thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent ;
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C ;
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y `2 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `2 = b `2 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from from from o1 , o2 ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , v be Vertex of G ;
let G be _Graph , v be Vertex of G ;
let a be VECTOR of V ;
let x be element ;
let x be element ;
let C be FormalContext , A , B be Subset of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a ;
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= REAL ;
let y be element ;
r2 in X ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = \bf 2 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
the function exp is differentiable in Z ;
j < i2 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b2 ;
f2 is one-to-one ;
support p = {} ;
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r1 ;
let e be Real , x be Real ;
not r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E ;
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster downarrow x -> closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
HH >= s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X , a be Point of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V , A be Subset of V ;
assume x in M ;
k < s . a ;
not t in { p } ;
let Y be many sorted set , f , g be Function of Y , BOOLEAN ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `2 = a + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= iH ;
1 <= iH ;
p9 c= cos . x ;
1 <= i-15 ;
1 <= i-15 ;
w in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
seq is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Y1 ;
assume w = 0. V ;
assume x in A . i ;
g in ComplexBoundedFunctions X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
if C c= f holds Let C c= f ;
x4 is increasing ;
let e1 be element ;
- b divides b ;
F c= \mathclose { \tau ( F ) ;
seq is non-decreasing ;
seq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is linearly-independent & finite is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X in X ;
y in rng f ;
let s , I be set , A be non-empty ManySortedSet of I ;
b `2 c= b9 `2 ;
assume not x in NAT + 1 ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q ] in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { \boldmath $ n $ } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 ;
g is continuous & g is continuous ;
assume O is symmetric and transitive is transitive ;
let x , y be element ;
let j2 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , Y be non empty TopSpace , f be Function of X , Y ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable of f , A , B ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( NAT * ( NAT + 1 ) ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> object one-to-one ;
let R be non empty doubleLoopStr , A , B be Subset of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be void non empty id of M , v be Element of M ;
let N be non empty Subset of M ;
let R be RelStr , n be finite LSeg of R ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v2 ;
x <= c2 . x ;
x in F ` ` ;
cluster S --> T -> a11 ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Nat ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : A2 <> {} & A2 <> {} ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A ;
i < len M + 1 ;
assume not - \infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom sec ;
assume [ x , y ] in R ;
set d = sqrt ( x ^2 - y ^2 ) ;
1 <= len g1 & g1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q9 .| = 1 ;
let s be SortSymbol of S ;
order ( i , n ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be Line of V ;
cluster m * n -> square ;
let k3 be Nat ;
i - 1 > m - 1 ;
R is transitive transitive & R is transitive ;
set F = <* u , w *> ;
p-2 c= P3 & CP c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is | ;
i <= len f2 & i <= len f2 ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R = n * r ;
cluster f . x -> complex-valued for Function ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= X0 & CX c= X ;
n2 <= ( ( 2 |^ n ) * ( 2 |^ n ) ) ;
A /\ ( P /\ Q ) c= A ` ;
cluster x -valued -> x -valued for Function ;
let Q be Subset-Family of S , A be Subset of T ;
assume n in dom g2 ;
let a be Element of R ;
t `2 in dom e1 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , Y ;
i . y in rng i ;
REAL c= dom f ;
f . x in rng f ;
\mathbb t <= sqrt ( r ^2 - 2 ) ;
s2 in { r } & s2 in { r } ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non empty real number ;
let m be Element of M ;
f in union rng ( F1 . n ) ;
let K be add-associative right_zeroed right_complementable associative associative associative associative associative associative non empty doubleLoopStr , A , B be Subset of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom x ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster T . X -> E for set ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S29 ;
b = sup ( dom f ) ;
x in Seg len q ;
reconsider X = [: D , D :] as set ;
[ a , c ] in [: E , E :] ;
assume n in dom h2 ;
w + 1 = a1 ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support q = Support p ;
assume X is complete \bf \bf \bf 2 ;
assume that f = g and p = q ;
n1 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng s2 ;
x0 < x0 + 1 ;
len ( L (#) F ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued finite sequence of NAT ;
let k be Element of NAT ;
Integral ( P , A ) < +infty ;
let n be Element of NAT ;
assume z in One \tt -> t of A ( ) ;
let i be set ;
n - 1 = n - 1 ;
len ( n |-> an ) = n ;
(# Z , c #) c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f , g be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E -tuples_on E ;
let B1 be Basis of x , y ;
Carrier ( 3 ) /\ L2 = {} ;
L1 /\ L2 = {} ;
assume \mathopen { \downarrow x } = \mathopen { \downarrow x } ;
assume b , c , b is_collinear ;
LIN q , c , c ;
x in rng ( f | A ) ;
set n8 = n + j ;
let IT be non empty set , A , B be Subset of D ;
let K be add-associative right_zeroed right_complementable associative associative associative associative non empty doubleLoopStr , A , B be Matrix of K ;
assume f `2 = f & h `2 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open Subset of ( TOP-REAL 2 ) | K1 ;
assume a , b ] is maximal in C ;
a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f , M = dom g ;
cluster strict -> \vert -> n\vert for \vert ;
not u in { \hbox { \boldmath $ g } } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster strict for RelStr ;
r (#) H is U -defined ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 in dom p ;
F . i is stable Subset of M ;
r-35 in reconsider 'not' ( y ) as Element of Q ( ) ;
let x , y be Element of X ;
A , I be a11 carrier of X ;
[ y , z ] in [: O , O :] ;
arity Macro ( i ) = 1 ;
rng ( Sgm A ) = A ;
q |- |- |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D `2 ) ^2 = {} ;
n + 1 + 1 <= len g ;
a in [: Al ( ) , Al ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `2 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster non empty associative for Group ;
x in support ( ( support t ) \ ( support t ) ) ;
assume a in [: G1 , G2 :] ;
i `2 <= len ( y `2 ) & ( y `2 ) `2 <= ( y `2 ) `2 ;
assume p divides b1 + b2 ;
M <= sup ( M1 * ( - M2 ) ) ;
assume x in ( W-min X ) .: ( X ) ;
j in dom ( z | ( dom z ) ) ;
let x be Element of D ( ) ;
IC Comput ( 5 , s , 5 ) = l1 + 1 ;
a = {} or a = { x } ;
set u9 = Vertices G , u9 = Vertices G , E = G \ { {} } ;
seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= hF & hF c= hF ;
]. a , b .[ c= Z ;
X1 , X2 are_equipotent implies X1 , X2 are_equipotent
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k ;
cluster -> INT -valued for Relation ;
ex v st C = v + W ;
let IT be non empty doubleLoopStr , A , B be Subset of V ;
assume V is Abelian add-associative right_zeroed right_complementable associative associative associative associative associative associative associative ;
[: X , Y :] \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive transitive RelStr , X be Subset of L , Y be Subset of L ;
R is reflexive & R is transitive ;
E , g |= All ( x , H ) ;
dom G ' /. y = a ;
sqrt ( 1 - 4 ) >= - r ;
G . x0 in rng G ;
let x be Element of F , y be Element of F ;
D [ ( ( P [ ) , 0 ] ) `1 , 0 ] ;
z in dom id ( the carrier of B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng ( f | [: dom f , rng g :] ) c= NAT ;
j + 1 in dom s1 ;
A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h ;
P . k1 in rng P ;
M = ( A +* ( {} , {} ) ) +* {} ;
let p be FinSequence of REAL ;
f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
ind [. a , b .] = b-a ;
assume that the distance of V , Q and Q is open ;
let a be Element of ^ V ;
let s be Element of P ( ) ;
let PP be non empty RelStr ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM ( R ) ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS K ;
l <= ( IC F ) . j ;
assume x in \mathopen { \downarrow } [ s , t ] } ;
( x `2 ) ^2 in ]. t `1 , t `2 .] ;
