (X,Ix) is Relation-like NAT -defined REAL -valued Function-like V33() V40( len X) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL (len X)
REAL (len X) is non empty functional FinSequence-membered M10( REAL )
K361((len X),REAL) is functional FinSequence-membered M10( REAL )
len (normsequence (X,Ix)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Iy is Relation-like NAT -defined Function-like carr Y -compatible Element of product (carr Y)
normsequence (Y,Iy) is Relation-like NAT -defined REAL -valued Function-like V33() V40( len Y) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL (len Y)
REAL (len Y) is non empty functional FinSequence-membered M10( REAL )
K361((len Y),REAL) is functional FinSequence-membered M10( REAL )
len (normsequence (Y,Iy)) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
||.x1.|| ^2 is V11() real ext-real Element of REAL
||.y1.|| ^2 is V11() real ext-real Element of REAL
<*(||.x1.|| ^2),(||.y1.|| ^2)*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum <*(||.x1.|| ^2),(||.y1.|| ^2)*> is V11() real ext-real Element of REAL
sqrt (Sum <*(||.x1.|| ^2),(||.y1.|| ^2)*>) is V11() real ext-real Element of REAL
(||.x1.|| ^2) + (||.y1.|| ^2) is V11() real ext-real Element of REAL
sqrt ((||.x1.|| ^2) + (||.y1.|| ^2)) is V11() real ext-real Element of REAL
sqr (normsequence (X,Ix)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of K361((len X),REAL)
Sum (sqr (normsequence (X,Ix))) is V11() real ext-real Element of REAL
sqr (normsequence (Y,Iy)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of K361((len Y),REAL)
Sum (sqr (normsequence (Y,Iy))) is V11() real ext-real Element of REAL
|.(normsequence (X,Ix)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (X,Ix)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr (normsequence (X,Ix))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (X,Ix)))) is V11() real ext-real Element of REAL
|.(normsequence (X,Ix)).| ^2 is V11() real ext-real Element of REAL
|.(normsequence (Y,Iy)).| is V11() real ext-real non negative Element of REAL
sqr (normsequence (Y,Iy)) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr (normsequence (Y,Iy))) is V11() real ext-real Element of REAL
sqrt (Sum (sqr (normsequence (Y,Iy)))) is V11() real ext-real Element of REAL
|.(normsequence (Y,Iy)).| ^2 is V11() real ext-real Element of REAL
(len (normsequence (X,Ix))) + (len (normsequence (Y,Iy))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
(normsequence (X,Ix)) ^ (normsequence (Y,Iy)) is Relation-like NAT -defined REAL -valued Function-like V33() V40((len X) + (len Y)) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(len X) + (len Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
len ((normsequence (X,Ix)) ^ (normsequence (Y,Iy))) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom (normsequence ((X ^ Y),Iv)) is countable Element of bool NAT
dom ((normsequence (X,Ix)) ^ (normsequence (Y,Iy))) is countable Element of bool NAT
k is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(normsequence ((X ^ Y),Iv)) . k is V11() real ext-real set
((normsequence (X,Ix)) ^ (normsequence (Y,Iy))) . k is V11() real ext-real set
Seg (len (normsequence ((X ^ Y),Iv))) is V33() V40( len (normsequence ((X ^ Y),Iv))) countable Element of bool NAT
dom (X ^ Y) is non empty countable Element of bool NAT
k1 is Element of dom (X ^ Y)
(X ^ Y) . k1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of ((X ^ Y) . k1) is non empty Relation-like the carrier of ((X ^ Y) . k1) -defined REAL -valued Function-like V26( the carrier of ((X ^ Y) . k1)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((X ^ Y) . k1),REAL:]
the carrier of ((X ^ Y) . k1) is non empty set
[: the carrier of ((X ^ Y) . k1),REAL:] is non empty Relation-like set
bool [: the carrier of ((X ^ Y) . k1),REAL:] is non empty set
Iv . k1 is set
the normF of ((X ^ Y) . k1) . (Iv . k1) is V11() real ext-real set
len (carr X) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len (carr X)) is V33() V40( len (carr X)) countable Element of bool NAT
len (carr Y) is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Seg (len (carr Y)) is V33() V40( len (carr Y)) countable Element of bool NAT
dom X is non empty countable Element of bool NAT
dom Y is non empty countable Element of bool NAT
dom (normsequence (X,Ix)) is countable Element of bool NAT
Iv . k is set
Ix1 . k is set
n is Element of dom X
X . n is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of (X . n) is non empty Relation-like the carrier of (X . n) -defined REAL -valued Function-like V26( the carrier of (X . n)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X . n),REAL:]
the carrier of (X . n) is non empty set
[: the carrier of (X . n),REAL:] is non empty Relation-like set
bool [: the carrier of (X . n),REAL:] is non empty set
Iv . n is set
the normF of (X . n) . (Iv . n) is V11() real ext-real set
(normsequence (X,Ix)) . n is V11() real ext-real set
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real set
(len X) + n is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
dom (normsequence (Y,Iy)) is countable Element of bool NAT
len Ix1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
Iv . k is set
Iy1 . n is set
n1 is Element of dom Y
Y . n1 is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
the normF of (Y . n1) is non empty Relation-like the carrier of (Y . n1) -defined REAL -valued Function-like V26( the carrier of (Y . n1)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (Y . n1),REAL:]
the carrier of (Y . n1) is non empty set
[: the carrier of (Y . n1),REAL:] is non empty Relation-like set
bool [: the carrier of (Y . n1),REAL:] is non empty set
the normF of (Y . n1) . (Iv . k1) is V11() real ext-real set
(normsequence (Y,Iy)) . n1 is V11() real ext-real set
(sqr (normsequence (X,Ix))) ^ (sqr (normsequence (Y,Iy))) is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
I . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
||.(I . v).|| is V11() real ext-real non negative Element of REAL
the normF of (product (X ^ Y)) is non empty Relation-like the carrier of (product (X ^ Y)) -defined REAL -valued Function-like V26( the carrier of (product (X ^ Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product (X ^ Y)),REAL:]
[: the carrier of (product (X ^ Y)),REAL:] is non empty Relation-like set
bool [: the carrier of (product (X ^ Y)),REAL:] is non empty set
the normF of (product (X ^ Y)) . (I . v) is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of ((product X),(product Y)) is non empty Relation-like the carrier of ((product X),(product Y)) -defined REAL -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((product X),(product Y)),REAL:]
[: the carrier of ((product X),(product Y)),REAL:] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)),REAL:] is non empty set
the normF of ((product X),(product Y)) . v is V11() real ext-real Element of REAL
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I is set
J is set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
[K,K] is Element of [: the carrier of X, the carrier of Y:]
I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
J is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K is left_complementable right_complementable complementable Element of the carrier of X
r is left_complementable right_complementable complementable Element of the carrier of Y
[K,r] is Element of [: the carrier of X, the carrier of Y:]
I + J is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,J) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[I,J] is set
the addF of (X,Y) . [I,J] is set
K + K is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,K) is left_complementable right_complementable complementable Element of the carrier of X
[K,K] is set
the addF of X . [K,K] is set
v + r is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (v,r) is left_complementable right_complementable complementable Element of the carrier of Y
[v,r] is set
the addF of Y . [v,r] is set
[(K + K),(v + r)] is Element of [: the carrier of X, the carrier of Y:]
I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
- I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J is left_complementable right_complementable complementable Element of the carrier of X
- J is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
- K is left_complementable right_complementable complementable Element of the carrier of Y
[(- J),(- K)] is Element of [: the carrier of X, the carrier of Y:]
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I + K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (I,K) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[I,K] is set
the addF of (X,Y) . [I,K] is set
J + (- J) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (J,(- J)) is left_complementable right_complementable complementable Element of the carrier of X
[J,(- J)] is set
the addF of X . [J,(- J)] is set
K + (- K) is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(- K)) is left_complementable right_complementable complementable Element of the carrier of Y
[K,(- K)] is set
the addF of Y . [K,(- K)] is set
[(J + (- J)),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
[(0. X),(K + (- K))] is Element of [: the carrier of X, the carrier of Y:]
I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
K is V11() real ext-real set
K * I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (K,I) is set
[K,I] is set
the Mult of (X,Y) . [K,I] is set
K * J is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (K,J) is set
[K,J] is set
the Mult of X . [K,J] is set
K * K is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (K,K) is set
[K,K] is set
the Mult of Y . [K,K] is set
[(K * J),(K * K)] is Element of [: the carrier of X, the carrier of Y:]
v is V11() real ext-real Element of REAL
(X,Y) . (v,I) is set
[v,I] is set
(X,Y) . [v,I] is set
I is left_complementable right_complementable complementable Element of the carrier of (X,Y)
||.I.|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . I is V11() real ext-real Element of REAL
J is left_complementable right_complementable complementable Element of the carrier of X
||.J.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . J is V11() real ext-real Element of REAL
K is left_complementable right_complementable complementable Element of the carrier of Y
[J,K] is Element of [: the carrier of X, the carrier of Y:]
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.J.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
(X,Y) . (J,K) is V11() real ext-real Element of REAL
[J,K] is set
(X,Y) . [J,K] is V11() real ext-real set
K is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.K.| is V11() real ext-real non negative Element of REAL
sqr K is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr K) is V11() real ext-real Element of REAL
sqrt (Sum (sqr K)) is V11() real ext-real Element of REAL
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product <*X,Y*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*X,Y*>) is non empty set
the carrier of X is non empty set
the carrier of Y is non empty set
0. (product <*X,Y*>) is V52( product <*X,Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the ZeroF of (product <*X,Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
<*(0. X),(0. Y)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
(X,Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
[: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I is non empty Relation-like the carrier of (X,Y) -defined the carrier of (product <*X,Y*>) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):]
I . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J is set
rng I is non empty Element of bool the carrier of (product <*X,Y*>)
bool the carrier of (product <*X,Y*>) is non empty set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
[K,v] is Element of [: the carrier of X, the carrier of Y:]
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
I . (r,yy) is set
[r,yy] is set
I . [r,yy] is set
<*r,yy*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,K] is Element of [: the carrier of X, the carrier of Y:]
I . [K,K] is set
I . (K,K) is set
[K,K] is set
I . [K,K] is set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J + K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . (J,K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[J,K] is set
the addF of (product <*X,Y*>) . [J,K] is set
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of X
K + v is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,v) is left_complementable right_complementable complementable Element of the carrier of X
[K,v] is set
the addF of X . [K,v] is set
r is left_complementable right_complementable complementable Element of the carrier of Y
<*K,r*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
yy is left_complementable right_complementable complementable Element of the carrier of Y
<*v,yy*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
r + yy is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (r,yy) is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is set
the addF of Y . [r,yy] is set
<*(K + v),(r + yy)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,r] is Element of [: the carrier of X, the carrier of Y:]
[v,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
y1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
x1 + y1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (x1,y1) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[x1,y1] is set
