:: MATRIX14 semantic presentation

REAL is set
NAT is non empty V21() V22() V23() V28() V33() V34() Element of bool REAL

{} is set

1 is non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative Element of NAT
{{},1} is set
NAT is non empty V21() V22() V23() V28() V33() V34() set
bool NAT is V28() set
bool NAT is V28() set
K183(NAT) is V45() set

RAT is set
INT is set
[:1,1:] is Relation-like set
bool [:1,1:] is set
[:[:1,1:],1:] is Relation-like set
bool [:[:1,1:],1:] is set

bool [:[:1,1:],REAL:] is set

2 is non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative Element of NAT
[:2,2:] is Relation-like set

bool [:[:2,2:],REAL:] is set

K386(2) is V166() L15()
the carrier of K386(2) is set
0 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
3 is non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative Element of NAT
Seg 1 is non empty trivial V28() V35(1) Element of bool NAT
{1} is non empty trivial V35(1) set
Seg 2 is non empty V28() V35(2) Element of bool NAT
{1,2} is set
n is non empty ZeroStr
the carrier of n is non empty set
K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
0. n is V83(n) Element of the carrier of n

K -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
n is non empty ZeroStr
K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of n is non empty set
0. n is V83(n) Element of the carrier of n

K -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n is non empty set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
{ b1 where b1 is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like Element of the carrier of n * : len b1 = len K } is set
the carrier of n is non empty set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len (K + A) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

dom (B2 + B3) is V35( len K) Element of bool NAT
Seg (len K) is V28() V35( len K) Element of bool NAT
the carrier of n is non empty set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len (K - A) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

dom (B2 - B3) is V35( len K) Element of bool NAT
Seg (len K) is V28() V35( len K) Element of bool NAT
n is non empty multLoopStr
the carrier of n is non empty set

dom (mlt (K,A)) is Element of bool NAT
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(mlt (K,A)) . B is set
K /. B is Element of the carrier of n
A /. B is Element of the carrier of n
(K /. B) * (A /. B) is Element of the carrier of n
the multF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n . ((K /. B),(A /. B)) is Element of the carrier of n
(mlt (K,A)) /. B is Element of the carrier of n
the multF of n .: (K,A) is Relation-like Function-like set
rng A is set
dom the multF of n is set
rng K is set
[:(rng K),(rng A):] is Relation-like set
dom K is Element of bool NAT
dom A is Element of bool NAT
(dom K) /\ (dom A) is Element of bool NAT
A . B is set
K . B is set
the carrier of n is non empty set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
(K + A) . B is set
K /. B is Element of the carrier of n
A /. B is Element of the carrier of n
(K /. B) + (A /. B) is Element of the carrier of n
the addF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . ((K /. B),(A /. B)) is Element of the carrier of n
(K - A) . B is set
(K /. B) - (A /. B) is Element of the carrier of n
- (A /. B) is Element of the carrier of n
(K /. B) + (- (A /. B)) is Element of the carrier of n
the addF of n . ((K /. B),(- (A /. B))) is Element of the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

B2 /. B is Element of the carrier of n
B2 . B is set

B3 /. B is Element of the carrier of n
B3 . B is set
Seg (len K) is V28() V35( len K) Element of bool NAT
len (K + A) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (len (K + A)) is V28() V35( len (K + A)) Element of bool NAT
dom (K + A) is Element of bool NAT
len (K - A) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (len (K - A)) is V28() V35( len (K - A)) Element of bool NAT
dom (K - A) is Element of bool NAT

the carrier of n is non empty non trivial set
K is Element of the carrier of n
- K is Element of the carrier of n

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len A) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