x in JumpParts InsCode ( InsCode ( T ) ) ;
let h be Morphism of c , a ;
Y c= [: \bf 1 , \bf 1 } :] ;
A2 \/ A1 \/ A2 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 ] in Y & x2 in Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q1 is_collinear ;
dom ( M1 * M2 ) = Seg n & dom ( M * M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ x1 , x2 ] ;
R , Q be ManySortedSet of A ;
set d = sqrt ( 1 + n ) ;
rng g2 c= dom W & rng g2 c= dom W implies ( g * ( f + g ) ) is one-to-one
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , A be Subset of V ;
let I be Program of SCM+FSA , J be Program of SCM+FSA ;
assume x in rng ( L * R ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 + ( W1 + W2 ) ;
assume support L misses rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in [: a9 , b9 :] ;
dom ( A * e ) = NAT ;
a , b be Vertex of G ;
let x be Element of product ( M . i ) ;
0 <= 2 * PI / 2 ;
o , a9 // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( ( curry f ) . i ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom ( Ser ( G ) ) = NAT & rng ( G . n ) c= NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & x in the carrier of f ;
conv ( @ A ) c= conv ( A ) ;
reconsider B = b as Element of the carrier of T ;
J , v |= _ { P \lbrack l \rbrack } ;
cluster J . i -> non empty for TopSpace ;
sup Y1 \/ Y in T & sup Y in T ;
W1 is_well field W1 & W2 is_Lin field W1 implies W1 is non empty
assume x in the carrier of R & y in the carrier of S ;
dom ( n |-> union ( n , 1 ) ) = Seg n ;
( ( s + 4 ) * ( s + 4 ) ) misses ( ( s + 4 ) * ( s + 4 ) ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f * ( f * ( a , b ) ) ) ;
assume that that that that Directed I c= J and Directed I c= K and J c= K ;
Im ( ( lim seq ) ^\ k ) = 0 ;
( ( - sin ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos is_differentiable_in x & cos . x = 1 / ( cos . x ) ^2
6 . n = 5 . n .= t . n ;
dom ( tan * ( exp_R + exp_R ) ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W .last() \/ W .last() ;
k-11 <= len ( v | i ) & k <= len ( v | i ) ;
x * a \equiv y * ( a mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= ( proj2 .: P ) .: P ;
h . p3 = g2 . I .= g2 . I ;
Gik = U * ( 1 , j ) `1 .= G * ( 1 , j ) `1 ;
f . r1 in rng f & f . r2 in rng f ;
i + 1 + 1 <= len One ;
rng F = rng ( F | ( rng F ) ) ;
mode non empty doubleLoopStr is non empty doubleLoopStr ;
[ x , y ] in A [: { a } , { a } :] ;
x1 . o in L2 . o ;
the carrier of support m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ k1 is bounded & lim seq = ( lim seq ) - ( lim seq ) ;
len ( F . n ) = len I & len ( F . n ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) . x = s . x ;
k <= k + 1 + 1 ;
reconsider c = {} as Element of L ;
let Y be with_symmetrical connected Subset of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or y1 = z ;
attr a <> {} means : Def6 : sqrt a = 1 ;
assume that cf a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial non trivial FinSequence of D ;
let F be non empty set , G be non empty set ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F . ( F . 1 ) ;
reconsider p9 = x as Subset of m ;
A , B , C be Element of R ;
cluster non empty strict for 19 of G ;
rng c `2 misses rng e ( ) \/ rng e ( ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( ( - cot ) `| Z ) ) /\ dom ( ( - cot ) `| Z ) ;
the component of Q c= ( UBD A ) ` & ( UBD A ) ` c= ( UBD A ) ` ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) ;
hence f = u * v & a * f = a * u ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b are_congruent_mod n ;
assume that S = S2 and p = p2 and p = p3 ;
gcd ( n1 , n2 ) = 1 & gcd ( n2 , n2 ) = 1 ;
set o9 = ( a * b ) * ( a * b ) ;
seq . n < |. r1 .| & |. r1 - r2 .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 is_collinear & b9 , c9 , a9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 .= 0 ;
4 .first() = ( W . 1 ) `1 .= ( W . 1 ) `1 ;
cluster -> trivial for subgraph of G ;
reconsider u = u as Element of Bags X ;
A in B ^ A implies A , B are_equipotent
x in { [ 2 * n + 3 , k ] } ;
1 >= sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) ;
f1 is_reconsider f2 is_reconsider reconsider f3 , f2 = f1 , f3 = f2 as Element of f2 ;
( f . q ) `2 <= ( q `2 ) ^2 ;
h is in the carrier of Cage ( C , n ) ;
( b `2 ) ^2 <= ( p `2 ) ^2 / ( p `2 ) ^2 ;
let f , g be X -defined Function of X , Y ;
S /. k , S /. k are_equipotent ;
x in dom ( max ( f , g ) ) ;
p2 in [: N , N :] . ( p1 , p2 ) ;
len ( H ) < len ( H ) & ( H ) . i < len H ;
F [ A , F ( ) . A ] ;
consider Z such that y in Z and Z in X ;
hence 1 in C implies A c= C |^ ( n + 1 )
assume that r1 <> 0 or r2 <> 0 ;
rng q1 c= rng ( C * ) & rng ( C * ) c= rng ( C * ) ;
A1 , A2 , L , L is_collinear & A2 , A1 , L is_collinear implies A1 , A2 , L is_collinear
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in \bf L ( p , S , SS ) ;
then S is negative means : Def6 : P [ S ] ;
Cl ( Int [#] T ) = [#] ( T | A ) ;
f12 | ( A2 /\ A1 ) = f2 | ( A2 /\ A1 ) ;
0. M in the carrier of W & the carrier of W c= the carrier of V ;
v , v `2 be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V \ Z ;
let X be Subset of S , T ;
consider H1 such that H = 'not' H1 and H1 in Free H1 ;
{ 1_ K } c= ( can t ) * ( can t ) ;
0 * a = 0. R .= a * 0 .= 0 * 0 ;
A |^ 2 , 2 |^ 2 ] = A |^ 2 ;
set vn|[ n , m ]| = ( v /. n ) `2 ;
r = 0. ( \langle \cal E , \Vert * \Vert \rangle ) ;
( f . p3 ) `1 >= 0 & ( f . p3 ) `1 >= 0 ;
len W = len ( W .last() ) .= len ( W .first() ) ;
f /* ( s * G ) is divergent to \hbox { - \infty $ } ;
consider l being Nat such that m = F . l ;
( t /. \rm mod ( b1 /. i ) ) does not empty empty ;
reconsider Y1 = X1 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( the carrier of L ) . x ;
( \sigma T ) \/ omega is Basis of T ;
for x being element st x in X holds x in Y ;
cluster [ x1 , x2 ] -> pair for element ;
downarrow a /\ \mathopen { \downarrow } t is Ideal of T ;
let X be with_NAT -defined NAT , NAT , f be NAT -defined Function of NAT , NAT ;
rng f = Funcs ( S , X ) ;
let p be Element of B , x be the \it Boolean Element of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N1 ;
0. X <= ( b |^ m ) * ( ( m + 1 ) * ( n + 1 ) ) ;
assume that i in I and R . i = R . i ;
i = j1 & p1 = q1 & p1 = q2 & q1 = q2 implies q1 = q2
assume g\mathfrak g in the right of g & f in the carrier' of g ;
let A1 , A2 be Point of S , A2 be Point of T ;
x in h " ( P /\ [#] ( T | P ) ) /\ [#] ( T | P ) ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X1 = X as non empty Subset of [: T , T :] ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Target of G ) -valued ;
n1 <= i2 + len g2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier of G2 & v in the carrier of G2 ;
y = Re ( y + Im ( y ) ) + ( Im ( y ) ) ;
( ( - 1 ) |^ p ) gcd ( - 1 ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( f `| ]. a , b .] ) . x <= ( f `| ]. a , b .] ) . x ;
rng ( M * ( M2 * F ) ) c= rng ( ( M * F ) * ( M * F ) ) ;
for p being Real st p in Z holds p >= a
\bf X \bf Y \bf Y \bf 1 } = proj1 ( f ) * ( f ) ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -b9 ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( ( mod P ) ) . ( ( mod P ) . i ) ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
for V being Subspace of V holds V is Subspace of V
reconsider i-7 = i - 1 as Element of NAT ;
dom f c= [: C , D :] & dom f = [: D , D :] ;
x in ( the Sorts of B ) . n ;
len a11 in Seg len ( f2 | i ) & len f2 = len f2 ;
pp c= the topology of T & the topology of T c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 ;
G * ( B * A ) = \mathord { id o1 } * id o1 ;
assume that p , u ] , u ] and u , v , v , w is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , $1 = $1 .. S ;
LIN a1 , a3 , b1 & LIN a3 , b1 , b3 ;
f " ( f .: x ) = { x } ;
dom ( w2 . i ) = dom ( r . i ) ;
assume that 1 <= i and i <= n and j <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I1 * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
q9 . x in rng q9 /\ rng q ;
Carrier ( H ) misses ( Carrier ( H ) ) ` ;
consider c being element such that [ a , c ] in G ;
assume N\lbrack o , o1 .] = o & o = o1 & o1 = o2 ;
q . ( j + 1 ) = q /. j + 1 ;
rng F c= ( F . C ) .: ( C . D ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
hence 0 <= x & x <= 1 & x <= 1 ;
p `2 <> 0. TOP-REAL 2 & q `2 <> 0. TOP-REAL 2 ;
cluster [: S , T :] -> non empty ;
let x be Element of S , T ;
<^ F , F ( a , b ) ^> is one-to-one ;
|. i .| <= - ( 2 |^ n ) / 2 ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
n * ( n + 1 ) ! > 0 * PI ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A1 ) ;
a3 , a4 // a3 , a4 & a3 , a4 // a4 , a4 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G ;
set v2 = ( v2 /. i ) `2 , v2 = ( v2 /. i ) `2 ;
x = r . n .= ( r . n ) . x .= ( r . n ) . x ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d2 * ( A * ( B * C ) ) ) = [: A , A * ( B * C ) :] ;
0 < sqrt ( p `1 - ||. z .|| + 1 ) ;
e . ( m + 1 ) <= e . m ;
B \ominus X \/ B \ominus Y c= B \ominus X
- \infty < Integral ( M , ( Im g ) | B ) ;
cluster O \tt F -> U -defined for OperSymbol of X ;