the addF of (X,Y) . [x1,y1] is set
[(K + v),(r + yy)] is Element of [: the carrier of X, the carrier of Y:]
I . ((K + v),(r + yy)) is set
[(K + v),(r + yy)] is set
I . [(K + v),(r + yy)] is set
I . (K,r) is set
[K,r] is set
I . [K,r] is set
I . (v,yy) is set
[v,yy] is set
I . [v,yy] is set
I . ((0. X),(0. Y)) is set
[(0. X),(0. Y)] is set
I . [(0. X),(0. Y)] is set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
- K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
- K is left_complementable right_complementable complementable Element of the carrier of Y
<*(- K),(- K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J + v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . (J,v) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[J,v] is set
the addF of (product <*X,Y*>) . [J,v] is set
K + (- K) is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (K,(- K)) is left_complementable right_complementable complementable Element of the carrier of X
[K,(- K)] is set
the addF of X . [K,(- K)] is set
K + (- K) is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (K,(- K)) is left_complementable right_complementable complementable Element of the carrier of Y
[K,(- K)] is set
the addF of Y . [K,(- K)] is set
<*(K + (- K)),(K + (- K))*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
<*(0. X),(K + (- K))*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of X
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
v is V11() real ext-real set
v * J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the Mult of (product <*X,Y*>) is non empty Relation-like [:REAL, the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([:REAL, the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[:REAL, the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the Mult of (product <*X,Y*>) . (v,J) is set
[v,J] is set
the Mult of (product <*X,Y*>) . [v,J] is set
v * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (v,K) is set
[v,K] is set
the Mult of X . [v,K] is set
v * K is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (v,K) is set
[v,K] is set
the Mult of Y . [v,K] is set
<*(v * K),(v * K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
[K,K] is Element of [: the carrier of X, the carrier of Y:]
I . (K,K) is set
[K,K] is set
I . [K,K] is set
yy is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v * yy is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (v,yy) is set
[v,yy] is set
the Mult of (X,Y) . [v,yy] is set
I . (v * yy) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
r is V11() real ext-real Element of REAL
r * K is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X . (r,K) is set
[r,K] is set
the Mult of X . [r,K] is set
r * K is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y . (r,K) is set
[r,K] is set
the Mult of Y . [r,K] is set
I . ((r * K),(r * K)) is set
[(r * K),(r * K)] is set
I . [(r * K),(r * K)] is set
<*(r * K),(r * K)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
||.J.|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) is non empty Relation-like the carrier of (product <*X,Y*>) -defined REAL -valued Function-like V26( the carrier of (product <*X,Y*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*X,Y*>),REAL:]
[: the carrier of (product <*X,Y*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*X,Y*>),REAL:] is non empty set
the normF of (product <*X,Y*>) . J is V11() real ext-real Element of REAL
K is left_complementable right_complementable complementable Element of the carrier of X
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . K is V11() real ext-real Element of REAL
K is left_complementable right_complementable complementable Element of the carrier of Y
<*K,K*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . K is V11() real ext-real Element of REAL
<*||.K.||,||.K.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
[K,K] is Element of [: the carrier of X, the carrier of Y:]
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . v is V11() real ext-real Element of REAL
r is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.r.| is V11() real ext-real non negative Element of REAL
sqr r is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr r) is V11() real ext-real Element of REAL
sqrt (Sum (sqr r)) is V11() real ext-real Element of REAL
I . v is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
I . (K,K) is set
[K,K] is set
I . [K,K] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like complete NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like complete NORMSTR
(X,Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
<*X,Y*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
dom <*X,Y*> is non empty countable Element of bool NAT
I is Element of dom <*X,Y*>
<*X,Y*> . I is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
product <*X,Y*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (X,Y) is non empty set
the carrier of (product <*X,Y*>) is non empty set
[: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):] is non empty set
0. (product <*X,Y*>) is V52( product <*X,Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the ZeroF of (product <*X,Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I is non empty Relation-like the carrier of (X,Y) -defined the carrier of (product <*X,Y*>) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (product <*X,Y*>):]
I . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
J is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J - K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
- K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J + (- K) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (J,(- K)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[J,(- K)] is set
the addF of (X,Y) . [J,(- K)] is set
I . (J - K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(- 1) * K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . ((- 1),K) is set
[(- 1),K] is set
the Mult of (X,Y) . [(- 1),K] is set
J + ((- 1) * K) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) . (J,((- 1) * K)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[J,((- 1) * K)] is set
the addF of (X,Y) . [J,((- 1) * K)] is set
I . (J + ((- 1) * K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
I . J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
I . ((- 1) * K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(I . J) + (I . ((- 1) * K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) is non empty Relation-like [: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*X,Y*>), the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the addF of (product <*X,Y*>) . ((I . J),(I . ((- 1) * K))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(I . J),(I . ((- 1) * K))] is set
the addF of (product <*X,Y*>) . [(I . J),(I . ((- 1) * K))] is set
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(- 1) * (I . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the Mult of (product <*X,Y*>) is non empty Relation-like [:REAL, the carrier of (product <*X,Y*>):] -defined the carrier of (product <*X,Y*>) -valued Function-like V26([:REAL, the carrier of (product <*X,Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):]
[:REAL, the carrier of (product <*X,Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*X,Y*>):], the carrier of (product <*X,Y*>):] is non empty set
the Mult of (product <*X,Y*>) . ((- 1),(I . K)) is set
[(- 1),(I . K)] is set
the Mult of (product <*X,Y*>) . [(- 1),(I . K)] is set
(I . J) + ((- 1) * (I . K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((I . J),((- 1) * (I . K))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(I . J),((- 1) * (I . K))] is set
the addF of (product <*X,Y*>) . [(I . J),((- 1) * (I . K))] is set
(I . J) - (I . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- (I . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(I . J) + (- (I . K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((I . J),(- (I . K))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(I . J),(- (I . K))] is set
the addF of (product <*X,Y*>) . [(I . J),(- (I . K))] is set
J is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I . J is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(I . J) - (I . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- (I . K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(I . J) + (- (I . K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((I . J),(- (I . K))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(I . J),(- (I . K))] is set
the addF of (product <*X,Y*>) . [(I . J),(- (I . K))] is set
||.((I . J) - (I . K)).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) is non empty Relation-like the carrier of (product <*X,Y*>) -defined REAL -valued Function-like V26( the carrier of (product <*X,Y*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*X,Y*>),REAL:]
[: the carrier of (product <*X,Y*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*X,Y*>),REAL:] is non empty set
the normF of (product <*X,Y*>) . ((I . J) - (I . K)) is V11() real ext-real Element of REAL
J - K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
- K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J + (- K) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) . (J,(- K)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[J,(- K)] is set
the addF of (X,Y) . [J,(- K)] is set
I . (J - K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
||.(I . (J - K)).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) . (I . (J - K)) is V11() real ext-real Element of REAL
||.(J - K).|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . (J - K) is V11() real ext-real Element of REAL
[:NAT, the carrier of (X,Y):] is non empty Relation-like set
bool [:NAT, the carrier of (X,Y):] is non empty set
J is non empty Relation-like NAT -defined the carrier of (X,Y) -valued Function-like V26( NAT ) quasi_total Element of bool [:NAT, the carrier of (X,Y):]
[:NAT, the carrier of (product <*X,Y*>):] is non empty Relation-like set
bool [:NAT, the carrier of (product <*X,Y*>):] is non empty set
I * J is non empty Relation-like NAT -defined the carrier of (product <*X,Y*>) -valued Function-like V26( NAT ) quasi_total Element of bool [:NAT, the carrier of (product <*X,Y*>):]
K is V11() real ext-real Element of REAL
v is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
yy is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
J . yy is left_complementable right_complementable complementable Element of the carrier of (X,Y)
J . x1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
(J . yy) - (J . x1) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
- (J . x1) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
(J . yy) + (- (J . x1)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) . ((J . yy),(- (J . x1))) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[(J . yy),(- (J . x1))] is set
the addF of (X,Y) . [(J . yy),(- (J . x1))] is set
||.((J . yy) - (J . x1)).|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) . ((J . yy) - (J . x1)) is V11() real ext-real Element of REAL
dom J is non empty countable Element of bool NAT
K is non empty Relation-like NAT -defined the carrier of (product <*X,Y*>) -valued Function-like V26( NAT ) quasi_total Element of bool [:NAT, the carrier of (product <*X,Y*>):]
K . yy is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
I . (J . yy) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
K . x1 is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
I . (J . x1) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . yy) - (K . x1) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- (K . x1) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . yy) + (- (K . x1)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((K . yy),(- (K . x1))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . yy),(- (K . x1))] is set
the addF of (product <*X,Y*>) . [(K . yy),(- (K . x1))] is set
||.((K . yy) - (K . x1)).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) . ((K . yy) - (K . x1)) is V11() real ext-real Element of REAL
I " is Relation-like Function-like set
dom (I ") is set
rng I is non empty Element of bool the carrier of (product <*X,Y*>)
bool the carrier of (product <*X,Y*>) is non empty set
rng (I ") is set
dom I is non empty Element of bool the carrier of (X,Y)
bool the carrier of (X,Y) is non empty set
lim K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(I ") . (lim K) is set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
v is V11() real ext-real Element of REAL
r is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
yy is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
x1 is epsilon-transitive epsilon-connected ordinal natural V11() real ext-real Element of NAT
J . x1 is left_complementable right_complementable complementable Element of the carrier of (X,Y)
(J . x1) - K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
- K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
(J . x1) + (- K) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) . ((J . x1),(- K)) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[(J . x1),(- K)] is set
the addF of (X,Y) . [(J . x1),(- K)] is set
||.((J . x1) - K).|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) . ((J . x1) - K) is V11() real ext-real Element of REAL