1_ n is Element of the carrier of n
1. n is V83(n) Element of the carrier of n
- (1_ n) is Element of the carrier of n
(- (1_ n)) * B3 is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len A) FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(- (1_ n)) * K is Element of the carrier of n
the multF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n . ((- (1_ n)),K) is Element of the carrier of n
((- (1_ n)) * K) * B2 is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len A) FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(1_ n) * K is Element of the carrier of n
the multF of n . ((1_ n),K) is Element of the carrier of n
- ((1_ n) * K) is Element of the carrier of n
(- ((1_ n) * K)) * B2 is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len A) FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
(- (1_ n)) * B2 is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len A) FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
K * ((- (1_ n)) * B2) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len A) FinSequence-like FinSubsequence-like Element of (len A) -tuples_on the carrier of n
n is non empty multLoopStr
the carrier of n is non empty set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

mlt ((K ^ B),(A ^ B2)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n

(mlt (K,A)) ^ (mlt (B,B2)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the multF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n .: ((K ^ B),(A ^ B2)) is Relation-like Function-like set
<:(K ^ B),(A ^ B2):> is Relation-like Function-like set
<:(K ^ B),(A ^ B2):> * the multF of n is Relation-like set
dom (K ^ B) is Element of bool NAT
len (K ^ B) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (len (K ^ B)) is V28() V35( len (K ^ B)) Element of bool NAT
rng (A ^ B2) is set
dom the multF of n is set
rng (K ^ B) is set
[:(rng (K ^ B)),(rng (A ^ B2)):] is Relation-like set
dom (<:(K ^ B),(A ^ B2):> * the multF of n) is set
dom (A ^ B2) is Element of bool NAT
(dom (K ^ B)) /\ (dom (A ^ B2)) is Element of bool NAT
len (A ^ B2) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len A) + (len B2) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (len (A ^ B2)) is V28() V35( len (A ^ B2)) Element of bool NAT
dom (mlt ((K ^ B),(A ^ B2))) is Element of bool NAT
the multF of n .: (B,B2) is Relation-like Function-like set

<:B,B2:> * the multF of n is Relation-like set
rng B is set
rng B2 is set
[:(rng B),(rng B2):] is Relation-like set
dom (<:B,B2:> * the multF of n) is set
dom B is Element of bool NAT
dom B2 is Element of bool NAT
(dom B) /\ (dom B2) is Element of bool NAT
Seg (len B) is V28() V35( len B) Element of bool NAT
dom (mlt (B,B2)) is Element of bool NAT
len (mlt (B,B2)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
the multF of n .: (K,A) is Relation-like Function-like set

<:K,A:> * the multF of n is Relation-like set
rng K is set
rng A is set
[:(rng K),(rng A):] is Relation-like set
dom (<:K,A:> * the multF of n) is set
dom K is Element of bool NAT
dom A is Element of bool NAT
(dom K) /\ (dom A) is Element of bool NAT
(len K) + (len B) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (len K) is V28() V35( len K) Element of bool NAT
dom (mlt (K,A)) is Element of bool NAT
len (mlt ((K ^ B),(A ^ B2))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B3 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
(mlt ((K ^ B),(A ^ B2))) . B3 is set
((mlt (K,A)) ^ (mlt (B,B2))) . B3 is set
Seg (len (mlt ((K ^ B),(A ^ B2)))) is V28() V35( len (mlt ((K ^ B),(A ^ B2)))) Element of bool NAT
(K ^ B) /. B3 is Element of the carrier of n
(K ^ B) . B3 is set
(A ^ B2) /. B3 is Element of the carrier of n
(A ^ B2) . B3 is set
A . B3 is set
(mlt (K,A)) . B3 is set
K /. B3 is Element of the carrier of n
A /. B3 is Element of the carrier of n
(K /. B3) * (A /. B3) is Element of the carrier of n
the multF of n . ((K /. B3),(A /. B3)) is Element of the carrier of n
K . B3 is set
((K ^ B) /. B3) * ((A ^ B2) /. B3) is Element of the carrier of n
the multF of n . (((K ^ B) /. B3),((A ^ B2) /. B3)) is Element of the carrier of n
B3 -' (len K) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B3 - (len K) is V47() V48() ext-real set
(len K) + (B3 -' (len K)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
((len K) + (len B)) - (len K) is V47() V48() ext-real set
0 + 1 is non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative Element of NAT
len (mlt (K,A)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len (mlt (K,A))) + (B3 -' (len K)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(mlt (B,B2)) . (B3 -' (len K)) is set
B /. (B3 -' (len K)) is Element of the carrier of n
B2 /. (B3 -' (len K)) is Element of the carrier of n
(B /. (B3 -' (len K))) * (B2 /. (B3 -' (len K))) is Element of the carrier of n
the multF of n . ((B /. (B3 -' (len K))),(B2 /. (B3 -' (len K)))) is Element of the carrier of n
B . (B3 -' (len K)) is set
B2 . (B3 -' (len K)) is set
((K ^ B) /. B3) * ((A ^ B2) /. B3) is Element of the carrier of n
the multF of n . (((K ^ B) /. B3),((A ^ B2) /. B3)) is Element of the carrier of n
((K ^ B) /. B3) * ((A ^ B2) /. B3) is Element of the carrier of n
the multF of n . (((K ^ B) /. B3),((A ^ B2) /. B3)) is Element of the carrier of n
((K ^ B) /. B3) * ((A ^ B2) /. B3) is Element of the carrier of n
the multF of n . (((K ^ B) /. B3),((A ^ B2) /. B3)) is Element of the carrier of n
len (mlt (K,A)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(len (mlt (K,A))) + (len (mlt (B,B2))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len ((mlt (K,A)) ^ (mlt (B,B2))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of n is non empty non trivial set