let U1 , U2 be non-empty MSAlgebra over S , o be OperSymbol of S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z is_collinear & x , y , z is_collinear ;
reconsider p9 = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in Lin ( A ) ;
let I , J be Program of SCM+FSA ;
assume - a is lower & b is lower ;
Int Cl Cl Int Cl Int Cl A c= Cl Int Cl Int Cl Int Cl Int A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( [ x , y ] , r ) ;
( p2 `2 ) ^2 <= ( p2 `2 ) ^2 / ( p2 `2 ) ^2 ;
Cl Q ` ` = [#] ( ( TOP-REAL 2 ) | P ) ;
set S = the carrier of T , T = the carrier of T ;
set I1 = IC ( f |^ n ) , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2
len p - n = len p - n + n ;
A is Permutation of Funcs ( A , x ) ;
reconsider nn6 = nn6 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s . j ) ;
q9 , q9 , x9 , y9 is_collinear & q9 , x9 , y9 is_collinear implies ex q st q , x9 , q9 is_collinear & q , q9 , x9 is_collinear & q , x9 , q9 is_collinear & q , x9 , q9 is_collinear & q , x9 , q9 is_collinear
a9 in the carrier of S1 & b9 in the carrier of S1 & c9 in the carrier of S2 ;
c1 /. ( n1 + 1 ) = c1 . ( n1 + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( f * ( S . x ) ) . x ;
consider x being element such that x in " " A ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( TOP-REAL n ) .. h , h = ( TOP-REAL n ) .. h ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M , k ) . i = M . i ;
reconsider m = sqrt ( x ^2 - 2 ) as Element of ( sqrt 2 ) ;
U1 , U2 be non-empty Subspace of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + len p3 + len p3 + len p3 ;
T1 , T2 be Scott Scott Scott TopAugmentation of L ;
then x <= y & ( r <= y ) & ( r <= x ) implies ( x <= y ) ;
set M = n -tuples_on ( m , n ) ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the Sorts of A1 ) * the Arity of S ) c= dom H & rng ( the Sorts of A2 ) c= dom H ;
z1 " " = z1 " * ( z1 " ) " .= z1 " * ( z1 " ) " ;
x0 - r in L /\ dom f /\ dom f ;
then w is strict \rm y1 , ( S , X ) /\ \mathop { \rm AllSymbolsOf S } ;
set xX = ( x ^ y ) ^ <* Z *> , Z ^ <* Z *> ;
len w1 in Seg len w1 & w1 in Seg len w2 implies w1 + w2 in dom w2
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = sqrt ( |. a . n .| ) * ( |. a .| ) ;
( p `1 ) ^2 <= ( G * ( i , j ) ) `1 ;
rng ( g | ( L~ g ) ) c= L~ ( g | ( L~ g ) ) ;
reconsider k = i-1 * j + l * j as Nat ;
for n being Nat holds F . n is with_non empty set ;
reconsider x9 = x as VECTOR of M ;
dom ( f | X ) = X /\ dom f ;
p , a // p , c & b , a // c , a ;
reconsider x1 = x as Element of REAL m -tuples_on REAL m ;
assume i in dom ( a * p ^ q ) ;
m . \hbox { \boldmath $ g $ } = p . ( \hbox { $ g $ } , h . ( \hbox { \boldmath $ g $ } ) ) ;
a |^ ( s . m ) <= 1 / ( a |^ n ) ;
S . ( n + k ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 \/ C2 & C1 \/ C2 = C1 \/ C2 ;
X . i = { x1 , x2 , x3 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
||. 0. R .|| = a & b0. R = b ;
F is closed implies Q is closed & Q is closed & Q is closed & not Q is closed & Q is closed & not ex t st t in Q & t in Q & t in Q & t in Q & t in Q & t in Q &
set T = \vert \vert \vert \vert ( X , x0 ) ;
Int Cl Cl Cl ( Cl R ) c= Cl Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F . x ) = { F . x } & rng ( F . x ) = { F . x } ;
G " { c } " { c } c= B \/ S ;
f is_orders [: X , Y :] & f is_^ X ;
set RF = the => of P , RF = the => of P ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT ;
reconsider pxK = u as Element of GF ( n ) ;
g . x in dom f & x in dom g implies g . x in dom g ;
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / N ;
len ( ( P . i ) `2 ) <= len ( P . i ) `2 ;
x " in the carrier of A1 & x " in the carrier of A2 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple function of S
f . x = a . i .= a1 . i .= a1 . i ;
let f be PartFunc of REAL i , REAL , i , j be Element of NAT ;
rng f = the carrier of Lin ( A ) & rng f is Subset of Lin ( A ) ;
assume s1 = sqrt ( 2 * ( p `1 - r ) ) ;
attr a > 1 & b > 0 implies a |^ b > 1 / a |^ b ;
let A , B , C be Subset of Lin ( I ) ;
reconsider X0 = X , X0 = Y as real number ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
t-3 , tt1 , t2 , t1 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2
Q [ e1 \/ { v } , f . ( e1 + 1 ) ] ;
g \circlearrowleft ( L~ z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vv1 ;
- f . w = ( - L ) * w ;
z - y <= x iff z <= x + y & z <= z + y ;
sqrt ( 7 * p1 + ( 1 - e ) * p2 ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom ( tan * tan ) ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) ;
[. a , b .] = 1_ G .= 1_ G ;
let V , W be non empty VectSpStr over K ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom ( f2 | [. 0 , 1 .] ) = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: ( X /\ Y ) .= proj2 .: ( X /\ Y ) ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - a < x0 - a . n + a . n ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
SIC S ^2 = ( S . g ) ^2 * ( S . g ) ^2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) . a in dom ( Initialized ( p ) ) ;
i1 , i2 , a3 , a4 , a5 , a6 , x5 , 6 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , 8
( ( #Z n ) + ( cos * ( r + 0 ) ) ) ^2 = ( cos * ( r + 0 ) ) ^2 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x implies f2 is_differentiable_in x
reconsider q2 = sqrt ( q `1 ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + ( j + 1 ) ;
assume f in the carrier of [: X , Y :] ;
F . a = H _ ( ( x , y ) \leftarrow ( y , x ) ) ;
true ( T , u ) = TRUE ( TRUE , u ) .= TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in [. 0 , 1 .] ;
( p2 `2 - x1 ) `2 - x1 `2 > - g `2 - g `2 ;
|. r1 - p .| = |. a1 .| * |. q1 - p .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b ;
DWWWWWWWWWWWWWWWWWWW} + 1 = 1 ;
i1 = ( - n ) + n & i2 = ( - n ) + n ;
f . a [= f . ( f .: O ) "\/" f . ( f .: O ) ;
attr f = v & g = u + v ;
I . n = Integral ( M , F . n ) | E ;
( \raise .4ex \hbox { $ \chi $ } } , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b2 as Element of NAT ;
( Comput ( P , s , 4 ) ) . SBP = 0 ;
L~ M1 meets L~ ( R + S ) /\ L~ R ;
set h = the continuous Function of X , R ;
set A = { L . ( k + 1 ) where k is Element of NAT : not contradiction } ;
for H st H is negative holds P [ H ] ;
set b13 = S5 ^ ( i , -' j ) , yy2 = ( i + j ) + ( j + j ) ;
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 + n ) < sqrt ( 1 + s ) " ;
( [ l , cod l ] ) `2 = [ [ l , cod l ] `2 ] `2 .= [ l , cod l ] `2 .= [ l , cod l ] `2 ;
y +* ( i , y ) /. i in dom g ;
let p be Element of Al ( ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( p1 .. p1 ) ) \/ rng ( f /^ ( p2 .. p1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) <= len D2 ;
assume x in ( ( ( ( ( ( ( TOP-REAL 2 ) ) | K1 ) ) ) | K1 ) ) /\ K1 ) /\ K1 ) ;
- 1 <= ( f2 . O ) `2 & ( f2 . O ) `2 <= ( f2 . O ) `2 ;
f , g as Function of I[01] , TOP-REAL 2 , R^1 ;
k1 - k2 = k1 - k2 - k1 + k2 - k2 - k2 + k2 - k2 + k2 - k2 + k2 - k2 + k2 - k2 + k2 - k2 = ( k1 - k2 ) + ( k2 - k2 ) - ( k1 - k2 ) + ( k2 - k2 ) -
rng ( seq + c ) c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 + r .[ \/ ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K .= - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A st a = Sum A or a is Ordinal of A ;
Cl ( Cl ( H ) ) = union ( Cl ( H ) ) .= union ( Cl ( H ) ) ;
len t = len t1 + len t2 + len t1 .= len t1 + len t2 + len t1 ;
v = v + w |-- ( v + A ) ;
v <> DataLoc ( ( t . GBP ) . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( y . s ) ;
{ s : s < t } in NAT & t in { {} } ;
s ` \ s = s ` \ ( 0. X ) \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A ;
( 319 + 1 ) ! = 319 ! * ( 319 + 1 ) ;
U ( succ A ) = U ( U ( ) ) .= U ( ) ;
reconsider y = y as Element of COMPLEX ( len y ) ;
consider i2 being Integer such that y2 = p * i2 and i2 in A and x = i2 * i2 ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) \! \mathop { N } ;
consider Z be set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , R^1 ;
( 1 , A ) . [ n + i , A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , a be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. w .| ;
consider y being Element of S such that z <= y and y in X ;
a 'or' ( b 'or' c ) = 'not' ( a 'or' b ) 'or' 'not' ( a 'or' c ) ;
||. ( ( x - g ) - ( g - h ) ) . m .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 - ( k1 + 1 ) & k1 + 1 - ( k2 + 1 ) = k2 + ( k2 + 1 ) ;
sqrt ( ( p `1 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) >= 0 ;
sqrt ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) < 0 ;
W-min ( C ) in Support ( ( R /. 1 ) * ( i , 1 ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( F | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a `2 // a `1 , b `2 or p `1 , a `2 `2 = b ;
g . n = a * Sum ( f | n ) .= f . n * f . n ;
consider f being Subset of X such that e = f and f is 1-element ;
F | ( N2 , S ) = ( CircleMap * F ) * ( N2 , S ) ;
q in LSeg ( q , v ) \/ LSeg ( p , q ) ;
Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of ( V ) = { 0. V } & the carrier of ( V ) = { 0. V } ;
rng ( ( ( - 1 ) (#) ( cos - f ) ) `| [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable ;