K . x1 is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . x1) - (lim K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
- (lim K) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
(K . x1) + (- (lim K)) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
the addF of (product <*X,Y*>) . ((K . x1),(- (lim K))) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
[(K . x1),(- (lim K))] is set
the addF of (product <*X,Y*>) . [(K . x1),(- (lim K))] is set
||.((K . x1) - (lim K)).|| is V11() real ext-real non negative Element of REAL
the normF of (product <*X,Y*>) . ((K . x1) - (lim K)) is V11() real ext-real Element of REAL
I . (J . x1) is left_complementable right_complementable complementable Element of the carrier of (product <*X,Y*>)
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
<*(product X),(product Y)*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product <*(product X),(product Y)*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*(product X),(product Y)*>) is non empty set
X ^ Y is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product (X ^ Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product (X ^ Y)) is non empty set
[: the carrier of (product <*(product X),(product Y)*>), the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [: the carrier of (product <*(product X),(product Y)*>), the carrier of (product (X ^ Y)):] is non empty set
the carrier of (product X) is non empty set
the carrier of (product Y) is non empty set
0. (product <*(product X),(product Y)*>) is V52( product <*(product X),(product Y)*>) left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
the ZeroF of (product <*(product X),(product Y)*>) is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
0. (product (X ^ Y)) is V52( product (X ^ Y)) left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the ZeroF of (product (X ^ Y)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
((product X),(product Y)) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
[: the carrier of (product X), the carrier of (product Y):] is non empty Relation-like set
((product X),(product Y)) is Element of [: the carrier of (product X), the carrier of (product Y):]
0. (product X) is V52( product X) left_complementable right_complementable complementable Element of the carrier of (product X)
the ZeroF of (product X) is left_complementable right_complementable complementable Element of the carrier of (product X)
0. (product Y) is V52( product Y) left_complementable right_complementable complementable Element of the carrier of (product Y)
the ZeroF of (product Y) is left_complementable right_complementable complementable Element of the carrier of (product Y)
[(0. (product X)),(0. (product Y))] is Element of [: the carrier of (product X), the carrier of (product Y):]
((product X),(product Y)) is non empty Relation-like [:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of (product Y):],[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
((product X),(product Y)) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of (product Y):]:] -defined [: the carrier of (product X), the carrier of (product Y):] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of (product Y):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:]
[:REAL,[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of (product Y):]:],[: the carrier of (product X), the carrier of (product Y):]:] is non empty set
((product X),(product Y)) is non empty Relation-like [: the carrier of (product X), the carrier of (product Y):] -defined REAL -valued Function-like V26([: the carrier of (product X), the carrier of (product Y):]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of (product X), the carrier of (product Y):],REAL:]
[:[: the carrier of (product X), the carrier of (product Y):],REAL:] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of (product Y):],REAL:] is non empty set
NORMSTR(# [: the carrier of (product X), the carrier of (product Y):],((product X),(product Y)),((product X),(product Y)),((product X),(product Y)),((product X),(product Y)) #) is strict NORMSTR
the carrier of ((product X),(product Y)) is non empty set
[: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):] is non empty set
0. ((product X),(product Y)) is V52(((product X),(product Y))) left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the ZeroF of ((product X),(product Y)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
K is non empty Relation-like the carrier of ((product X),(product Y)) -defined the carrier of (product (X ^ Y)) -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total Element of bool [: the carrier of ((product X),(product Y)), the carrier of (product (X ^ Y)):]
K . (0. ((product X),(product Y))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[: the carrier of ((product X),(product Y)), the carrier of (product <*(product X),(product Y)*>):] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)), the carrier of (product <*(product X),(product Y)*>):] is non empty set
v is non empty Relation-like the carrier of ((product X),(product Y)) -defined the carrier of (product <*(product X),(product Y)*>) -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total Element of bool [: the carrier of ((product X),(product Y)), the carrier of (product <*(product X),(product Y)*>):]
v . (0. ((product X),(product Y))) is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
rng v is non empty Element of bool the carrier of (product <*(product X),(product Y)*>)
bool the carrier of (product <*(product X),(product Y)*>) is non empty set
[: the carrier of (product <*(product X),(product Y)*>), the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [: the carrier of (product <*(product X),(product Y)*>), the carrier of ((product X),(product Y)):] is non empty set
v " is Relation-like Function-like set
dom (v ") is set
rng (v ") is set
dom v is non empty Element of bool the carrier of ((product X),(product Y))
bool the carrier of ((product X),(product Y)) is non empty set
r is non empty Relation-like the carrier of (product <*(product X),(product Y)*>) -defined the carrier of ((product X),(product Y)) -valued Function-like V26( the carrier of (product <*(product X),(product Y)*>)) quasi_total Element of bool [: the carrier of (product <*(product X),(product Y)*>), the carrier of ((product X),(product Y)):]
K * r is non empty Relation-like the carrier of (product <*(product X),(product Y)*>) -defined the carrier of (product (X ^ Y)) -valued Function-like V26( the carrier of (product <*(product X),(product Y)*>)) quasi_total Element of bool [: the carrier of (product <*(product X),(product Y)*>), the carrier of (product (X ^ Y)):]
yy is non empty Relation-like the carrier of (product <*(product X),(product Y)*>) -defined the carrier of (product (X ^ Y)) -valued Function-like V26( the carrier of (product <*(product X),(product Y)*>)) quasi_total Element of bool [: the carrier of (product <*(product X),(product Y)*>), the carrier of (product (X ^ Y)):]
yy . (0. (product <*(product X),(product Y)*>)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
rng K is non empty Element of bool the carrier of (product (X ^ Y))
bool the carrier of (product (X ^ Y)) is non empty set
rng yy is non empty Element of bool the carrier of (product (X ^ Y))
x1 is left_complementable right_complementable complementable Element of the carrier of (product X)
y1 is left_complementable right_complementable complementable Element of the carrier of (product Y)
<*x1,y1*> is non empty Relation-like NAT -defined Function-like V33() V40(2) FinSequence-like FinSubsequence-like countable set
yy . <*x1,y1*> is set
K . (x1,y1) is set
[x1,y1] is set
K . [x1,y1] is set
xx2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
yy2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
xx2 ^ yy2 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
v . (x1,y1) is set
v . [x1,y1] is set
[x1,y1] is Element of [: the carrier of (product X), the carrier of (product Y):]
v . [x1,y1] is set
r . (v . [x1,y1]) is set
K . (r . (v . [x1,y1])) is set
x1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
y1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
x1 + y1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
the addF of (product <*(product X),(product Y)*>) is non empty Relation-like [: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):] -defined the carrier of (product <*(product X),(product Y)*>) -valued Function-like V26([: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):]) quasi_total Element of bool [:[: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):]
[: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):] is non empty Relation-like set
[:[: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*(product X),(product Y)*>), the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):] is non empty set
the addF of (product <*(product X),(product Y)*>) . (x1,y1) is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
[x1,y1] is set
the addF of (product <*(product X),(product Y)*>) . [x1,y1] is set
yy . (x1 + y1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
yy . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
yy . y1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(yy . x1) + (yy . y1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) is non empty Relation-like [: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ Y)), the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the addF of (product (X ^ Y)) . ((yy . x1),(yy . y1)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(yy . x1),(yy . y1)] is set
the addF of (product (X ^ Y)) . [(yy . x1),(yy . y1)] is set
xx2 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v . xx2 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
yy2 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v . yy2 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
xx2 + yy2 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the addF of ((product X),(product Y)) is non empty Relation-like [: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product Y)), the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the addF of ((product X),(product Y)) . (xx2,yy2) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
[xx2,yy2] is set
the addF of ((product X),(product Y)) . [xx2,yy2] is set
r . x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
r . y1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v . (xx2 + yy2) is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
r . (v . (xx2 + yy2)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
dom r is non empty Element of bool the carrier of (product <*(product X),(product Y)*>)
K . (xx2 + yy2) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . xx2 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . yy2 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
(K . xx2) + (K . yy2) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) . ((K . xx2),(K . yy2)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[(K . xx2),(K . yy2)] is set
the addF of (product (X ^ Y)) . [(K . xx2),(K . yy2)] is set
(K * r) . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . (r . y1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
((K * r) . x1) + (K . (r . y1)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the addF of (product (X ^ Y)) . (((K * r) . x1),(K . (r . y1))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
[((K * r) . x1),(K . (r . y1))] is set
the addF of (product (X ^ Y)) . [((K * r) . x1),(K . (r . y1))] is set
x1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
yy . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
y1 is V11() real ext-real Element of REAL
y1 * x1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
the Mult of (product <*(product X),(product Y)*>) is non empty Relation-like [:REAL, the carrier of (product <*(product X),(product Y)*>):] -defined the carrier of (product <*(product X),(product Y)*>) -valued Function-like V26([:REAL, the carrier of (product <*(product X),(product Y)*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):]
[:REAL, the carrier of (product <*(product X),(product Y)*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*(product X),(product Y)*>):], the carrier of (product <*(product X),(product Y)*>):] is non empty set
the Mult of (product <*(product X),(product Y)*>) . (y1,x1) is set
[y1,x1] is set
the Mult of (product <*(product X),(product Y)*>) . [y1,x1] is set
yy . (y1 * x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
y1 * (yy . x1) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the Mult of (product (X ^ Y)) is non empty Relation-like [:REAL, the carrier of (product (X ^ Y)):] -defined the carrier of (product (X ^ Y)) -valued Function-like V26([:REAL, the carrier of (product (X ^ Y)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):]
[:REAL, the carrier of (product (X ^ Y)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (X ^ Y)):], the carrier of (product (X ^ Y)):] is non empty set
the Mult of (product (X ^ Y)) . (y1,(yy . x1)) is set
[y1,(yy . x1)] is set
the Mult of (product (X ^ Y)) . [y1,(yy . x1)] is set
xx2 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v . xx2 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
r . x1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
dom r is non empty Element of bool the carrier of (product <*(product X),(product Y)*>)
y1 * xx2 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