the carrier of n is non empty non trivial set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

B is Element of the carrier of n

mlt ((B * K),A) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
B * (mlt (K,A)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n

mlt (K,(B * A)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

mlt (B3,B2) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
B * (mlt (B3,B2)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n

the carrier of n is non empty non trivial set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
K "*" A is Element of the carrier of n
B is Element of the carrier of n

(B * K) "*" A is Element of the carrier of n
B * (K "*" A) is Element of the carrier of n
the multF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n . (B,(K "*" A)) is Element of the carrier of n
mlt ((B * K),A) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((B * K),A)) is Element of the carrier of n

B * (mlt (K,A)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (B * (mlt (K,A))) is Element of the carrier of n
Sum (mlt (K,A)) is Element of the carrier of n
B * (Sum (mlt (K,A))) is Element of the carrier of n
the multF of n . (B,(Sum (mlt (K,A)))) is Element of the carrier of n

the carrier of n is non empty non trivial set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
K "*" A is Element of the carrier of n
B is Element of the carrier of n

K "*" (B * A) is Element of the carrier of n
B * (K "*" A) is Element of the carrier of n
the multF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the multF of n . (B,(K "*" A)) is Element of the carrier of n
mlt (K,(B * A)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt (K,(B * A))) is Element of the carrier of n

B * (mlt (K,A)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (B * (mlt (K,A))) is Element of the carrier of n
Sum (mlt (K,A)) is Element of the carrier of n
B * (Sum (mlt (K,A))) is Element of the carrier of n
the multF of n . (B,(Sum (mlt (K,A)))) is Element of the carrier of n

the carrier of n is non empty non trivial set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

K "*" (A + B) is Element of the carrier of n
K "*" A is Element of the carrier of n
K "*" B is Element of the carrier of n
(K "*" A) + (K "*" B) is Element of the carrier of n
the addF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . ((K "*" A),(K "*" B)) is Element of the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
mlt (K,(A + B)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt (K,(A + B))) is Element of the carrier of n

mlt (B3,B2) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n

mlt (B3,B) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(mlt (B3,B2)) + (mlt (B3,B)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
Sum ((mlt (B3,B2)) + (mlt (B3,B))) is Element of the carrier of n
Sum (mlt (B3,B2)) is Element of the carrier of n
Sum (mlt (B3,B)) is Element of the carrier of n
(Sum (mlt (B3,B2))) + (Sum (mlt (B3,B))) is Element of the carrier of n
the addF of n . ((Sum (mlt (B3,B2))),(Sum (mlt (B3,B)))) is Element of the carrier of n

Sum (mlt (K,A)) is Element of the carrier of n
(Sum (mlt (K,A))) + (K "*" B) is Element of the carrier of n
the addF of n . ((Sum (mlt (K,A))),(K "*" B)) is Element of the carrier of n

the carrier of n is non empty non trivial set

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

len B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

(K ^ B) "*" (A ^ B2) is Element of the carrier of n
K "*" A is Element of the carrier of n
B "*" B2 is Element of the carrier of n
(K "*" A) + (B "*" B2) is Element of the carrier of n
the addF of n is Relation-like Function-like V18([: the carrier of n, the carrier of n:], the carrier of n) Element of bool [:[: the carrier of n, the carrier of n:], the carrier of n:]
[: the carrier of n, the carrier of n:] is Relation-like set
[:[: the carrier of n, the carrier of n:], the carrier of n:] is Relation-like set
bool [:[: the carrier of n, the carrier of n:], the carrier of n:] is set
the addF of n . ((K "*" A),(B "*" B2)) is Element of the carrier of n