||. ( ( vseq . n ) - ( vseq . m ) ) . n .|| < e ;
set g = O --> 1 ;
reconsider t2 = t as 0 -started string of S2 ;
reconsider xx0 = ( seq ^\ k ) . n as sequence of REAL n ;
assume means : L~ Cage ( C , n ) meets L~ Cage ( C , n ) & not Cage ( C , n ) meets L~ Cage ( C , n ) ;
- ( Cl Cl 1 ) < F . n - ( r . x ) ;
set d1 = \bf \bf L ( x1 , y1 , z1 ) , d2 = dist ( y1 , z1 ) ;
2 |^ ( ( sqrt 5 ) - 1 ) = ( 2 |^ ( ( sqrt 5 ) - 1 ) ) / ( 2 |^ ( sqrt 5 ) ) ;
dom ( v . ( len ( v . ( len v ) ) ) ) = Seg len ( v . ( len v ) ) ;
set x1 = ( - ( k2 - k1 ) ) / ( k2 - k1 ) ;
assume for n being Element of NAT holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= ( TH ) . i and ( TH ) . i <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L2 ) + Carrier ( L2 ) ) c= I /\ dom ( L2 ) ;
'not' All ( x , p ) => 'not' All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal normal normal w.r.t. A ;
Z c= dom ( ( ( ( ( ( exp_R * f1 ) ) `| Z ) ) (#) ( ( exp_R * f1 ) `| Z ) ) ) ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r / 2 ;
\bf IC \bf \bf \bf \bf \bf \bf ( A , succ d ) c= \bf IC ( d , \bf IC \bf SCM ( A ) ) ;
E = dom ( L . n ) & ( L . n ) is_measurable_on E ;
C |^ ( A + B ) = C |^ ( B + C ) ;
the carrier of W2 c= the carrier of ( V ) & the carrier of ( V ) c= the carrier of ( V ) ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
attr x > 0 means : Def6 : x = x / 2 & x < 1 / 2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) /\ LSeg ( f , i ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c are_connected & - C , - C are_homotopic implies - C , - C are_isomorphic
assume f = id ( the carrier of O ) & f is one-to-one ;
consider v such that v <> 0. V and f . v = L . v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \/ the carrier of V ) ;
reconsider g = f " as Function of [: U2 , U2 :] , U2 ;
A1 in the points of ( k , X ) . ( X . i ) ;
|. - x .| = - ( x - x ) .= - x + x .= - x ;
set S = 1GateCircStr ( x , y , c ) ;
Fib ( n ) * ( 5 * 5 ) - ( 5 * 5 ) >= 4 * 5 ;
v3 /. ( k + 1 ) = v3 . ( k + 1 ) ;
0 mod i = ( - ( i mod ( 0 qua Nat ) ) ) * ( i mod ( 0 qua Nat ) ) ;
Indices ( - M1 ) = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
Line ( SX2 , j ) . j = SX2 . j .= Line ( cX2 , j ) . j ;
h . ( x1 , y1 ) = [ y1 , y2 ] .= [ y1 , y2 ] ;
|. f .| is_integrable_on ( Re ( f (#) ( |. b .| (#) h ) ) ) is non-negative ;
assume x = ( a1 ^ <* x1 *> ) ^ <* y1 *> ^ <* y2 *> ;
M is closed on IExec ( I , P , s ) , P & M is halting implies M is closed
DataLoc ( 4 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| + |. y .| = - x + y ;
LIN c , q , b & LIN c , q , q ;
f13 . ( 0 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + y2 ) ;
fya1 . a = ( f . a ) . a & v in InputVertices ( S . a ) ;
( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 ;
consider p such that p = pO and s1 < p and p < s2 and p < s2 ;
|. ( f /* ( s * F ) ) . l - ( f /* ( s * F ) ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N & width N = width N ;
f1 /* ( f1 /* ( s ^\ k ) ) is convergent & f2 /* ( s ^\ k ) is convergent ;
f . x1 = y1 & f . y1 = y2 & f . y2 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & dom ( Proj ( i , n ) * s ) = REAL m ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 as Subset of X ( ) ;
1 in the carrier of [. 1 / 2 , 1 .] & [. 1 / 2 , 1 .] c= [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds lattice \langle L , L \rangle , L \rangle is isomorphic ;
[ gi , gj ] in [: I , I :] \ [: I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f1 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 } & s . 1 = 5 & s . 3 = 4 ;
( min ( g , min ( f , g ) ) ) . c <= h . c ;
set G2 = the subgraph of G , G2 = the subgraph of G , v = the Vertex of G , w = the Vertex of G , w = the Vertex of G , n = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n ;
|. s1 . m - p . ( m + 1 ) .| < d / ( p . m - p . n ) ;
for x being element st x in Q holds x in ( iff x in Q & x in Q ) & x in Q & x in Q ;
P = the carrier of ( TOP-REAL n ) | ( P | ( P | ( P | ( P | P ) ) ) ) ;
assume that p1 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) = {} ;
( 0. X \ x ) |^ m = 0. X \ ( m + 1 ) ;
let g be Element of Hom ( cod f , cod g ) ;
2 * a * b + ( 2 * c ) * d <= 2 * ( C + D ) * ( C + E ) ;
f , g be PartFunc of the carrier of X , the carrier of Y ;
set h = Hom ( a , g ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg n ;
H * ( g " * a ) in the carrier of H * ( g " * a ) ;
x in dom ( ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) ) ) `| dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) (
cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , p2 , P , p1 , p2 & LE q2 , p1 , p2 implies LE q2 , p1 , p2
attr B is component means : Def6 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng $2 ( ) ;
n + - n < len ( p ^ <* n *> ^ <* n *> ) + - n ;
attr a <> 0. K means : Def6 : rk = rk ( a ) & rk ( a ) = rk ( a ) ;
consider j such that j in dom |^ m and I = len an + j ;
consider x1 such that z in x1 and x1 in ( P . x1 ) & y1 in ( P . x1 ) & y2 in ( P . x2 ) ;
for n ex r being Element of REAL st X [ n , r ]
set CC1 = Comput ( P2 , s2 , i + 1 ) ;
set \cal v = 3 / ( a , b ) , w = 3 / ( a , b ) , y = 3 / ( a , b ) , z = 4 / ( a , b ) ;
conv ( F .: W ) c= union ( F .: ( E " ( W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) ) ) ) ) ) ) ) ;
s3 <= ( s2 + ( r2 - ( s2 - ( s2 - s1 ) ) ) ) / 2 + ( s2 - ( s2 - s1 ) ) / 2 ;
dom ( f * ( f1 + f2 ) ) = dom f /\ dom ( f1 + f2 ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k /\ Seg k ;
rng ( s ^\ k ) c= dom f1 \ ( f1 /* ( s ^\ k ) ) ;
reconsider g2 = gp as Point of ( TOP-REAL n ) | LSeg ( p , q ) ;
( T * h . s ) . x = T . ( h . s ) ;
I . ( L . J ) = ( I * L ) . ( J . x ) ;
y in dom ( ( the 19 implies implies implies ( the Sorts of A ) . o ) ) . y = ( the Sorts of A ) . y ;
for I being non degenerated commutative commutative commutative commutative commutative commutative commutative associative non empty doubleLoopStr holds the carrier of I is commutative
set s2 = s +* ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC Comput ( P1 , s1 , i ) = P1 . IC Comput ( P1 , s1 , i ) .= P1 . IC Comput ( P1 , s1 , i ) ;
lim S1 in the carrier of [. a , b .] & lim S1 = a & lim S1 <= b ;
v . i = ( v *' ) . i .= ( v *' ) . i ;
consider n being element such that n in NAT and x = seq . n ;
consider x be Element of c such that F1 . x <> F2 ( x ) and F2 . x <> F2 ( x ) ;
X ( ) , x1 ( ) , x2 ( ) are_equipotent ( ) = { E ( ) } ;
j + ( 2 * k1 ) + ( 2 * k1 ) > j + ( 2 * k1 ) ;
{ s , t } on on on on Q & { t , t } on Q ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n4 , n4 , n4
g1 . ( HT ( g2 , T ) ) = 0. L ;
then H1 , H2 are_isomorphic & card H1 , H2 are_isomorphic & card H1 , H2 are_equipotent implies card H1 , H2 .] is R1
( ( N-min L~ f ) .. f ) .. f > 1 & ( GoB f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 1 .] /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) ;
let f1 , f2 be PartFunc of REAL , REAL , f2 be PartFunc of REAL , REAL ;
DigA ( t-23 , z1 ) is Element of k -tuples_on ( k -tuples_on BOOLEAN ) ;
I \lbrack d , k1 , I , I , J , k2 , I , J , K , M , M , N , N , M , N , N , M , N , N , N , M , N , N , N , M , N , N , N , M , N , N , K , M , N , N
u9 \times { u } = { [ a , u ] } & [: a , u ] , [: b , u :] :] = { [ a , u ] } ;
( w | p ) | ( p | ( w | p ) ) = p ;
consider u2 such that u2 in W2 and x = v + u and u in W2 and x = v + u ;
for y st y in rng F ex n st y = a |^ n & y in rng F ;
dom ( ( g * ( ( ( ( ( g ) ) \dot \to C ) ) ) ) ) ) ) = K ;
ex x being element st x in ( [#] U2 ) \/ ( A \/ B ) ;
ex x being element st x in ( ( UAp O ) . s ) . s & x in ( the Sorts of U1 ) . s ;
f . x in the carrier of [. - r , r .] & f . x in the carrier of [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 union X2 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 } /\ LSeg ( p1 , p2 ) ;
sqrt ( b + ( bs ) ) < { r : a < r & r < b } ;
sup { x , y } "\/" x = sup { x , y } & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of Gsuch that z = y and P [ z ] ;
( the real of ( X ) ) . e <= e ;
len ( w ^ w2 ) + 1 = len w + ( len w + 1 ) ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & q = ( TOP-REAL 2 ) | K1 ;
f | ( E ` ) ` = g | ( E ` ) ` ` .= g | ( E ` ) ` ` .= g | ( E ` ) ` ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 as Element of NAT ;
( a * A ) ` = ( a * A ) ` ` .= ( a * A ) ` ;
assume ex x0 be Element of NAT st f |^ x0 is < x0 & x0 < x0 + r ;
Seg len ( ( ( ( Sum f2 ) ) * ( f | i ) ) ) = dom ( ( Sum ( f2 ) ) * ( f | i ) ) ) ;
( ( Complement ( A ) ) . m ) c= ( ( Complement ( A ) ) . n ) . m ;
f1 . p = p8 & g1 . p = d & g1 . ( p . q ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( |. x .| ) <= sqrt ( ( r ^2 ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F . n ) = dom g & dom ( F . n ) = dom g ;
assume for x , y st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W2 and W1 is Subspace of W2 ;
||. ( ( t . x ) - ( t . x ) ) .|| = lim ( ||. ( t . x - t . x ) .|| ) ;
assume that i in dom D and f | A is bounded and g | A is bounded ;