the Mult of ((product X),(product Y)) is non empty Relation-like [:REAL, the carrier of ((product X),(product Y)):] -defined the carrier of ((product X),(product Y)) -valued Function-like V26([:REAL, the carrier of ((product X),(product Y)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):]
[:REAL, the carrier of ((product X),(product Y)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),(product Y)):], the carrier of ((product X),(product Y)):] is non empty set
the Mult of ((product X),(product Y)) . (y1,xx2) is set
[y1,xx2] is set
the Mult of ((product X),(product Y)) . [y1,xx2] is set
v . (y1 * xx2) is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
r . (v . (y1 * xx2)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
K . (r . (v . (y1 * xx2))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . (y1 * xx2) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . xx2 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
y1 * (K . xx2) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
the Mult of (product (X ^ Y)) . (y1,(K . xx2)) is set
[y1,(K . xx2)] is set
the Mult of (product (X ^ Y)) . [y1,(K . xx2)] is set
dom r is non empty Element of bool the carrier of (product <*(product X),(product Y)*>)
r . (v . (0. ((product X),(product Y)))) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
K . (r . (v . (0. ((product X),(product Y))))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
x1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
yy . x1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
||.(yy . x1).|| is V11() real ext-real non negative Element of REAL
the normF of (product (X ^ Y)) is non empty Relation-like the carrier of (product (X ^ Y)) -defined REAL -valued Function-like V26( the carrier of (product (X ^ Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product (X ^ Y)),REAL:]
[: the carrier of (product (X ^ Y)),REAL:] is non empty Relation-like set
bool [: the carrier of (product (X ^ Y)),REAL:] is non empty set
the normF of (product (X ^ Y)) . (yy . x1) is V11() real ext-real Element of REAL
||.x1.|| is V11() real ext-real non negative Element of REAL
the normF of (product <*(product X),(product Y)*>) is non empty Relation-like the carrier of (product <*(product X),(product Y)*>) -defined REAL -valued Function-like V26( the carrier of (product <*(product X),(product Y)*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*(product X),(product Y)*>),REAL:]
[: the carrier of (product <*(product X),(product Y)*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*(product X),(product Y)*>),REAL:] is non empty set
the normF of (product <*(product X),(product Y)*>) . x1 is V11() real ext-real Element of REAL
y1 is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
v . y1 is left_complementable right_complementable complementable Element of the carrier of (product <*(product X),(product Y)*>)
dom r is non empty Element of bool the carrier of (product <*(product X),(product Y)*>)
r . (v . y1) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product Y))
K . (r . (v . y1)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
K . y1 is left_complementable right_complementable complementable Element of the carrier of (product (X ^ Y))
||.y1.|| is V11() real ext-real non negative Element of REAL
the normF of ((product X),(product Y)) is non empty Relation-like the carrier of ((product X),(product Y)) -defined REAL -valued Function-like V26( the carrier of ((product X),(product Y))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((product X),(product Y)),REAL:]
[: the carrier of ((product X),(product Y)),REAL:] is non empty Relation-like set
bool [: the carrier of ((product X),(product Y)),REAL:] is non empty set
the normF of ((product X),(product Y)) . y1 is V11() real ext-real Element of REAL
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
(X,Y) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y) #) is non empty strict RLSStruct
the carrier of (X,Y) is non empty set
<*Y*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product <*Y*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*Y*>) is non empty functional with_common_domain product-like set
zeros <*Y*> is Relation-like NAT -defined Function-like carr <*Y*> -compatible Element of product (carr <*Y*>)
addop <*Y*> is Relation-like Function-like BinOps of carr <*Y*>
[:(addop <*Y*>):] is non empty Relation-like [:(product (carr <*Y*>)),(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:(product (carr <*Y*>)),(product (carr <*Y*>)):]) quasi_total Element of bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:(product (carr <*Y*>)),(product (carr <*Y*>)):] is non empty Relation-like set
[:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
multop <*Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*Y*>
[:(multop <*Y*>):] is non empty Relation-like [:REAL,(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:REAL,(product (carr <*Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:REAL,(product (carr <*Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
RLSStruct(# (product (carr <*Y*>)),(zeros <*Y*>),[:(addop <*Y*>):],[:(multop <*Y*>):] #) is non empty strict RLSStruct
(X,(product <*Y*>)) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (product <*Y*>) is non empty set
[: the carrier of X, the carrier of (product <*Y*>):] is non empty Relation-like set
(X,(product <*Y*>)) is Element of [: the carrier of X, the carrier of (product <*Y*>):]
0. (product <*Y*>) is V52( product <*Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the ZeroF of (product <*Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[(0. X),(0. (product <*Y*>))] is Element of [: the carrier of X, the carrier of (product <*Y*>):]
(X,(product <*Y*>)) is non empty Relation-like [:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:] -defined [: the carrier of X, the carrier of (product <*Y*>):] -valued Function-like V26([:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:]
[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty set
(X,(product <*Y*>)) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:] -defined [: the carrier of X, the carrier of (product <*Y*>):] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:]
[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty set
RLSStruct(# [: the carrier of X, the carrier of (product <*Y*>):],(X,(product <*Y*>)),(X,(product <*Y*>)),(X,(product <*Y*>)) #) is non empty strict RLSStruct
the carrier of (X,(product <*Y*>)) is non empty set
[: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):] is non empty set
[:NAT, the carrier of Y:] is non empty Relation-like set
bool [:NAT, the carrier of Y:] is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
0. (X,(product <*Y*>)) is V52((X,(product <*Y*>))) left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the ZeroF of (X,(product <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
[: the carrier of Y, the carrier of (product <*Y*>):] is non empty Relation-like set
bool [: the carrier of Y, the carrier of (product <*Y*>):] is non empty set
I is non empty Relation-like the carrier of Y -defined the carrier of (product <*Y*>) -valued Function-like V26( the carrier of Y) quasi_total Element of bool [: the carrier of Y, the carrier of (product <*Y*>):]
I . (0. Y) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
J is set
K is set
<*K*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[J,<*K*>] is set
K is left_complementable right_complementable complementable Element of the carrier of Y
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
<*K*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):] is non empty set
J is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):]
K is non empty Relation-like the carrier of (X,Y) -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):]
K . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K is set
v is set
K . K is set
K . v is set
r is set
yy is set
[r,yy] is set
x1 is set
y1 is set
[x1,y1] is set
<*yy*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[r,<*yy*>] is set
K . (r,yy) is set
K . [r,yy] is set
K . (x1,y1) is set
K . [x1,y1] is set
<*y1*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[x1,<*y1*>] is set
K is set
v is set
r is set
[v,r] is set
rng I is non empty Element of bool the carrier of (product <*Y*>)
bool the carrier of (product <*Y*>) is non empty set
yy is set
I . yy is set
[v,yy] is set
<*yy*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
K . (v,yy) is set
K . [v,yy] is set
x1 is Element of [: the carrier of X, the carrier of Y:]
K . x1 is set
rng K is non empty Element of bool the carrier of (X,(product <*Y*>))
bool the carrier of (X,(product <*Y*>)) is non empty set
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
K . (K,v) is set
[K,v] is set
K . [K,v] is set
<*v*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[K,<*v*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K + v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (K,v) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[K,v] is set
the addF of (X,Y) . [K,v] is set
K . (K + v) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K . K is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K . v is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
(K . K) + (K . v) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the addF of (X,(product <*Y*>)) is non empty Relation-like [: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):]) quasi_total Element of bool [:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):]
[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
[:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty set
the addF of (X,(product <*Y*>)) . ((K . K),(K . v)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
[(K . K),(K . v)] is set
the addF of (X,(product <*Y*>)) . [(K . K),(K . v)] is set
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is left_complementable right_complementable complementable Element of the carrier of X
y1 is left_complementable right_complementable complementable Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
r + x1 is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (r,x1) is left_complementable right_complementable complementable Element of the carrier of X
[r,x1] is set
the addF of X . [r,x1] is set
yy + y1 is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (yy,y1) is left_complementable right_complementable complementable Element of the carrier of Y
[yy,y1] is set
the addF of Y . [yy,y1] is set
K . ((r + x1),(yy + y1)) is set
[(r + x1),(yy + y1)] is set
K . [(r + x1),(yy + y1)] is set
<*(yy + y1)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[(r + x1),<*(yy + y1)*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K . (r,yy) is set
[r,yy] is set
K . [r,yy] is set
K . (x1,y1) is set
[x1,y1] is set
K . [x1,y1] is set
<*yy*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[r,<*yy*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
<*y1*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[x1,<*y1*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
I . yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . y1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . (yy + y1) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
xx2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
yy2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
xx2 + yy2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the addF of (product <*Y*>) is non empty Relation-like [: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([: the carrier of (product <*Y*>), the carrier of (product <*Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the addF of (product <*Y*>) . (xx2,yy2) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[xx2,yy2] is set
the addF of (product <*Y*>) . [xx2,yy2] is set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . K is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
v is V11() real ext-real Element of REAL
v * K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (v,K) is set
[v,K] is set
the Mult of (X,Y) . [v,K] is set
K . (v * K) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
v * (K . K) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the Mult of (X,(product <*Y*>)) is non empty Relation-like [:REAL, the carrier of (X,(product <*Y*>)):] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([:REAL, the carrier of (X,(product <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):]
[:REAL, the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty set
the Mult of (X,(product <*Y*>)) . (v,(K . K)) is set
[v,(K . K)] is set
the Mult of (X,(product <*Y*>)) . [v,(K . K)] is set
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
v * r is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (v,r) is set
[v,r] is set
the Mult of X . [v,r] is set
v * yy is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (v,yy) is set
[v,yy] is set
the Mult of Y . [v,yy] is set
K . ((v * r),(v * yy)) is set
[(v * r),(v * yy)] is set
K . [(v * r),(v * yy)] is set
<*(v * yy)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[(v * r),<*(v * yy)*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K . (r,yy) is set
[r,yy] is set
K . [r,yy] is set
<*yy*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[r,<*yy*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
I . yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . (v * yy) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
x1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
v * x1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the Mult of (product <*Y*>) is non empty Relation-like [:REAL, the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([:REAL, the carrier of (product <*Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[:REAL, the carrier of (product <*Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the Mult of (product <*Y*>) . (v,x1) is set
[v,x1] is set
the Mult of (product <*Y*>) . [v,x1] is set
<*(0. Y)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
K . ((0. X),(0. Y)) is set
[(0. X),(0. Y)] is set
K . [(0. X),(0. Y)] is set
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product X is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr X is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr X) is non empty functional with_common_domain product-like set
zeros X is Relation-like NAT -defined Function-like carr X -compatible Element of product (carr X)
addop X is Relation-like Function-like BinOps of carr X
[:(addop X):] is non empty Relation-like [:(product (carr X)),(product (carr X)):] -defined product (carr X) -valued Function-like V26([:(product (carr X)),(product (carr X)):]) quasi_total Element of bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):]
[:(product (carr X)),(product (carr X)):] is non empty Relation-like set
[:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:(product (carr X)),(product (carr X)):],(product (carr X)):] is non empty set
multop X is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr X
[:(multop X):] is non empty Relation-like [:REAL,(product (carr X)):] -defined product (carr X) -valued Function-like V26([:REAL,(product (carr X)):]) quasi_total Element of bool [:[:REAL,(product (carr X)):],(product (carr X)):]
[:REAL,(product (carr X)):] is non empty Relation-like set
[:[:REAL,(product (carr X)):],(product (carr X)):] is non empty Relation-like set
bool [:[:REAL,(product (carr X)):],(product (carr X)):] is non empty set
RLSStruct(# (product (carr X)),(zeros X),[:(addop X):],[:(multop X):] #) is non empty strict RLSStruct
the carrier of (product X) is non empty set
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
((product X),Y) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of Y is non empty set
[: the carrier of (product X), the carrier of Y:] is non empty Relation-like set
((product X),Y) is Element of [: the carrier of (product X), the carrier of Y:]
0. (product X) is V52( product X) left_complementable right_complementable complementable Element of the carrier of (product X)
the ZeroF of (product X) is left_complementable right_complementable complementable Element of the carrier of (product X)
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. (product X)),(0. Y)] is Element of [: the carrier of (product X), the carrier of Y:]
((product X),Y) is non empty Relation-like [:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:] -defined [: the carrier of (product X), the carrier of Y:] -valued Function-like V26([:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:]
[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty set
((product X),Y) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of Y:]:] -defined [: the carrier of (product X), the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:]
[:REAL,[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty set
RLSStruct(# [: the carrier of (product X), the carrier of Y:],((product X),Y),((product X),Y),((product X),Y) #) is non empty strict RLSStruct
the carrier of ((product X),Y) is non empty set
<*Y*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
X ^ <*Y*> is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding set
product (X ^ <*Y*>) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr (X ^ <*Y*>) is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr (X ^ <*Y*>)) is non empty functional with_common_domain product-like set
zeros (X ^ <*Y*>) is Relation-like NAT -defined Function-like carr (X ^ <*Y*>) -compatible Element of product (carr (X ^ <*Y*>))
addop (X ^ <*Y*>) is Relation-like Function-like BinOps of carr (X ^ <*Y*>)
[:(addop (X ^ <*Y*>)):] is non empty Relation-like [:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):] -defined product (carr (X ^ <*Y*>)) -valued Function-like V26([:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):]) quasi_total Element of bool [:[:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):]
[:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):] is non empty Relation-like set
[:[:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):] is non empty Relation-like set
bool [:[:(product (carr (X ^ <*Y*>))),(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):] is non empty set
multop (X ^ <*Y*>) is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr (X ^ <*Y*>)
[:(multop (X ^ <*Y*>)):] is non empty Relation-like [:REAL,(product (carr (X ^ <*Y*>))):] -defined product (carr (X ^ <*Y*>)) -valued Function-like V26([:REAL,(product (carr (X ^ <*Y*>))):]) quasi_total Element of bool [:[:REAL,(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):]
[:REAL,(product (carr (X ^ <*Y*>))):] is non empty Relation-like set
[:[:REAL,(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):] is non empty Relation-like set
bool [:[:REAL,(product (carr (X ^ <*Y*>))):],(product (carr (X ^ <*Y*>))):] is non empty set
RLSStruct(# (product (carr (X ^ <*Y*>))),(zeros (X ^ <*Y*>)),[:(addop (X ^ <*Y*>)):],[:(multop (X ^ <*Y*>)):] #) is non empty strict RLSStruct
the carrier of (product (X ^ <*Y*>)) is non empty set
[: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):] is non empty set
0. ((product X),Y) is V52(((product X),Y)) left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the ZeroF of ((product X),Y) is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
0. (product (X ^ <*Y*>)) is V52( product (X ^ <*Y*>)) left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the ZeroF of (product (X ^ <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
product <*Y*> is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
carr <*Y*> is non empty Relation-like non-empty NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
product (carr <*Y*>) is non empty functional with_common_domain product-like set
zeros <*Y*> is Relation-like NAT -defined Function-like carr <*Y*> -compatible Element of product (carr <*Y*>)
addop <*Y*> is Relation-like Function-like BinOps of carr <*Y*>
[:(addop <*Y*>):] is non empty Relation-like [:(product (carr <*Y*>)),(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:(product (carr <*Y*>)),(product (carr <*Y*>)):]) quasi_total Element of bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:(product (carr <*Y*>)),(product (carr <*Y*>)):] is non empty Relation-like set
[:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:(product (carr <*Y*>)),(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
multop <*Y*> is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable MultOps of REAL , carr <*Y*>
[:(multop <*Y*>):] is non empty Relation-like [:REAL,(product (carr <*Y*>)):] -defined product (carr <*Y*>) -valued Function-like V26([:REAL,(product (carr <*Y*>)):]) quasi_total Element of bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):]
[:REAL,(product (carr <*Y*>)):] is non empty Relation-like set
[:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty Relation-like set
bool [:[:REAL,(product (carr <*Y*>)):],(product (carr <*Y*>)):] is non empty set
RLSStruct(# (product (carr <*Y*>)),(zeros <*Y*>),[:(addop <*Y*>):],[:(multop <*Y*>):] #) is non empty strict RLSStruct
((product X),(product <*Y*>)) is non empty left_complementable right_complementable complementable strict Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() RLSStruct
the carrier of (product <*Y*>) is non empty set
[: the carrier of (product X), the carrier of (product <*Y*>):] is non empty Relation-like set
((product X),(product <*Y*>)) is Element of [: the carrier of (product X), the carrier of (product <*Y*>):]
0. (product <*Y*>) is V52( product <*Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the ZeroF of (product <*Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[(0. (product X)),(0. (product <*Y*>))] is Element of [: the carrier of (product X), the carrier of (product <*Y*>):]
((product X),(product <*Y*>)) is non empty Relation-like [:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:] -defined [: the carrier of (product X), the carrier of (product <*Y*>):] -valued Function-like V26([:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:]
[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty set
((product X),(product <*Y*>)) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:] -defined [: the carrier of (product X), the carrier of (product <*Y*>):] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:]
[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty set
RLSStruct(# [: the carrier of (product X), the carrier of (product <*Y*>):],((product X),(product <*Y*>)),((product X),(product <*Y*>)),((product X),(product <*Y*>)) #) is non empty strict RLSStruct
the carrier of ((product X),(product <*Y*>)) is non empty set
[: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):] is non empty set
[:NAT, the carrier of Y:] is non empty Relation-like set
bool [:NAT, the carrier of Y:] is non empty set
0. ((product X),(product <*Y*>)) is V52(((product X),(product <*Y*>))) left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the ZeroF of ((product X),(product <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
I is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):]
I . (0. ((product X),Y)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
[: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty set
J is non empty Relation-like the carrier of ((product X),(product <*Y*>)) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),(product <*Y*>))) quasi_total Element of bool [: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):]
J . (0. ((product X),(product <*Y*>))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J * I is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):]
K is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):]
K . (0. ((product X),Y)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
rng J is non empty Element of bool the carrier of (product (X ^ <*Y*>))
bool the carrier of (product (X ^ <*Y*>)) is non empty set
rng I is non empty Element of bool the carrier of ((product X),(product <*Y*>))
bool the carrier of ((product X),(product <*Y*>)) is non empty set
rng (J * I) is non empty Element of bool the carrier of (product (X ^ <*Y*>))
J .: the carrier of ((product X),(product <*Y*>)) is Element of bool the carrier of (product (X ^ <*Y*>))
v is left_complementable right_complementable complementable Element of the carrier of (product X)
r is left_complementable right_complementable complementable Element of the carrier of Y
<*r*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
K . (v,r) is set
[v,r] is set
K . [v,r] is set
I . (v,r) is set
I . [v,r] is set
[v,<*r*>] is Element of [: the carrier of (product X),(bool [:NAT, the carrier of Y:]):]
[: the carrier of (product X),(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
[v,r] is Element of [: the carrier of (product X), the carrier of Y:]
yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
J . (v,yy) is set
[v,yy] is set
J . [v,yy] is set
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
x1 ^ y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I . [v,r] is set
J . (I . [v,r]) is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
r is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
v + r is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the addF of ((product X),Y) is non empty Relation-like [: the carrier of ((product X),Y), the carrier of ((product X),Y):] -defined the carrier of ((product X),Y) -valued Function-like V26([: the carrier of ((product X),Y), the carrier of ((product X),Y):]) quasi_total Element of bool [:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):]
[: the carrier of ((product X),Y), the carrier of ((product X),Y):] is non empty Relation-like set
[:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty Relation-like set
bool [:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty set
the addF of ((product X),Y) . (v,r) is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
[v,r] is set
the addF of ((product X),Y) . [v,r] is set
K . (v + r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
K . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
K . r is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
(K . v) + (K . r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) is non empty Relation-like [: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):] -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26([: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):]
[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty set
the addF of (product (X ^ <*Y*>)) . ((K . v),(K . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(K . v),(K . r)] is set
the addF of (product (X ^ <*Y*>)) . [(K . v),(K . r)] is set
I . (v + r) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
J . (I . (v + r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