(mlt (K,A)) ^ (mlt (B,B2)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum ((mlt (K,A)) ^ (mlt (B,B2))) is Element of the carrier of n
Sum (mlt (K,A)) is Element of the carrier of n
Sum (mlt (B,B2)) is Element of the carrier of n
(Sum (mlt (K,A))) + (Sum (mlt (B,B2))) is Element of the carrier of n
the addF of n . ((Sum (mlt (K,A))),(Sum (mlt (B,B2)))) is Element of the carrier of n
mlt ((K ^ B),(A ^ B2)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
Sum (mlt ((K ^ B),(A ^ B2))) is Element of the carrier of n
(Sum (mlt (K,A))) + (B "*" B2) is Element of the carrier of n
the addF of n . ((Sum (mlt (K,A))),(B "*" B2)) is Element of the carrier of n
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V83(K) Element of the carrier of K

mlt (A,(n |-> (0. K))) is Relation-like NAT -defined the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K

(0. K) * (K,n) is Relation-like NAT -defined the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
mlt (A,((0. K) * (K,n))) is Relation-like NAT -defined the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
mlt (A,(K,n)) is Relation-like NAT -defined the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K
(0. K) * (mlt (A,(K,n))) is Relation-like NAT -defined the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Element of n -tuples_on the carrier of K

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

0 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is V83(n) Element of the carrier of n

0 -tuples_on (0 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 0 -tuples_on the carrier of n
len (1. (n,0)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

0 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
0. n is V83(n) Element of the carrier of n

0 -tuples_on (0 -tuples_on the carrier of n) is functional non empty FinSequence-membered FinSequenceSet of 0 -tuples_on the carrier of n

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K

(A * B) * B2 is Relation-like NAT -defined the carrier of K * -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular Matrix of n,n, the carrier of K

A * (B * B2) is Relation-like NAT -defined the carrier of K * -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular Matrix of n,n, the carrier of K
width B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
width A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K

n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K

the carrier of n is non empty non trivial set
0. n is V83(n) Element of the carrier of n

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(n,(len K)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
K "*" (n,(len K)) is Element of the carrier of n

mlt (B,(n,(len K))) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
Sum (mlt (B,(n,(len K)))) is Element of the carrier of n
Sum (n,(len K)) is Element of the carrier of n

the carrier of n is non empty non trivial set
0. n is V83(n) Element of the carrier of n

len K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(n,(len K)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(len K) -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
(len K) |-> (0. n) is Relation-like NAT -defined the carrier of n -valued Function-like V28() V35( len K) FinSequence-like FinSubsequence-like Element of (len K) -tuples_on the carrier of n
(n,(len K)) "*" K is Element of the carrier of n
K "*" (n,(len K)) is Element of the carrier of n

the carrier of n is non empty non trivial set
0. n is V83(n) Element of the carrier of n

K is Element of the carrier of n

<*(0. n)*> "*" <*K*> is Element of the carrier of n
len <*K*> is non empty V21() V22() V23() V27() V28() V33() V47() V48() ext-real positive non negative Element of NAT
(n,1) is Relation-like NAT -defined the carrier of n -valued Function-like non empty trivial V28() V35(1) FinSequence-like FinSubsequence-like Element of 1 -tuples_on the carrier of n
1 -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

(n,1) "*" <*K*> is Element of the carrier of n

the carrier of n is non empty non trivial set
K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
0. n is V83(n) Element of the carrier of n

K -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
1. n is V83(n) Element of the carrier of n
Replace ((K |-> (0. n)),A,(1. n)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n

K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
(n,K,A) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
the carrier of n is non empty non trivial set
0. n is V83(n) Element of the carrier of n

K -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
1. n is V83(n) Element of the carrier of n
Replace ((K |-> (0. n)),A,(1. n)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
len (n,K,A) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len (Replace ((K |-> (0. n)),A,(1. n))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len (K |-> (0. n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