sqrt ( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) <= sqrt ( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) ;
g | Ball ( p , r ) = id Ball ( p , r ) ;
set Nmin = ( Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable iff the topology of T is countable
width B |-> 0. K = Line ( B , i ) .= width ( B @ ) .= width ( B @ ) .= width ( B @ ) .= width ( B @ ) ;
attr a <> 0 means : Def2 : ( A \subseteq B iff a \in ( A ++ B ) c= ( A ++ a ) ;
then f is_partial u0 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u0 , 1 ;
assume that a > 0 and a <> 1 and b <> 0 and c <> 0 and a <> 0 and b <> 0 ;
w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w2 , w1 , w1 , w2 , w1 , w2 , w1
p2 /. IC Comput ( p2 , s2 , i ) = p2 . IC Comput ( p2 , s2 , i ) .= ( p2 . IC Comput ( p2 , s2 , i ) ) .= ( p2 . IC Comput ( p2 , s2 , i ) ) ;
ind ( ( T | b ) | b ) = ind b .= ind b .= ind b .= ind b ;
[ a , A ] in the Points of Line ( A , 1 ) & [ a , A ] in the Sorts of Line ( A , 1 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( 'not' Ex ( a , CompF ( PA , G ) ) ) . z = TRUE ;
reconsider \varphi = \varphi /. 11 , \varphi = l /. ( 11 + 1 ) as Element of phi . ( 11 + 1 ) ;
len s1 - ( len s2 ) * ( len s2 - 1 ) + 1 > 0 + 1 ;
\delta ( f * ( f . ( sup A ) - f . ( lower_bound A ) ) ) < r ;
[ f , f21 ] in the carrier of A & [ f , f22 ] in the carrier of A & [ f , f22 ] in the carrier of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = g2 . z ;
[#] ( V1 ) = { 0. V } .= the carrier of ( V1 ) \/ the carrier of ( V1 ) ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P is one-to-one and P is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ^ <* p3 *> ) .= h ^ h ;
c /. [ b , c ] = c /. [ a , c ] .= c /. [ a , c ] .= c /. [ a , c ] ;
reconsider t1 = p1 , t2 = p2 , t2 = p1 as Point of C ( ) ;
sqrt ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 1 ) ) ) ) ) / ) ) ) ) ) ) ^2 ) )
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D ) `2 .= C * ( p1 `2 + D ) `2 + D * ( p1 `2 + E ) `2 .= C * ( p1 `2 + E ) `2 + E * ( p1 `2 + E `2 ) `2 ;
R . b = 2 * PI .= 2 * PI .= 2 * PI .= PI ;
consider y2 such that B = 1- ( 1 - ( y1 * y2 ) ) + 0 and 0 <= y1 and y1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of B ) ) ) ;
[ P . ( l ) , P . ( l ) ] in => ( ( T . ( l ) ) => ( T . ( l ) ) ) ;
set s2 = Initialize ( s ) , P1 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as Matrix of n , REAL ;
y in product ( ( Carrier J ) +* ( { 1 } ) ) ) ;
1 / |[ 0 , 1 ]| = 1 / ( 0 , 1 / ( 1 + ( 1 / 2 ) ) ) ]| & 1 / ( 1 + ( 1 / 2 ) ) = 0 ;
assume x in the left left of g or x in the left of g or x in the right of f & x in the right of g ;
consider M being strict Subgroup of A such that a = M and T is Subspace of A and M is Subspace of A ;
for x st x in Z holds ( ( ( ( exp_R + f ) + exp_R ) ) `| Z ) . x <> 0 & ( ( exp_R + f ) `| Z ) . x <> 0 ;
len W1 + len W2 + m = 1 + len W1 + len W2 + len W2 + m + m + m + m + n + n ;
reconsider h1 = ( v . n ) - t . n as Lipschitzian LinearOperator from X , Y ;
( len p + len q ) + 1 in dom ( p + q ) ;
assume that s2 is conjunctive and F is the |= of ( the Sorts of s1 ) & F is the Sorts of s2 ;
( ( gcd ( x , y ) ) ) , ( ( gcd ( x , y ) ) ) ] = gcd ( x , y ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + ( p + m ) . u
for B being Subset of u st B in E holds A = B or A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = [: p , q :] , W1 = [: p , q :] , W2 = [: p , q :] ;
x in { X where X is Subset of L : ex Y being Subset of L st Y is Subset of L & X is closed } ;
the carrier of W1 /\ ( W1 /\ W2 ) c= the carrier of ( W1 + W2 ) /\ ( the carrier of W2 ) ;
[ [ a + b ] , id a ] * id [: a , b :] = id [: a , b :] * id [: a , b :] ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( Al ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= sqrt ( ( PI / 2 ) |^ n ) + 1 ;
( reproj ( 1 , z ) ) . x in dom ( f1 (#) f2 ) & ( f1 (#) f2 ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = { c2 } ;
ex P st a1 on P & a2 on P & a3 on P & a2 on P & a3 on Q & a4 on P & a4 on P & a4 on Q ;
reconsider gf = g `2 * f `2 as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( \mathopen { \downarrow v1 ) ` and v1 in ( \mathopen { \downarrow } v1 ) ` ;
n in { i where i is Nat : i < n + 1 } ;
( F /. i , j ) `2 >= ( F /. ( m + k ) ) `2 ;
assume K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn & ( p `2 / |. p .| - sn ) / ( 1 + sn ) >= sn } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = P +* ( ( a , intloc 0 ) , I2 = P +* ( a , intloc 0 ) ;
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. ( i + 1 ) <> z /. ( i + 1 )
X c= ( the carrier of L1 ) & the carrier of L2 c= the carrier of L2 & the carrier of L1 c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a |^ 3 ;
reconsider e1 = e1 , e1 = e1 , e1 = f as Element of D ( ) ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 . m ;
f * g is_differentiable_in ( proj ( i , m ) ) . x & f * reproj ( i , m ) . x = ( proj ( i , m ) ) . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ A + ( $1 + 1 ) ;
the left options of - g = the right of g & the right of g = the right of g & the right of g = the carrier' of g ;
reconsider pX2 = x , pX2 = y as Point of ( TOP-REAL 2 ) | P ;
consider g2 such that g2 = y and x <= g2 and x <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and x0 <= g2 and g2 <= x0 and x0 < g2 and g2 <= x0 and x0 < g2 and g2 <= x0 and x0 < g2 and x0 < g2 and g2 < x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ;
for x being element st x in X holds x in the set of the set of \HM { n } & x in the set of \HM { n } & x in the set of \HM { n }
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func ( 1 ) -tuples_on X -> set equals ( \mathop { \rm id X ) * ( id X ) ) * ( id X ) ;
len ( ( ( ( C /. 1 ) * ( D /. 1 ) ) ) * ( ( C /. 1 ) * ( D /. 1 ) ) ) ) <= len ( ( ( C /. 1 ) * ( D /. 1 ) ) ) ;
attr K is has a & a <> 0. K implies v . ( a |^ i ) = i * a |^ ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o in the carrier' of S ;
for x st x in X ex y st x c= y & y in X & x is fixpoint of f . y holds y is fixpoint of f . y
IC Comput ( P1 , Comput ( P1 , s1 , k ) ) in dom ( I +* J ) ;
attr q < s & r < s & ]. p , q .] c= ]. p , q .] ;
consider c be Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
the ResultSort of S2 = id the carrier' of S2 & the ResultSort of S2 = id the carrier' of S2 ;
set yz = [ <* y , z *> , f2 ] , yz = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( ( ( exp_R * ( f + exp_R ) ) ) ) `| Z ) ) `| Z ) ) /\ dom ( ( ( exp_R * ( f + exp_R ) ) `| Z ) ) ;
r-7 in Int cell ( GoB f , i , j ) \ L~ f \ LSeg ( f , i ) ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 - len f1 + len f1 +
for n ex x st x in N & x in N1 & h . n = x0 & h . x = x0 ;
set s = ( > 0 ) , p = ( > 0 ) , s = ( 0 , I ) , p = ( 0 , I ) , q = ( 0 , I ) , r = ( 0 , I ) , s = ( 0 , I ) , M = ( 0 , I ) , M = ( 0 , I ) ;
p . k = 1 or ( p . k ) = 1 or ( p . k ) = 1 or ( p . k ) = 0 ;
u + Sum ( L \ { u } ) in ( U \ { u } ) \/ ( Carrier L ) ;
consider x9 being set such that x in x9 and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( m1 + ( len p ) ) . m1 ;
g + h = gg + h & g + h = g + h + ( g + h ) ;
L1 is distributive & L2 is distributive & L1 is distributive implies L1 "\/" L2 is distributive
attr x in rng f & y in rng ( f | x ) & x in rng ( f | y ) ;
assume that 1 < p and sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = 1 and 0 <= p `2 and p `2 <= 0 ;
F* ( f , H ) = rpoly ( 1 , H ) + t .= ( 1 , t ) *' ;
for X being set , A being Subset of X holds A ` = {} implies A ` = {} X
( ( ( N-min X ) /. i ) `1 ) `1 <= ( ( ( N-min X ) /. i ) `1 ;
for c being Element of the bound of A holds c <> a & c <> a implies c <> a
s1 . DataLoc ( ( s2 . GBP ) , 3 ) = ( Exec ( i2 , s2 ) ) . DataLoc ( ( s2 . SBP ) , 3 ) . DataLoc ( ( s2 . a ) , 3 ) .= ( s2 . a ) . DataLoc ( ( s2 . a ) , 3 ) .= ( s2 . a ) . DataLoc ( s2 . a , 3 ) ;
for a , b being Real holds [ a , b ] in ( y iff a >= 0 ) & b >= 0 implies a >= 0
for x , y being Element of X holds x ` \ y ` \ x = ( x \ y ) ` \ y `
mode BCK-algebra of i , j , m , n , m , n , k , m , n , m , n , n , m ;
set x2 = ( Re ( y ) ) | ( ( Re y ) | ( ( Im y ) | ( dom y ) ) ) ;
[ y , x ] in dom ( u . y ) & u . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) - lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) .| < e / 2 ;
( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) <= ( - ( q `2 / |. q .| - sn ) ) / ( 1 + sn ) ;
set A = sqrt 2 ;
for x , y being set st x in R & y in R holds x , y are_\hbox { $ \subseteq $ }
deffunc F ( Nat ) = b . ( $1 * M ) * ( b . $1 ) ;
for s being element holds s in |= ( f 'or' g ) iff s in |= ( f 'or' g ) \/ ( f 'or' h )
for S being non empty non void partial with_b1 b1 b1 b1 b1 b1 b1 b1 b1 b1 b1 b1 b2 b2 for b2 , b2 , b3 be Point of S st S is connected & b2 is connected holds S is connected
max ( degree , degree ) >= 0 & degree ( z ) >= 0 & degree ( z ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) ;
set n-15 = nnnmeet ( M , x ) , M = ( n -tuples_on the carrier of M ) , CnF = ( n -tuples_on the carrier of M ) ;
f " ( V ) in ( id the carrier of X ) & f " ( V ) in D ( ) & f " ( V ) in D ( ) ;