I . v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
I . r is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
(I . v) + (I . r) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the addF of ((product X),(product <*Y*>)) is non empty Relation-like [: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):] -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26([: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):]
[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty set
the addF of ((product X),(product <*Y*>)) . ((I . v),(I . r)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
[(I . v),(I . r)] is set
the addF of ((product X),(product <*Y*>)) . [(I . v),(I . r)] is set
J . ((I . v) + (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
(J . (I . v)) + (J . (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) . ((J . (I . v)),(J . (I . r))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(J . (I . v)),(J . (I . r))] is set
the addF of (product (X ^ <*Y*>)) . [(J . (I . v)),(J . (I . r))] is set
(K . v) + (J . (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) . ((K . v),(J . (I . r))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(K . v),(J . (I . r))] is set
the addF of (product (X ^ <*Y*>)) . [(K . v),(J . (I . r))] is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r is V11() real ext-real Element of REAL
r * v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the Mult of ((product X),Y) is non empty Relation-like [:REAL, the carrier of ((product X),Y):] -defined the carrier of ((product X),Y) -valued Function-like V26([:REAL, the carrier of ((product X),Y):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):]
[:REAL, the carrier of ((product X),Y):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty set
the Mult of ((product X),Y) . (r,v) is set
[r,v] is set
the Mult of ((product X),Y) . [r,v] is set
K . (r * v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r * (K . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the Mult of (product (X ^ <*Y*>)) is non empty Relation-like [:REAL, the carrier of (product (X ^ <*Y*>)):] -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26([:REAL, the carrier of (product (X ^ <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):]
[:REAL, the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty set
the Mult of (product (X ^ <*Y*>)) . (r,(K . v)) is set
[r,(K . v)] is set
the Mult of (product (X ^ <*Y*>)) . [r,(K . v)] is set
I . (r * v) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
J . (I . (r * v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
I . v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
r * (I . v) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the Mult of ((product X),(product <*Y*>)) is non empty Relation-like [:REAL, the carrier of ((product X),(product <*Y*>)):] -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26([:REAL, the carrier of ((product X),(product <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):]
[:REAL, the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty set
the Mult of ((product X),(product <*Y*>)) . (r,(I . v)) is set
[r,(I . v)] is set
the Mult of ((product X),(product <*Y*>)) . [r,(I . v)] is set
J . (r * (I . v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r * (J . (I . v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the Mult of (product (X ^ <*Y*>)) . (r,(J . (I . v))) is set
[r,(J . (I . v))] is set
the Mult of (product (X ^ <*Y*>)) . [r,(J . (I . v))] is set
X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
(X,Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of X is non empty set
the carrier of Y is non empty set
[: the carrier of X, the carrier of Y:] is non empty Relation-like set
(X,Y) is Element of [: the carrier of X, the carrier of Y:]
0. X is V52(X) left_complementable right_complementable complementable Element of the carrier of X
the ZeroF of X is left_complementable right_complementable complementable Element of the carrier of X
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. X),(0. Y)] is Element of [: the carrier of X, the carrier of Y:]
(X,Y) is non empty Relation-like [:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of Y:],[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of Y:]:] -defined [: the carrier of X, the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:]
[:REAL,[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of Y:]:],[: the carrier of X, the carrier of Y:]:] is non empty set
(X,Y) is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of Y:],REAL:]
[:[: the carrier of X, the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of Y:],(X,Y),(X,Y),(X,Y),(X,Y) #) is strict NORMSTR
the carrier of (X,Y) is non empty set
<*Y*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product <*Y*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
(X,(product <*Y*>)) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*Y*>) is non empty set
[: the carrier of X, the carrier of (product <*Y*>):] is non empty Relation-like set
(X,(product <*Y*>)) is Element of [: the carrier of X, the carrier of (product <*Y*>):]
0. (product <*Y*>) is V52( product <*Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the ZeroF of (product <*Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[(0. X),(0. (product <*Y*>))] is Element of [: the carrier of X, the carrier of (product <*Y*>):]
(X,(product <*Y*>)) is non empty Relation-like [:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:] -defined [: the carrier of X, the carrier of (product <*Y*>):] -valued Function-like V26([:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:]
[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:[: the carrier of X, the carrier of (product <*Y*>):],[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty set
(X,(product <*Y*>)) is non empty Relation-like [:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:] -defined [: the carrier of X, the carrier of (product <*Y*>):] -valued Function-like V26([:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:]
[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of X, the carrier of (product <*Y*>):]:],[: the carrier of X, the carrier of (product <*Y*>):]:] is non empty set
(X,(product <*Y*>)) is non empty Relation-like [: the carrier of X, the carrier of (product <*Y*>):] -defined REAL -valued Function-like V26([: the carrier of X, the carrier of (product <*Y*>):]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of X, the carrier of (product <*Y*>):],REAL:]
[:[: the carrier of X, the carrier of (product <*Y*>):],REAL:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of (product <*Y*>):],REAL:] is non empty set
NORMSTR(# [: the carrier of X, the carrier of (product <*Y*>):],(X,(product <*Y*>)),(X,(product <*Y*>)),(X,(product <*Y*>)),(X,(product <*Y*>)) #) is strict NORMSTR
the carrier of (X,(product <*Y*>)) is non empty set
[: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):] is non empty set
[:NAT, the carrier of Y:] is non empty Relation-like set
bool [:NAT, the carrier of Y:] is non empty set
0. (X,Y) is V52((X,Y)) left_complementable right_complementable complementable Element of the carrier of (X,Y)
the ZeroF of (X,Y) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
0. (X,(product <*Y*>)) is V52((X,(product <*Y*>))) left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the ZeroF of (X,(product <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
[: the carrier of Y, the carrier of (product <*Y*>):] is non empty Relation-like set
bool [: the carrier of Y, the carrier of (product <*Y*>):] is non empty set
I is non empty Relation-like the carrier of Y -defined the carrier of (product <*Y*>) -valued Function-like V26( the carrier of Y) quasi_total Element of bool [: the carrier of Y, the carrier of (product <*Y*>):]
I . (0. Y) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
J is set
K is set
<*K*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[J,<*K*>] is set
K is left_complementable right_complementable complementable Element of the carrier of Y
I . K is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
<*K*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):] is non empty set
J is non empty Relation-like [: the carrier of X, the carrier of Y:] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([: the carrier of X, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of X, the carrier of Y:], the carrier of (X,(product <*Y*>)):]
K is non empty Relation-like the carrier of (X,Y) -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26( the carrier of (X,Y)) quasi_total Element of bool [: the carrier of (X,Y), the carrier of (X,(product <*Y*>)):]
K . (0. (X,Y)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K is set
v is set
K . K is set
K . v is set
r is set
yy is set
[r,yy] is set
x1 is set
y1 is set
[x1,y1] is set
<*yy*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[r,<*yy*>] is set
K . (r,yy) is set
K . [r,yy] is set
K . (x1,y1) is set
K . [x1,y1] is set
<*y1*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[x1,<*y1*>] is set
K is set
v is set
r is set
[v,r] is set
rng I is non empty Element of bool the carrier of (product <*Y*>)
bool the carrier of (product <*Y*>) is non empty set
yy is set
I . yy is set
<*yy*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable set
[v,yy] is set
K . (v,yy) is set
K . [v,yy] is set
x1 is Element of [: the carrier of X, the carrier of Y:]
K . x1 is set
rng K is non empty Element of bool the carrier of (X,(product <*Y*>))
bool the carrier of (X,(product <*Y*>)) is non empty set
K is left_complementable right_complementable complementable Element of the carrier of X
v is left_complementable right_complementable complementable Element of the carrier of Y
K . (K,v) is set
[K,v] is set
K . [K,v] is set
<*v*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[K,<*v*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K + v is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the addF of (X,Y) is non empty Relation-like [: the carrier of (X,Y), the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([: the carrier of (X,Y), the carrier of (X,Y):]) quasi_total Element of bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):]
[: the carrier of (X,Y), the carrier of (X,Y):] is non empty Relation-like set
[:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[: the carrier of (X,Y), the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the addF of (X,Y) . (K,v) is left_complementable right_complementable complementable Element of the carrier of (X,Y)
[K,v] is set
the addF of (X,Y) . [K,v] is set
K . (K + v) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K . K is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
K . v is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
(K . K) + (K . v) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the addF of (X,(product <*Y*>)) is non empty Relation-like [: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):]) quasi_total Element of bool [:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):]
[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
[:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of (X,(product <*Y*>)), the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty set
the addF of (X,(product <*Y*>)) . ((K . K),(K . v)) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
[(K . K),(K . v)] is set
the addF of (X,(product <*Y*>)) . [(K . K),(K . v)] is set
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
x1 is left_complementable right_complementable complementable Element of the carrier of X
y1 is left_complementable right_complementable complementable Element of the carrier of Y
[x1,y1] is Element of [: the carrier of X, the carrier of Y:]
r + x1 is left_complementable right_complementable complementable Element of the carrier of X
the addF of X is non empty Relation-like [: the carrier of X, the carrier of X:] -defined the carrier of X -valued Function-like V26([: the carrier of X, the carrier of X:]) quasi_total Element of bool [:[: the carrier of X, the carrier of X:], the carrier of X:]
[: the carrier of X, the carrier of X:] is non empty Relation-like set
[:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[: the carrier of X, the carrier of X:], the carrier of X:] is non empty set
the addF of X . (r,x1) is left_complementable right_complementable complementable Element of the carrier of X
[r,x1] is set
the addF of X . [r,x1] is set
yy + y1 is left_complementable right_complementable complementable Element of the carrier of Y
the addF of Y is non empty Relation-like [: the carrier of Y, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([: the carrier of Y, the carrier of Y:]) quasi_total Element of bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:]
[: the carrier of Y, the carrier of Y:] is non empty Relation-like set
[:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[: the carrier of Y, the carrier of Y:], the carrier of Y:] is non empty set
the addF of Y . (yy,y1) is left_complementable right_complementable complementable Element of the carrier of Y
[yy,y1] is set
the addF of Y . [yy,y1] is set
K . ((r + x1),(yy + y1)) is set
[(r + x1),(yy + y1)] is set
K . [(r + x1),(yy + y1)] is set
<*(yy + y1)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[(r + x1),<*(yy + y1)*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K . (r,yy) is set