1. n is V83(n) Element of the carrier of n
the carrier of n is non empty non trivial set
K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
(n,A,K) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
0. n is V83(n) Element of the carrier of n

A -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
Replace ((A |-> (0. n)),K,(1. n)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,A,K) . K is set
len (A |-> (0. n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len (Replace ((A |-> (0. n)),K,(1. n))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(Replace ((A |-> (0. n)),K,(1. n))) /. K is Element of the carrier of n

0. n is V83(n) Element of the carrier of n
the carrier of n is non empty non trivial set
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
(n,B,K) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n

B -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
1. n is V83(n) Element of the carrier of n
Replace ((B |-> (0. n)),K,(1. n)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
(n,B,K) . A is set
Seg B is V28() V35(B) Element of bool NAT
len (B |-> (0. n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
len (Replace ((B |-> (0. n)),K,(1. n))) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(Replace ((B |-> (0. n)),K,(1. n))) /. A is Element of the carrier of n
(B |-> (0. n)) /. A is Element of the carrier of n
(B |-> (0. n)) . A is set

K is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set

the carrier of n is non empty non trivial set
the carrier of n * is functional non empty FinSequence-membered FinSequenceSet of the carrier of n

(n,A,K) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
0. n is V83(n) Element of the carrier of n

A -tuples_on the carrier of n is functional non empty FinSequence-membered FinSequenceSet of the carrier of n
1. n is V83(n) Element of the carrier of n
Replace ((A |-> (0. n)),K,(1. n)) is Relation-like NAT -defined the carrier of n -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of n
[K,1] is set
Indices (1. (n,A)) is set
dom (1. (n,A)) is Element of bool NAT
width (1. (n,A)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (width (1. (n,A))) is V28() V35( width (1. (n,A))) Element of bool NAT
[:(dom (1. (n,A))),(Seg (width (1. (n,A)))):] is Relation-like set
(1. (n,A)) * (K,1) is Element of the carrier of n

B . 1 is set
len (1. (n,A)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
rng (1. (n,A)) is set
len B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
B . B2 is set
(n,A,K) . B2 is set
[K,B2] is set
(1. (n,A)) * (K,B2) is Element of the carrier of n
(1. (n,A)) * (K,K) is Element of the carrier of n
1_ n is Element of the carrier of n

B3 . B2 is set

B3 . B2 is set
len (n,A,K) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set

the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(K,n,A) is Relation-like NAT -defined the carrier of K -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of K
0. K is V83(K) Element of the carrier of K

n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
1. K is V83(K) Element of the carrier of K
Replace ((n |-> (0. K)),A,(1. K)) is Relation-like NAT -defined the carrier of K -valued Function-like V28() FinSequence-like FinSubsequence-like FinSequence of the carrier of K
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
(1. (K,n)) * (A,B) is Element of the carrier of K
(K,n,A) . B is set
[A,B] is set
Indices (1. (K,n)) is set
dom (1. (K,n)) is Element of bool NAT
width (1. (K,n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (width (1. (K,n))) is V28() V35( width (1. (K,n))) Element of bool NAT
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like set

B2 . B is set
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K

n -tuples_on the carrier of K is functional non empty FinSequence-membered FinSequenceSet of the carrier of K
0. K is V83(K) Element of the carrier of K

n |-> (n |-> (0. K)) is Relation-like NAT -defined n -tuples_on the carrier of K -valued Function-like V28() V35(n) FinSequence-like FinSubsequence-like Function-yielding V40() Element of n -tuples_on (n -tuples_on the carrier of K)
n -tuples_on (n -tuples_on the carrier of K) is functional non empty FinSequence-membered FinSequenceSet of n -tuples_on the carrier of K