rng ( ( a , c ) ) +* ( ( a , b ) +* ( c , d ) ) c= { a , b } ;
consider y being subgraph of G1 such that y = y and dom y = y and dom y `2 = WWGraph( G1 ) ;
dom ( ( 1 / 2 ) (#) f ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ ;
Matrix ( i , j , n , r ) is Element of Matrix ( i , j , n ) , K ;
v ^ ( n |-> 0 ) in Lin ( rng ( ( B | ( support ( b1 | c1 ) ) ) ) ) ;
ex a , k1 , k2 st i = a := k1 & k2 = b := k2 & k1 <> k2 & k2 <> b & k2 <> b ;
t . [: NAT , NAT :] = ( [: NAT , NAT :] --> ( i , NAT ) ) . ( i , NAT ) .= ( ( the Sorts of A ) . ( i , NAT ) ) . ( i , NAT ) .= ( the Sorts of A ) . ( i , NAT ) .= ( the Sorts of A ) . ( i , NAT ) ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and rng p c= Seg ( n + 1 ) ;
not LIN b , b9 , a9 & not LIN a , b9 , c9 & not LIN a , a9 , b9 & not LIN a , b9 , c9 ;
( L1 -L2 ) => O c= ( L1 '&' L2 ) => ( L2 '&' O ) & ( L1 '&' L2 ) => ( L2 '&' O ) c= ( L1 '&' L2 ) => ( L2 '&' O )
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( w - v ) = b * ( y - w ) and 0 < a and 0 < b and a < 0 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) * |. $1 .| ;
u = cos ^ ( x , y ) . v * x + cos . ( x , y ) . v .= v + ( cos . x ) * y .= v + ( cos . y ) * x ;
dist ( ( seq . n ) + x , g ) <= dist ( ( seq . n ) + g ) + 0 ;
P [ p , |. p .| : |. p .| < |. p .| & |. p .| < |. p .| ;
consider X be Subset of Al ( ) such that X c= Y ( ) and X is finite and X is finite and X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( GoB Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g . l1 <= l & l <= h . l1 & l <= h . l1 } ;
( Partial_Sums ( G . n ) ) . n <= ( Partial_Sums ( G . n ) ) . ( n + 1 ) ;
f . y = x .= x * 0. L .= x * 0. L .= x * 0. L .= x * 0. L ;
NIC ( i1 , i2 ) = { i1 , i2 , i1 , i2 , i2 , i2 , j2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) .= {} ;
Product ( ( ( Carrier ( I ) ) +* ( i , { 1 } ) ) ) ) in Z . i ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) +* ( the carrier of S2 ) .= Following ( s1 , n ) +* ( the carrier of S2 ) ;
( W-min ( P ) ) `1 <= ( q `1 ) / 2 & ( q `2 ) / 2 <= ( q `2 ) / 2 ;
f /. i2 <> f /. ( i1 + len g -' 1 ) & f /. ( i1 + len g -' 1 ) = f /. ( i1 + len g -' 1 ) ;
M , v / ( ( x. 3 ) / ( x. 4 , ( x. 4 ) / ( x. 4 , x. 4 ) ) ) |= H / ( x. 4 , ( x. 4 ) / ( x. 4 , x. 4 ) ) ) ;
len ( ( P ^ ) ^ ( Q ^ ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) ;
A |^ ( m , n ) c= A |^ m & A |^ ( k , n ) c= A |^ ( k , n ) ;
R |^ n \ { q : |. q .| < a } c= { q : |. q .| < a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X be set such that X in Q and for Z st Z in Q & Z <> X holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = ||. ( ||. v .|| ) . v .|| & ||. v .|| = ||. ( ||. v .|| ) . v .||
for phi st phi in X holds phi in X & phi in X & phi in X ;
rng ( ( Sgm dom ( f | ( dom f ) ) ) ) c= dom ( ( Sgm dom ( f | ( dom f ) ) ) | ( dom ( f | ( dom f ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a . ( len c ) = c . ( len c ) ;
Args ( a , b ) = <* .: ( <* b , c *> , <* c , d *> , <* d , e *> , <* f , g *> , g *> *> ) ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 & a3 = b3 & a4 = b3 & a4 = b3 & a4 = b3 & a4 = 6 & a4 = 6 & a4 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 6 & 8 = 7 & 8 = 8 & 8 = 8 & 8 = 8 & 8 = 7 & 8 = 8 & 8 = 8
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D2 . ( indx ( D2 , D1 , n1 ) + 1 ) ;
f . ( |. r .| ) = <* |[ r , s ]| .| .= <* |[ r , s ]| *> . 1 .= <* r , s ]| . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds Cseq . m = Cseq . m ;
consider d being Real such that for a , b being Real st a in X & b in Y & a <= b holds a <= b ;
||. L /. ( h /. ( h /. ( h /. ( h + c ) ) - L /. ( h /. c ) ) .|| <= x0 + R /. ( h /. c ) - L /. ( h /. c ) ) ;
attr F is commutative associative means : Def6 : for b being Element of X holds F . b = f . b ;
p = ( r * p2 + 0 ) * ( p + 0. TOP-REAL 2 ) .= 1 * ( p + 0. TOP-REAL 2 ) .= 1 * ( p + 0. TOP-REAL 2 ) .= 1 * ( p + 0. TOP-REAL 2 ) .= 1 * ( p + 0. TOP-REAL 2 ) .= p + 0. TOP-REAL 2 ;
consider z1 such that b , x1 , z1 , z2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5
consider i such that Arg ( ( Rotate ( s , q ) ) ) = s + ( 2 * PI ) + ( 2 * PI ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g c= f . x and rng g c= f . x ;
assume that A = P2 \/ Q and ( P \/ Q ) misses ( P \/ Q ) and ( P \/ Q ) misses ( P \/ Q ) and ( P \/ Q ) misses ( P \/ Q ) ;
attr F is associative means : Def6 : F .: ( F .: ( f , g ) , h ) = F .: ( f , h ) ;
ex x being Element of NAT st m = x `1 & x `2 < z `1 & x `2 < z `2 or m in { i } ;
consider k2 be Nat such that k2 in dom ( P . ( k2 + 1 ) ) and l in P . ( k2 + 1 ) ;
seq = r * ( seq . n ) implies for n holds seq . n = r * ( seq . n )
F1 . [ [ id a , id a ] , [ a , a ] ] = [ f * ( id a ) , [ id a , id a ] ] ;
{ p } "\/" D2 = { p "\/" q where q is Element of L : q in D & q in D } ;
consider z being element such that z in dom ( ( dom F ) . 0 ) and ( ( dom F ) . 0 ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E ) and ( T | E ) . e = v ;
( F . b1 * b1 ) . x = ( ( ( Mx2Tran J ) . b1 , b2 ) ) . b1 .= ( ( ( Mx2Tran J ) . b1 , b2 ) ) . b2 ;
- ( 1 / ( - ( 1 / 2 ) ) ) = ( - ( 1 / 2 ) ) (#) ( - ( 1 / 2 ) ) .= ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) .= ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) .= ( - ( 1 / 2 ) ) * ( - ( 1 / 2 ) ) .= ( - ( 1 / 2 )
attr x in dom f /\ dom g & g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( 'not' a , A , G ) , B , G ) '<' All ( 'not' All ( 'not' a , B , G ) , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp ( Cage ( C , n ) , i + 1 ) ;
x \ ( a |^ m ) = x \ ( a |^ k ) .= ( x \ a ) |^ k .= ( x \ a ) |^ k ;
k -func func func func ( I ) -> Element of ( commute ( I ) ) . k equals ( commute ( I ) ) . k .= ( commute ( I ) ) . k .= ( commute ( I ) ) . k ;
for s being State of A2 holds Following ( s , n ) . ( n + 2 ) is stable ;
for x st x in Z holds ( f1 . x ) ^2 = a ^2 & ( f1 . x ) ^2 <> 0 & ( f1 . x ) ^2 < ( f1 . x ) ^2 ;
support ( ( support ( n ) ) \/ support ( ( support ( m ) ) ) ) c= support ( ( support ( n ) ) \/ support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier of B ) | ( the carrier of C ) ;
- sqrt ( ( a * sqrt ( 1 + b ^2 ) ) ^2 ) <= - ( ( - sqrt ( 1 + a ^2 ) ) ^2 ) ;
phi /. ( succ b1 ) = g . ( a . a ) & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 } \/ { x4 } ;
the Sorts of U1 /\ ( the Sorts of U2 ) c= the Sorts of U2 /\ ( the Sorts of U2 ) ;
( - ( 2 * a ) + sqrt ( 2 * a ) ) + ( - sqrt ( 2 * a ) ) ^2 ) > 0 ;
consider W00 such that for z being element holds z in W iff z in N & z in N & P [ z ] ;
assume that ( the ResultSort of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = r ;
Z = dom ( ( ( ( ( ( ( exp_R + arccot ) ) (#) ( exp_R + arccot ) ) ) (#) ( ( exp_R + exp_R ) (#) ( exp_R + exp_R ) ) ) ) `| Z ) ;
integral ( f , SS1 ) is convergent & lim ( integral ( f , S1 ) ) = integral ( f , S1 ) + integral ( f , S1 ) ;
X ( f . ( a , f . b ) => ( ( g . a ) => ( ( g . b ) => ( ( g . b ) => ( ( g . a ) => ( ( g . b ) => ( ( g . a ) => ( ( g . b ) => ( ( g . a ) => ( ( g . a ) => ( ( g . b ) => ( ( g . a ) => ( ( g . b ) => ( ( g
len ( ( - M2 ) * ( M - M2 ) ) = n & width ( - M2 ) * ( M - - M2 ) = n ;
attr X1 union X2 is open means : Def6 : X1 union X2 is open SubSpace of X & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open implies X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union X2 is open & X1 union
for L being lower-bounded antisymmetric non empty RelStr for X being non empty Subset of L holds X "\/" Y = { Bottom L } & X "\/" Y = { Bottom L }
reconsider f29 = ( F . b ) * ( F . b ) as Function of [: M , M :] , M ( ) ;
consider w being FinSequence of I such that the succ of M = <* s *> ^ <* w *> ^ <* w *> ^ <* w *> ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) . i & f . i = rpoly ( 1 , z ) . i ;
ex L being Subset of X st L = L & for K being Subset of X st K in C holds K /\ L <> {} ;
( the carrier of C1 ) /\ ( the carrier of C2 ) c= the carrier of C1 & the carrier of C2 /\ ( the carrier of C2 ) c= the carrier of C2 ;
reconsider o9 = o `2 as Element of TS ( ( the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) . ( o , the Sorts of A ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + 0 * x2 + 0 * x3 .= x1 + 0 * x2 + 0 * x3 .= 1 * x1 + 0 * x2 + 0 * x3 + 0 * x3 .= 1 * x1 + 0 * x3 + 0 * x3 + 0 * x4 ;
E " . 1 = ( E qua Function of E ) . ( 1 qua Function ) .= ( E qua Function ) . ( 1 qua Function ) .= ( E qua Function ) . ( 1 / E ) .= ( E . ( 1 / E ) ) . ( 1 / E ) .= E . ( 1 / E ) .= E . ( 1 / E ) .= E . ( 1 / E ) ;
reconsider u1 = the carrier of U1 /\ ( the carrier of U2 ) as non empty Subset of ( the carrier of U2 ) ;
( ( x "/\" z ) "\/" ( y "/\" z ) ) "\/" ( x "/\" y ) <= ( x "/\" z ) "\/" ( y "\/" z ) ;
|. f . ( ( s1 . l1 + 1 ) * ( s1 . l1 ) ) - ( s1 . m1 ) .| < \frac 1 / ( |. M .| + 1 ) ;
LSeg ( ( v1 , n ) * ( i , j ) ) is vertical & LSeg ( ( v1 , n ) * ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x0 - x0 ) ) + R /. ( x- ( x0 - x0 ) ) ;