[r,yy] is set
K . [r,yy] is set
K . (x1,y1) is set
[x1,y1] is set
K . [x1,y1] is set
<*yy*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[r,<*yy*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
<*y1*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[x1,<*y1*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
I . yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . y1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . (yy + y1) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
xx2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
yy2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
xx2 + yy2 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the addF of (product <*Y*>) is non empty Relation-like [: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([: the carrier of (product <*Y*>), the carrier of (product <*Y*>):]) quasi_total Element of bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):] is non empty Relation-like set
[:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[: the carrier of (product <*Y*>), the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the addF of (product <*Y*>) . (xx2,yy2) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[xx2,yy2] is set
the addF of (product <*Y*>) . [xx2,yy2] is set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . K is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
v is V11() real ext-real Element of REAL
v * K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
the Mult of (X,Y) is non empty Relation-like [:REAL, the carrier of (X,Y):] -defined the carrier of (X,Y) -valued Function-like V26([:REAL, the carrier of (X,Y):]) quasi_total Element of bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):]
[:REAL, the carrier of (X,Y):] is non empty Relation-like set
[:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,Y):], the carrier of (X,Y):] is non empty set
the Mult of (X,Y) . (v,K) is set
[v,K] is set
the Mult of (X,Y) . [v,K] is set
K . (v * K) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
v * (K . K) is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
the Mult of (X,(product <*Y*>)) is non empty Relation-like [:REAL, the carrier of (X,(product <*Y*>)):] -defined the carrier of (X,(product <*Y*>)) -valued Function-like V26([:REAL, the carrier of (X,(product <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):]
[:REAL, the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (X,(product <*Y*>)):], the carrier of (X,(product <*Y*>)):] is non empty set
the Mult of (X,(product <*Y*>)) . (v,(K . K)) is set
[v,(K . K)] is set
the Mult of (X,(product <*Y*>)) . [v,(K . K)] is set
r is left_complementable right_complementable complementable Element of the carrier of X
yy is left_complementable right_complementable complementable Element of the carrier of Y
[r,yy] is Element of [: the carrier of X, the carrier of Y:]
v * r is left_complementable right_complementable complementable Element of the carrier of X
the Mult of X is non empty Relation-like [:REAL, the carrier of X:] -defined the carrier of X -valued Function-like V26([:REAL, the carrier of X:]) quasi_total Element of bool [:[:REAL, the carrier of X:], the carrier of X:]
[:REAL, the carrier of X:] is non empty Relation-like set
[:[:REAL, the carrier of X:], the carrier of X:] is non empty Relation-like set
bool [:[:REAL, the carrier of X:], the carrier of X:] is non empty set
the Mult of X . (v,r) is set
[v,r] is set
the Mult of X . [v,r] is set
v * yy is left_complementable right_complementable complementable Element of the carrier of Y
the Mult of Y is non empty Relation-like [:REAL, the carrier of Y:] -defined the carrier of Y -valued Function-like V26([:REAL, the carrier of Y:]) quasi_total Element of bool [:[:REAL, the carrier of Y:], the carrier of Y:]
[:REAL, the carrier of Y:] is non empty Relation-like set
[:[:REAL, the carrier of Y:], the carrier of Y:] is non empty Relation-like set
bool [:[:REAL, the carrier of Y:], the carrier of Y:] is non empty set
the Mult of Y . (v,yy) is set
[v,yy] is set
the Mult of Y . [v,yy] is set
K . ((v * r),(v * yy)) is set
[(v * r),(v * yy)] is set
K . [(v * r),(v * yy)] is set
<*(v * yy)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[(v * r),<*(v * yy)*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
K . (r,yy) is set
[r,yy] is set
K . [r,yy] is set
<*yy*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
[r,<*yy*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
I . yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
I . (v * yy) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
x1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
v * x1 is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the Mult of (product <*Y*>) is non empty Relation-like [:REAL, the carrier of (product <*Y*>):] -defined the carrier of (product <*Y*>) -valued Function-like V26([:REAL, the carrier of (product <*Y*>):]) quasi_total Element of bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):]
[:REAL, the carrier of (product <*Y*>):] is non empty Relation-like set
[:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product <*Y*>):], the carrier of (product <*Y*>):] is non empty set
the Mult of (product <*Y*>) . (v,x1) is set
[v,x1] is set
the Mult of (product <*Y*>) . [v,x1] is set
<*(0. Y)*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
K . ((0. X),(0. Y)) is set
[(0. X),(0. Y)] is set
K . [(0. X),(0. Y)] is set
K is left_complementable right_complementable complementable Element of the carrier of (X,Y)
K . K is left_complementable right_complementable complementable Element of the carrier of (X,(product <*Y*>))
||.(K . K).|| is V11() real ext-real non negative Element of REAL
the normF of (X,(product <*Y*>)) is non empty Relation-like the carrier of (X,(product <*Y*>)) -defined REAL -valued Function-like V26( the carrier of (X,(product <*Y*>))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,(product <*Y*>)),REAL:]
[: the carrier of (X,(product <*Y*>)),REAL:] is non empty Relation-like set
bool [: the carrier of (X,(product <*Y*>)),REAL:] is non empty set
the normF of (X,(product <*Y*>)) . (K . K) is V11() real ext-real Element of REAL
||.K.|| is V11() real ext-real non negative Element of REAL
the normF of (X,Y) is non empty Relation-like the carrier of (X,Y) -defined REAL -valued Function-like V26( the carrier of (X,Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (X,Y),REAL:]
[: the carrier of (X,Y),REAL:] is non empty Relation-like set
bool [: the carrier of (X,Y),REAL:] is non empty set
the normF of (X,Y) . K is V11() real ext-real Element of REAL
v is left_complementable right_complementable complementable Element of the carrier of X
r is left_complementable right_complementable complementable Element of the carrier of Y
[v,r] is Element of [: the carrier of X, the carrier of Y:]
I . r is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
<*r*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of X is non empty Relation-like the carrier of X -defined REAL -valued Function-like V26( the carrier of X) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of X,REAL:]
[: the carrier of X,REAL:] is non empty Relation-like set
bool [: the carrier of X,REAL:] is non empty set
the normF of X . v is V11() real ext-real Element of REAL
||.r.|| is V11() real ext-real non negative Element of REAL
the normF of Y is non empty Relation-like the carrier of Y -defined REAL -valued Function-like V26( the carrier of Y) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of Y,REAL:]
[: the carrier of Y,REAL:] is non empty Relation-like set
bool [: the carrier of Y,REAL:] is non empty set
the normF of Y . r is V11() real ext-real Element of REAL
<*||.v.||,||.r.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
K . (v,r) is set
[v,r] is set
K . [v,r] is set
[v,<*r*>] is Element of [: the carrier of X,(bool [:NAT, the carrier of Y:]):]
[: the carrier of X,(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
||.yy.|| is V11() real ext-real non negative Element of REAL
the normF of (product <*Y*>) is non empty Relation-like the carrier of (product <*Y*>) -defined REAL -valued Function-like V26( the carrier of (product <*Y*>)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product <*Y*>),REAL:]
[: the carrier of (product <*Y*>),REAL:] is non empty Relation-like set
bool [: the carrier of (product <*Y*>),REAL:] is non empty set
the normF of (product <*Y*>) . yy is V11() real ext-real Element of REAL
<*||.v.||,||.yy.||*> is non empty Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
x1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.x1.| is V11() real ext-real non negative Element of REAL
sqr x1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr x1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr x1)) is V11() real ext-real Element of REAL
y1 is Relation-like NAT -defined REAL -valued Function-like V33() V40(2) FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable Element of REAL 2
|.y1.| is V11() real ext-real non negative Element of REAL
sqr y1 is Relation-like NAT -defined REAL -valued Function-like V33() FinSequence-like FinSubsequence-like complex-yielding V140() V141() countable FinSequence of REAL
Sum (sqr y1) is V11() real ext-real Element of REAL
sqrt (Sum (sqr y1)) is V11() real ext-real Element of REAL
X is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product X is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product X) is non empty set
Y is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive RealNormSpace-like NORMSTR
((product X),Y) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of Y is non empty set
[: the carrier of (product X), the carrier of Y:] is non empty Relation-like set
((product X),Y) is Element of [: the carrier of (product X), the carrier of Y:]
0. (product X) is V52( product X) left_complementable right_complementable complementable Element of the carrier of (product X)
the ZeroF of (product X) is left_complementable right_complementable complementable Element of the carrier of (product X)
0. Y is V52(Y) left_complementable right_complementable complementable Element of the carrier of Y
the ZeroF of Y is left_complementable right_complementable complementable Element of the carrier of Y
[(0. (product X)),(0. Y)] is Element of [: the carrier of (product X), the carrier of Y:]
((product X),Y) is non empty Relation-like [:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:] -defined [: the carrier of (product X), the carrier of Y:] -valued Function-like V26([:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:]
[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of Y:],[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty set
((product X),Y) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of Y:]:] -defined [: the carrier of (product X), the carrier of Y:] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of Y:]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:]
[:REAL,[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of Y:]:],[: the carrier of (product X), the carrier of Y:]:] is non empty set
((product X),Y) is non empty Relation-like [: the carrier of (product X), the carrier of Y:] -defined REAL -valued Function-like V26([: the carrier of (product X), the carrier of Y:]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of (product X), the carrier of Y:],REAL:]
[:[: the carrier of (product X), the carrier of Y:],REAL:] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of Y:],REAL:] is non empty set
NORMSTR(# [: the carrier of (product X), the carrier of Y:],((product X),Y),((product X),Y),((product X),Y),((product X),Y) #) is strict NORMSTR
the carrier of ((product X),Y) is non empty set
<*Y*> is non empty Relation-like NAT -defined Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
X ^ <*Y*> is non empty Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable RealLinearSpace-yielding RealNormSpace-yielding set
product (X ^ <*Y*>) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product (X ^ <*Y*>)) is non empty set
[: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):] is non empty set
0. ((product X),Y) is V52(((product X),Y)) left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the ZeroF of ((product X),Y) is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
0. (product (X ^ <*Y*>)) is V52( product (X ^ <*Y*>)) left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the ZeroF of (product (X ^ <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
product <*Y*> is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
((product X),(product <*Y*>)) is non empty left_complementable right_complementable complementable Abelian add-associative right_zeroed vector-distributive scalar-distributive scalar-associative scalar-unital V103() discerning reflexive strict RealNormSpace-like NORMSTR
the carrier of (product <*Y*>) is non empty set
[: the carrier of (product X), the carrier of (product <*Y*>):] is non empty Relation-like set
((product X),(product <*Y*>)) is Element of [: the carrier of (product X), the carrier of (product <*Y*>):]
0. (product <*Y*>) is V52( product <*Y*>) left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
the ZeroF of (product <*Y*>) is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
[(0. (product X)),(0. (product <*Y*>))] is Element of [: the carrier of (product X), the carrier of (product <*Y*>):]