0. (K,n,n) is Relation-like NAT -defined the carrier of K * -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular Matrix of n,n, the carrier of K
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
A * (B2,B) is Element of the carrier of K
[B2,B] is set
Indices (0. (K,n)) is set
dom (0. (K,n)) is Element of bool NAT
width (0. (K,n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg (width (0. (K,n))) is V28() V35( width (0. (K,n))) Element of bool NAT
[:(dom (0. (K,n))),(Seg (width (0. (K,n)))):] is Relation-like set
(0. (K,n)) + (0. (K,n)) is Relation-like NAT -defined the carrier of K * -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular Matrix of n,n, the carrier of K
((0. (K,n)) + (0. (K,n))) * (B2,B) is Element of the carrier of K
(0. (K,n)) * (B2,B) is Element of the carrier of K
((0. (K,n)) * (B2,B)) + ((0. (K,n)) * (B2,B)) is Element of the carrier of K
the addF of K is Relation-like Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K . (((0. (K,n)) * (B2,B)),((0. (K,n)) * (B2,B))) is Element of the carrier of K
((0. (K,n)) * (B2,B)) - ((0. (K,n)) * (B2,B)) is Element of the carrier of K
- ((0. (K,n)) * (B2,B)) is Element of the carrier of K
((0. (K,n)) * (B2,B)) + (- ((0. (K,n)) * (B2,B))) is Element of the carrier of K
the addF of K . (((0. (K,n)) * (B2,B)),(- ((0. (K,n)) * (B2,B)))) is Element of the carrier of K
((0. (K,n)) * (B2,B)) + (((0. (K,n)) * (B2,B)) - ((0. (K,n)) * (B2,B))) is Element of the carrier of K
the addF of K . (((0. (K,n)) * (B2,B)),(((0. (K,n)) * (B2,B)) - ((0. (K,n)) * (B2,B)))) is Element of the carrier of K
((0. (K,n)) * (B2,B)) + (0. K) is Element of the carrier of K
the addF of K . (((0. (K,n)) * (B2,B)),(0. K)) is Element of the carrier of K
len A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Indices A is set
dom A is Element of bool NAT
width A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg () is V28() V35( width A) Element of bool NAT
[:(dom A),(Seg ()):] is Relation-like set
Seg n is V28() V35(n) Element of bool NAT
[:(Seg n),(Seg n):] is Relation-like set
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
[B,B2] is set
A * (B,B2) is Element of the carrier of K
(A * (B,B2)) + (A * (B,B2)) is Element of the carrier of K
the addF of K is Relation-like Function-like V18([: the carrier of K, the carrier of K:], the carrier of K) Element of bool [:[: the carrier of K, the carrier of K:], the carrier of K:]
[: the carrier of K, the carrier of K:] is Relation-like set
[:[: the carrier of K, the carrier of K:], the carrier of K:] is Relation-like set
bool [:[: the carrier of K, the carrier of K:], the carrier of K:] is set
the addF of K . ((A * (B,B2)),(A * (B,B2))) is Element of the carrier of K
B3 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
A * (B3,B2) is Element of the carrier of K

0. (K,(len A),()) is Relation-like NAT -defined the carrier of K * -valued Function-like V28() FinSequence-like FinSubsequence-like Function-yielding V40() tabular Matrix of len A, width A, the carrier of K
n is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT

the carrier of K is non empty non trivial set
the carrier of K * is functional non empty FinSequence-membered FinSequenceSet of the carrier of K

1. K is V83(K) Element of the carrier of K
0. K is V83(K) Element of the carrier of K

Indices A is set
dom A is Element of bool NAT
width A is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Seg () is V28() V35( width A) Element of bool NAT
[:(dom A),(Seg ()):] is Relation-like set
Seg n is V28() V35(n) Element of bool NAT
[:(Seg n),(Seg n):] is Relation-like set
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
A * (B,B2) is Element of the carrier of K
IFEQ (B,B2,(1. K),(0. K)) is Element of the carrier of K
[B,B2] is set
len (1. (K,n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
width (1. (K,n)) is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
Indices (1. (K,n)) is set
dom (1. (K,n)) is Element of bool NAT
Seg (width (1. (K,n))) is V28() V35( width (1. (K,n))) Element of bool NAT
[:(dom (1. (K,n))),(Seg (width (1. (K,n)))):] is Relation-like set
B is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative set
[B,B2] is set
A * (B,B2) is Element of the carrier of K
(1. (K,n)) * (B,B2) is Element of the carrier of K
B3 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
B2 is V21() V22() V23() V27() V28() V33() V47() V48() ext-real non negative Element of NAT
A * (B3,B2) is Element of the carrier of K
IFEQ (B3,B2,(1. K),(