g . c * ( - ( g . c ) * f . c ) <= h . c * ( - ( f . c ) ) + ( - ( f . c ) ) * f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx ( f ) in the carrier of A and width ( f ) = width A and width ( f ) = width A and width ( f ) = width A and width ( f ) = width A and width ( f ) = width A and width ( f ) = width A ;
len ( - ( - ( - ( - ( - ( - ( - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - - - ( - - ( - - ( - - - - - ( - - ( - - ( - - ( -
for n , i being Nat st i + 1 < n & i + 1 < n holds [ i + 1 , n + 1 ] in the InternalRel of ( n + 1 )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 implies pdiff1 ( f1 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2
attr a <> 0 & b <> 0 & a <> 0 & a <> 0 & b <> 0 implies Arg ( a ) = Arg ( b ) & Arg ( a ) = Arg ( b ) & Arg ( a ) = Arg ( b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( ( the topology of X ) | ( the carrier of Y ) )
assume that V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed and V1 is closed ;
z * x1 + ( 1 - ( z * y1 ) ) * ( x1 + ( - z ) ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) + ( z - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 - y2 ) * ( y1 -
rng ( ( ( ( ( ( P qua Function ) * ( P * ( Q * ( Q * ( Q * R ) ) ) ) ) ) * ( ( P * ( Q * R ) ) ) ) ) ) ) = Seg ( ( ( P * R ) * ( Q * R ) ) ) .= ( ( P * R ) * ( Q * R ) ) * ( Q * R ) ) ;
consider s2 being complex number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n ;
h2 " . n = h2 . n " . n & 0 < h2 . n implies 0 < ( h2 " ) . n & 0 < ( h2 " ) . n & 0 < ( h2 " ) . n ;
( Partial_Sums ( |. r .| ) ) . m = ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m .= ( |. r .| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1 ) * v & ( - 1 ) * w = ( - 1 ) * w & ( - 1 ) * w = ( - 1 ) * w ;
sup ( ( k .: D ) .: ) = sup ( ( k .: D ) .: ) .= sup ( ( k .: D ) .: ) .= ( k .: D ) . ( ( k .: D ) . ( k , j ) ) .= ( k .: D ) . ( k , j ) .= ( k .: D ) . ( k , j ) .= ( k .: D ) . ( k , j ) .= ( k .: D ) . ( k , j ) .= ( k , j ) . ( k ,
A |^ k , l ) .. ( A |^ n ) = ( A |^ n ) |^ ( k + 1 ) .= ( A |^ n ) |^ ( k + 1 ) ;
for R being add-associative right_zeroed right_complementable associative associative associative associative associative associative associative non empty doubleLoopStr for I , J , K being Subset of R holds I + J = ( I + J ) + ( K + L )
( f . p ) `1 = sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ;
for a , b being non zero Nat st a , b are_congruent_mod p & a , b are_congruent_mod p & a , b are_congruent_mod p holds ( e mod p ) = ( ( e div p ) * ( a div p ) )
consider A5 being countable Al ( ) such that r is \vert \vert ( Al ( ) ) .| -\vert ( Al ( ) ) .| is \vert and |. A5 .| is \vert and |. A5 .| is \vert ;
for X being non empty addLoopStr for M , N being Subset of X for x being Point of X st x in M holds x + y in M + N
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ y1 , y2 ] } & { y1 , y2 } c= [: y1 , y2 :] ;
h . O = |[ A * ( f . O ) + B * ( f . O ) , C * ( f . O ) + D * ( f . O ) ]| ;
( Gauge ( C , n ) * ( k , i ) ) /. k in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime for Nat ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b being Element of L holds a \ b <= c implies a \ b <= c \ b
consider b being element such that b in dom ( H _ { ( x , y ) } ) and z = H _ ( ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 1 & W . 2 in G . 3 & W . 3 in G . 3 ;
( ( h (#) f ) /* ( h + c ) ) . n = ( ( h (#) f ) /* n ) . x + ( h (#) f ) . x ;
j + 1 = j + len ( h11 + 1 ) .= i + ( len h + 1 ) - ( len h + 1 ) + ( len h + 1 ) - ( len h + 1 ) ;
^ ( S /* ( f /* ( f /* ( f /* ( f /* ( f /* c ) ) ) ) ) ) = S /* ( ( f /* ( f /* ( f /* c ) ) ) ) .= ( S *' ( f /* ( f /* c ) ) ) .= ( S *' ( f /* c ) ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is extended gand p in R & q <> q implies ex P st P is R & P is R & p is R & q is R ;
dom ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= meet ( dom ( X --> f ) ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom (
sup ( ( proj2 .: ( ( ( TOP-REAL 2 ) | ( C ) ) ) | ( ( TOP-REAL 2 ) | ( C ) ) ) ) ) <= sup ( ( proj2 .: ( ( C ) | ( C ) ) ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r
i * ( f - ( f - ( i + 1 ) ) ) = i * ( f - ( i + 1 ) ) .= i * ( f - ( i + 1 ) ) .= i * ( f - ( i + 1 ) ) ;
consider f being Function of 2 , X such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] ( Y | C ) and g2 in ( X | C ) and g2 in ( X | C ) and [ g1 , g2 ] in [: X , Y :] ;
redefine func d \! \mathop { n } -> Nat means : Def1 : d |^ n = n |^ ( n + 1 ) & d |^ ( n + 1 ) divides n |^ ( n + 1 ) ;
fyy1 . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a * ( - P . x ) .= a * ( - P . x ) .= a * ( - P . x ) .= a * ( - P . x ) .= a * ( - P . x ) .= a * ( - P . x ) .= a * ( - P . x ) ;
t = h . D or t = h . E or t = h . E or t = h . F or t = h . J or t = h . J ;
consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( seq . n ) . n ) < 1 / ( ( seq . n ) . m ) ;
sqrt ( ( q `1 / q `2 ) ^2 + ( q `2 / q `2 ) ^2 ) <= sqrt ( ( q `1 / q `2 ) ^2 ) + ( q `2 / q `2 ) ^2 ) ;
h21 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 ) - h . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S ( ) , x2 being Element of X ( ) such that a = [ o , x2 ] and o <= x1 ( ) ;
for L being RelStr for a , b being Element of L holds a <= b iff a <= b & b <= a & a <= b & b <= a ;
||. h1 . n .|| = ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| .= ||. ( h1 . n ) .|| ;
( ( - ( ( exp_R - exp_R ) ) ) `| Z ) . x = f . x - ( exp_R . x ) / ( exp_R . x ) .= ( - exp_R . x ) / ( exp_R . x ) .= ( - exp_R . x ) / ( exp_R . x ) .= ( - exp_R . x ) / ( exp_R . x ) .= ( - exp_R ) . x ;
attr r = F .: ( p , q ) means : Def6 : len r = len ( p ^ q ) ;
sqrt ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= sqrt ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n & i in Seg n holds Det ( ( Det ( M ) ) ) = Sum ( ( Line ( M , i ) ) )
then a <> 0. R & a " * ( a * v ) = 1 * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " * a " *
( p . j - 1 ) * ( q . ( i -' 1 ) ) = Sum ( p . ( j -' 1 ) ) * ( q . ( j -' 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( R /* ( h ^\ n ) ) . $1 * ( R /* ( h ^\ n ) ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H2 ) ;
Args ( o , Free ( X ) ) = ( ( the Sorts of Free ( S ) ) * ( the Sorts of Free ( X , Y ) ) ) . o ;
H1 = n + 1 / ( 2 to_power ( n + 1 ) ) .= n + 1 / ( 2 to_power ( n + 1 ) ) .= n + 1 / ( 2 to_power ( n + 1 ) ) .= n + 1 / ( 2 to_power ( n + 1 ) ) ;
( O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 0 & ( O O ) . ( O , O ) = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = ( F1 | ( dom F1 /\ dom F2 ) ) .: ( dom F2 /\ dom F2 ) .= ( F1 | ( dom F2 /\ dom F2 ) ) .: ( dom F2 /\ ( dom F2 /\ dom F2 ) ) .= ( F1 | ( dom F2 /\ dom F2 ) ) .: ( dom F2 /\ ( dom F2 /\ dom F2 ) ) .= ( F1 | ( dom F2 ) ) .: ( dom F2 ) .= ( F1 | ( dom F2 ) ) .= ( F1 | ( dom F2 ) ) .= ( F1 | ( dom F2 ) ) .= ( F1
attr b <> 0 & d <> 0 & b <> 0 & d <> 0 implies sqrt ( a + b ) = sqrt ( b + sqrt ( b + c ) ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) .= dom ( ( f +* g ) | D ) .= dom ( ( f +* g ) | D ) .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i being set st i in dom g ex u being Element of B st g /. i = u * a & g /. i = u * a * a & g /. i = u * a * a ;
g `2 * P `2 = g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and ( not ( ex s st s = s1 & ( not s ) . i = s1 ) & ( not ( ex s st s = s1 ) & ( not ( ex s st s = s1 & s in s1 ) & ( not ( ex s st s in s1 ) ) & ( not ( ex s st s in s1 & s in s1 ) ) & ( not ( ex s st s in s1 ) ) & ( not ( not s in s1 ) & ( not ( not s in s1 ) & ( not ( not s in s1
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] .= g | ]. a , b .] .= g | [. a , b .] ;
[ s1 , t1 ] , [ s2 , t2 ] ] is connected & [ s2 , t2 ] , [ s2 , t2 ] ] in [: s1 , s2 :] & [ s2 , t2 ] in [: s1 , s2 :] ;
then H is negative means : Def6 : H is negative & H is negative & H is negative & H is negative ;
attr f1 is total means : Def6 : ( 1 / 2 ) (#) ( f1 /* ( h + c ) ) = ( 1 / 2 ) (#) ( f1 /* c ) ;
z1 in W2 " ( W2 ` ) or z1 in ( W2 ` ) & z2 in ( W2 ` ) & z in ( W2 ` ) & z in ( W2 ` ) & z in ( W2 ` ) & z in ( W2 ` ) ;
p = 1 * p .= a " * a " * p .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) ;
for r be Real for K be Real for K be Matrix of n , REAL st for n be Nat holds K . n <= K . ( n + 1 ) holds K . ( n + 1 ) <= K . ( n + 1 )
means TOP-REAL 2 meets L~ Cage ( C , n ) or L~ Cage ( C , n ) meets L~ Cage ( C , n ) or Index ( p , C ) meets L~ Cage ( C , n ) ;
||. ( f . ( g . k + 1 ) ) - ( g . ( g . k + 1 ) ) .|| <= ||. g . ( g . k + 0 ) .|| * ||. ( g . k + 0 ) .|| ;
assume h = ( ( B .--> ( C .--> D ) ) +* ( D .--> E ) +* ( F .--> J ) ) +* ( J .--> F ) +* ( J .--> F ) +* ( J .--> F ) +* ( J .--> J ) +* ( F .--> N ) ) +* ( F .--> N ) +* ( F .--> N ) assume assume h +* ( J .--> E ) +* ( F .--> N ) ) +* ( J .--> N ) ) ) ;
|. ( ( ( H . n ) || divset ( D , i ) ) ) . k - ( ( H . n ) `| divset ( D , i ) ) . k .| <= e * ( e - ( ( H . n ) * ( e - ( H . n ) ) ) ) . k ;
( ( ( the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , the Sorts of Free ( Al , v ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . e = [