((product X),(product <*Y*>)) is non empty Relation-like [:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:] -defined [: the carrier of (product X), the carrier of (product <*Y*>):] -valued Function-like V26([:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:]
[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:[: the carrier of (product X), the carrier of (product <*Y*>):],[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty set
((product X),(product <*Y*>)) is non empty Relation-like [:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:] -defined [: the carrier of (product X), the carrier of (product <*Y*>):] -valued Function-like V26([:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:]) quasi_total Element of bool [:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:]
[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
[:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty Relation-like set
bool [:[:REAL,[: the carrier of (product X), the carrier of (product <*Y*>):]:],[: the carrier of (product X), the carrier of (product <*Y*>):]:] is non empty set
((product X),(product <*Y*>)) is non empty Relation-like [: the carrier of (product X), the carrier of (product <*Y*>):] -defined REAL -valued Function-like V26([: the carrier of (product X), the carrier of (product <*Y*>):]) quasi_total complex-yielding V140() V141() Element of bool [:[: the carrier of (product X), the carrier of (product <*Y*>):],REAL:]
[:[: the carrier of (product X), the carrier of (product <*Y*>):],REAL:] is non empty Relation-like set
bool [:[: the carrier of (product X), the carrier of (product <*Y*>):],REAL:] is non empty set
NORMSTR(# [: the carrier of (product X), the carrier of (product <*Y*>):],((product X),(product <*Y*>)),((product X),(product <*Y*>)),((product X),(product <*Y*>)),((product X),(product <*Y*>)) #) is strict NORMSTR
the carrier of ((product X),(product <*Y*>)) is non empty set
[: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):] is non empty set
[:NAT, the carrier of Y:] is non empty Relation-like set
bool [:NAT, the carrier of Y:] is non empty set
0. ((product X),(product <*Y*>)) is V52(((product X),(product <*Y*>))) left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the ZeroF of ((product X),(product <*Y*>)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
I is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of ((product X),(product <*Y*>)):]
I . (0. ((product X),Y)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
[: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty set
J is non empty Relation-like the carrier of ((product X),(product <*Y*>)) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),(product <*Y*>))) quasi_total Element of bool [: the carrier of ((product X),(product <*Y*>)), the carrier of (product (X ^ <*Y*>)):]
J . (0. ((product X),(product <*Y*>))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J * I is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):]
K is non empty Relation-like the carrier of ((product X),Y) -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26( the carrier of ((product X),Y)) quasi_total Element of bool [: the carrier of ((product X),Y), the carrier of (product (X ^ <*Y*>)):]
K . (0. ((product X),Y)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
rng J is non empty Element of bool the carrier of (product (X ^ <*Y*>))
bool the carrier of (product (X ^ <*Y*>)) is non empty set
rng I is non empty Element of bool the carrier of ((product X),(product <*Y*>))
bool the carrier of ((product X),(product <*Y*>)) is non empty set
rng (J * I) is non empty Element of bool the carrier of (product (X ^ <*Y*>))
J .: the carrier of ((product X),(product <*Y*>)) is Element of bool the carrier of (product (X ^ <*Y*>))
v is left_complementable right_complementable complementable Element of the carrier of (product X)
r is left_complementable right_complementable complementable Element of the carrier of Y
<*r*> is non empty Relation-like NAT -defined the carrier of Y -valued Function-like V33() V40(1) FinSequence-like FinSubsequence-like countable FinSequence of the carrier of Y
K . (v,r) is set
[v,r] is set
K . [v,r] is set
I . (v,r) is set
I . [v,r] is set
[v,<*r*>] is Element of [: the carrier of (product X),(bool [:NAT, the carrier of Y:]):]
[: the carrier of (product X),(bool [:NAT, the carrier of Y:]):] is non empty Relation-like set
[v,r] is Element of [: the carrier of (product X), the carrier of Y:]
yy is left_complementable right_complementable complementable Element of the carrier of (product <*Y*>)
J . (v,yy) is set
[v,yy] is set
J . [v,yy] is set
x1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
x1 ^ y1 is Relation-like NAT -defined Function-like V33() FinSequence-like FinSubsequence-like countable set
I . [v,r] is set
J . (I . [v,r]) is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
r is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
v + r is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the addF of ((product X),Y) is non empty Relation-like [: the carrier of ((product X),Y), the carrier of ((product X),Y):] -defined the carrier of ((product X),Y) -valued Function-like V26([: the carrier of ((product X),Y), the carrier of ((product X),Y):]) quasi_total Element of bool [:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):]
[: the carrier of ((product X),Y), the carrier of ((product X),Y):] is non empty Relation-like set
[:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty Relation-like set
bool [:[: the carrier of ((product X),Y), the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty set
the addF of ((product X),Y) . (v,r) is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
[v,r] is set
the addF of ((product X),Y) . [v,r] is set
K . (v + r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
K . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
K . r is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
(K . v) + (K . r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) is non empty Relation-like [: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):] -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26([: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):]) quasi_total Element of bool [:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):]
[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
[:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of (product (X ^ <*Y*>)), the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty set
the addF of (product (X ^ <*Y*>)) . ((K . v),(K . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(K . v),(K . r)] is set
the addF of (product (X ^ <*Y*>)) . [(K . v),(K . r)] is set
I . (v + r) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
J . (I . (v + r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
I . v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
I . r is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
(I . v) + (I . r) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the addF of ((product X),(product <*Y*>)) is non empty Relation-like [: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):] -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26([: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):]) quasi_total Element of bool [:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):]
[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
[:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [:[: the carrier of ((product X),(product <*Y*>)), the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty set
the addF of ((product X),(product <*Y*>)) . ((I . v),(I . r)) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
[(I . v),(I . r)] is set
the addF of ((product X),(product <*Y*>)) . [(I . v),(I . r)] is set
J . ((I . v) + (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . r) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
(J . (I . v)) + (J . (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) . ((J . (I . v)),(J . (I . r))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(J . (I . v)),(J . (I . r))] is set
the addF of (product (X ^ <*Y*>)) . [(J . (I . v)),(J . (I . r))] is set
(K . v) + (J . (I . r)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the addF of (product (X ^ <*Y*>)) . ((K . v),(J . (I . r))) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
[(K . v),(J . (I . r))] is set
the addF of (product (X ^ <*Y*>)) . [(K . v),(J . (I . r))] is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r is V11() real ext-real Element of REAL
r * v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
the Mult of ((product X),Y) is non empty Relation-like [:REAL, the carrier of ((product X),Y):] -defined the carrier of ((product X),Y) -valued Function-like V26([:REAL, the carrier of ((product X),Y):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):]
[:REAL, the carrier of ((product X),Y):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),Y):], the carrier of ((product X),Y):] is non empty set
the Mult of ((product X),Y) . (r,v) is set
[r,v] is set
the Mult of ((product X),Y) . [r,v] is set
K . (r * v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r * (K . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the Mult of (product (X ^ <*Y*>)) is non empty Relation-like [:REAL, the carrier of (product (X ^ <*Y*>)):] -defined the carrier of (product (X ^ <*Y*>)) -valued Function-like V26([:REAL, the carrier of (product (X ^ <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):]
[:REAL, the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of (product (X ^ <*Y*>)):], the carrier of (product (X ^ <*Y*>)):] is non empty set
the Mult of (product (X ^ <*Y*>)) . (r,(K . v)) is set
[r,(K . v)] is set
the Mult of (product (X ^ <*Y*>)) . [r,(K . v)] is set
I . (r * v) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
J . (I . (r * v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
I . v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
r * (I . v) is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
the Mult of ((product X),(product <*Y*>)) is non empty Relation-like [:REAL, the carrier of ((product X),(product <*Y*>)):] -defined the carrier of ((product X),(product <*Y*>)) -valued Function-like V26([:REAL, the carrier of ((product X),(product <*Y*>)):]) quasi_total Element of bool [:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):]
[:REAL, the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
[:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty Relation-like set
bool [:[:REAL, the carrier of ((product X),(product <*Y*>)):], the carrier of ((product X),(product <*Y*>)):] is non empty set
the Mult of ((product X),(product <*Y*>)) . (r,(I . v)) is set
[r,(I . v)] is set
the Mult of ((product X),(product <*Y*>)) . [r,(I . v)] is set
J . (r * (I . v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
J . (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
r * (J . (I . v)) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
the Mult of (product (X ^ <*Y*>)) . (r,(J . (I . v))) is set
[r,(J . (I . v))] is set
the Mult of (product (X ^ <*Y*>)) . [r,(J . (I . v))] is set
v is left_complementable right_complementable complementable Element of the carrier of ((product X),Y)
K . v is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
||.(K . v).|| is V11() real ext-real non negative Element of REAL
the normF of (product (X ^ <*Y*>)) is non empty Relation-like the carrier of (product (X ^ <*Y*>)) -defined REAL -valued Function-like V26( the carrier of (product (X ^ <*Y*>))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of (product (X ^ <*Y*>)),REAL:]
[: the carrier of (product (X ^ <*Y*>)),REAL:] is non empty Relation-like set
bool [: the carrier of (product (X ^ <*Y*>)),REAL:] is non empty set
the normF of (product (X ^ <*Y*>)) . (K . v) is V11() real ext-real Element of REAL
||.v.|| is V11() real ext-real non negative Element of REAL
the normF of ((product X),Y) is non empty Relation-like the carrier of ((product X),Y) -defined REAL -valued Function-like V26( the carrier of ((product X),Y)) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((product X),Y),REAL:]
[: the carrier of ((product X),Y),REAL:] is non empty Relation-like set
bool [: the carrier of ((product X),Y),REAL:] is non empty set
the normF of ((product X),Y) . v is V11() real ext-real Element of REAL
I . v is left_complementable right_complementable complementable Element of the carrier of ((product X),(product <*Y*>))
J . (I . v) is left_complementable right_complementable complementable Element of the carrier of (product (X ^ <*Y*>))
||.(J . (I . v)).|| is V11() real ext-real non negative Element of REAL
the normF of (product (X ^ <*Y*>)) . (J . (I . v)) is V11() real ext-real Element of REAL
||.(I . v).|| is V11() real ext-real non negative Element of REAL
the normF of ((product X),(product <*Y*>)) is non empty Relation-like the carrier of ((product X),(product <*Y*>)) -defined REAL -valued Function-like V26( the carrier of ((product X),(product <*Y*>))) quasi_total complex-yielding V140() V141() Element of bool [: the carrier of ((product X),(product <*Y*>)),REAL:]
[: the carrier of ((product X),(product <*Y*>)),REAL:] is non empty Relation-like set
bool [: the carrier of ((product X),(product <*Y*>)),REAL:] is non empty set
the normF of ((product X),(product <*Y*>)) . (I . v) is V11() real ext-real Element of REAL