{ x1 , x1 , y1 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 } = { x1 , y1 , y2 , x4 } .= { x1 , x2 , x3 , x4 } ;
assume that A = [. 0 , PI * PI .] and integral ( ( ( cos * sin ) `| Z ) ) = 0 ;
p `2 is Permutation of dom ( f1 /. i ) & p `2 = ( ( Sgm Y ) /. i ) " * p " ;
for x , y st x in A holds |. ( 1 / 2 ) . x - ( 1 / 2 ) . y .| <= 1 * |. f . x - ( 1 / 2 ) .|
( p2 `2 ) ^2 = |. q2 .| * sqrt ( ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) ) ^2 ) .= ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) * ( ( q2 `2 ) ^2 ) .= ( ( q2 `2 ) ^2 ) * ( ( q2 `2 ) ^2 ) ;
for f being PartFunc of the carrier of C , REAL , g being PartFunc of the carrier of C , REAL st dom f = the carrier of C & dom g = the carrier of C & rng g c= the carrier of C holds f | X is continuous & g | X is continuous
assume for x be Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( 'not' All ( z , G ) ) . x = TRUE ;
consider F such that dom F = n1 and for k be Nat st k in dom F holds P [ k , F . k ] ;
ex u , u1 st u <> u1 & u , v // u1 , v1 & u , v // u1 , v1 & u , v // u1 , v1 & u , v // u1 , v1 & u , v // v1 , v2 & u , v // v1 , v2 & u , v // u1 , v1 implies u , v // u1 , v1 & u , v // v1 , v2
for G being non empty Group , N being non empty Subgroup of G holds ( N * N ) * ( N N * N ) = N N N N N N N * ( N * N )
for s be Real st s in dom F holds F . s = ( ( R ^ ) (#) ( f + g ) ) . s & ( F ^ ) . s = ( ( R ^ ) (#) ( f + g ) ) . x
width ( ( \lbrace f1 , b1 , b2 , b3 , b3 , b3 , b2 , b3 , b3 , b3 , 6 , 6 , 7 , 8 , 8 , 6 , 8 , 8 , 8 , 6 , 8 , 8 , 6 , 8 , 6 , 8 , 6 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 6 , 8 , 6 , 6 , 8 , 8 , 6 , 8 , 6 , 8 , 6 , 8 , 8 , 8 , 6 , 6 , 8 , 8 , 6 , 8 , 8 , 8 , 8 ,
f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ is continuous ;
assume that X is closed and a in X and a in X and y in X and x in X & y in X & x in X \/ Y ;
Z = dom ( ( ( ( ( ( ( exp_R * ( exp_R + exp_R ) ) ) ) (#) ( ( exp_R + exp_R * ( exp_R + exp_R ) ) ) ) ) `| Z ) /\ dom ( ( exp_R * ( exp_R + exp_R ) ) ) ) ;
func ^ ( l ) -> Subset of V means : Def6 : for k holds 1 <= k & k <= len l implies it . k = V . k ;
for L being non empty TopSpace , N being net of L , M being net of N , N being net of L holds M is net of N iff M is net of N & N is net of N
for s be Element of NAT holds ( ( ( ( the Sorts of seq ) . n ) + ( the Sorts of seq ) . n ) ) . s = ( ( ( the Sorts of seq ) . n ) . s
then z /. 1 = ( N-min L~ z ) .. z + ( ( N-min L~ z ) .. z ) .. z < ( ( N-min L~ z ) .. z + ( E-max L~ z ) .. z ;
len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len p + len <* 0 *> .= len p + len <* 0 *> .= len p + 1 ;
assume that Z c= dom ( ( ( - 1 ) (#) ( ( exp_R - f ) ) `| Z ) and for x st x in Z holds f . x = a * x + b / ( exp_R . x ) ^2 ) and for x st x in Z holds f . x = a * x + b / ( exp_R . x ) ^2 ;
for R being add-associative right_zeroed right_complementable associative associative associative associative associative associative associative associative associative non empty doubleLoopStr for I , J being Subset of R holds ( I + J ) /\ ( I /\ J ) c= I /\ J /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len x .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom ( x ^ y ) .= dom x ^ y ;
for S being category , B being Functor of C , B for c being object of C holds ( id the carrier of B ) . ( id the carrier of C ) = id ( ( the carrier of B ) . ( id the carrier of C )
ex a st a = a2 & a in f /\ g & f . a = f . ( a , a ) & f . ( a , a ) = f . ( a , a ) ;
a in Free ( H ) '&' ( ( ( H ) / ( ( ( ( H ) / ( ( ( H ) / ( ( H ) / ( ( H ) ) ) ) ) ) ) ) ) ) ) ;
for C1 , C2 being stable non empty set , f being Function of C1 , C2 holds ( ex g being stable Function of C2 , C2 st f = g & g is stable & f is stable ) iff f is stable & g is stable & f is stable & g is stable & f is stable & g is stable & f is stable & g is stable )
( W-min L~ Cage ( C , n ) ) `1 = ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 .= ( W-min L~ Cage ( C , n ) ) `1 ;
assume that u = <* x0 , y0 , y0 *> and f is partial & f is_continuous_in x0 & u = y0 & y0 = y0 & y0 = y0 & y0 = y0 ;
then ( t . {} ) `1 in ( C . {} ) `1 & ex x being Element of C ( ) st x in ( C . {} ) `1 & t . {} = ( C . {} ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a being Element of T st a = f . x & y = f . y holds a >= b ;
redefine func Class ( R , x ) -> Subset-Family of R means : Def6 : for A being Subset of R holds it . A = Class ( R , x ) ;
defpred P [ Nat ] means ( ( the Sorts of G ) . $1 ) `1 c= G . ( ( the Sorts of G ) . $1 ) `1 ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 ;
manon empty REAL ( m ) . {} = ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} .= m . {} ;
d11 = ( x ^ y ) ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .= f ^ ( y ^ z ) .=
consider g such that x = g and dom g = dom ( f | X ) and for x st x in dom ( f | X ) holds g . x = ( f | X ) . x ;
x + 0. ( n + len x ) = x + x .= x + x .= x + x + x .= x + x + x .= x + x + x .= x + x + x .= x + x + x .= x + x + x .= x + x + x + x .= x + x + x + x + x .= x + x + x + x + x + x + x .= x + x + x + x + x .= x + x + x + x + x + x .= x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x + x .= x + x + x + x
k11 - ( k -' ( k1 + 1 ) ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) + ( k -' 1 ) ) + ( k -' 1 ) + ( k -' 1 ) +
assume that P1 is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p2 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P is_an_arc_of p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1 , p2 and P c= p1
reconsider a1 = a , b1 = b , c1 = c , c2 = c , c2 = d as Element of A ( ) ;
reconsider FFFFFf = G1 . ( t * b ) as Morphism of ( G1 * F1 ) . ( t * b ) as Morphism of ( G1 * F1 ) . ( t * b ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 -' 1 + 1 -' 1 ) , f /. ( i + 1 -' 1 + 1 -' 1 ) ) ;
Integral ( P . m , P . m ) | dom ( ( P . n ) | E ) . m <= Integral ( M . n , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f1 & y in dom f2 holds f1 . x = f2 . y & f1 . y = f2 . x ;
consider v such that v = y and dist ( u , v ) < min ( ( r - ( G * ( i , 1 ) ) ) - ( r - ( G * ( i , 1 ) ) ) ) ;
for G being Group , H being Element of G holds a = b iff a |^ H = b |^ H & a |^ H = b |^ H implies a |^ H = b |^ H
consider B be Function of [: the carrier of V , the carrier of V :] , the carrier of V such that for x being element st x in the carrier of V holds P [ x , B . x ] ;
reconsider K1 = { p where p is Point of TOP-REAL 2 : ( p `1 <= 1 & p `2 <= - 1 } as Subset of TOP-REAL 2 ;
sqrt ( ( N - ( N - S ) ) / ( 2 * ( N - S ) ) / ( 2 * ( N - S ) ) ) <= sqrt ( ( N - S ) / ( 2 * ( N - S ) ) ) / ( 2 * ( N - S ) ) ) ;
for x be Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len ( @ @ p ^ q ) = len ( @ p ^ q ) + len ( @ q ^ q ) .= len ( @ p ^ q ) + len ( @ q ^ q ) .= len ( @ p ^ q ) + len ( @ q ) .= len ( @ p ^ q ) + len ( q ^ q ) .= len ( p ^ q ) + len ( q ^ q ) ;
v / ( ( x. 3 ) / ( x. 4 , m1 ) ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) ) ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) / ( x. 4 , m1 ) /
consider r be Element of M such that M , v / ( ( x. ( x , m ) ) . ( ( x. ( x , m ) ) . ( x , m ) ) ) |= r and m / ( x , m ) . ( x , m ) ) = r / ( x , m ) ;
redefine func Union w1 \ ( the Sorts of G ) \ ( the Sorts of ( the Sorts of G ) ) . ( ( the Sorts of G ) . ( the Sorts of G ) ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( ( the Sorts of G ) . ( o , the Sorts of G ) ) . ( o , ( the Sorts of G ) . ( o , the Sorts of G ) ) ) ) -> Element of Union : by { it : The , ( the Sorts of G ) . ( ( o , ( the Sorts of G ) . ( o , ( the Sorts of G ) . ( o , ( the Sorts of G ) .
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( ( s2 ) . b2 ) . b2 .= ( ( s2 ) . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 )
for n being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . n + ( Partial_Sums ( |. seq .| ) ) . n - ( Partial_Sums ( |. seq .| ) ) . n <= ( Partial_Sums ( |. seq .| ) ) . n - ( Partial_Sums ( |. seq .| ) ) . n
set F = S \! \mathop { 0 } ;
( Partial_Sums ( seq ) ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + Partial_Sums ( seq ) . n + ( ( Partial_Sums ( seq ) ) . n ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( f | Z ) . x0 ) + R . ( x- ( f | Z ) . x0 ) ;
the closed closed LE ( a , b , c , d ) , ( ( \HM { the } \HM { closed } \HM { Subset of TOP-REAL 2 ) | ( P ) ) , ( ( TOP-REAL 2 ) | Q ) , ( ( TOP-REAL 2 ) | Q ) ;
a * b ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 >= 6 * a * b * c + ( a * c ) ^2 + ( a * b ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 + ( a *
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) ;
( \HM { the } \HM { function } \HM { from } ) +* ( M , ( the carrier of M ) --> ( { TRUE } ) ) = ( ( the carrier of M ) --> TRUE ) +* ( ( the carrier of M ) --> TRUE ) .= ( the carrier of M ) --> TRUE .= ( the carrier of M ) --> TRUE .= ( the carrier of M ) --> TRUE ;
Sum ( F |^ n1 ) = ( r |^ n1 ) * Sum ( ( r |^ n1 ) ) .= ( ( r |^ n1 ) * ( r |^ n2 ) ) * ( ( r |^ n2 ) ) .= ( ( r |^ n2 ) * ( r |^ n2 ) ) * ( r |^ n2 ) .= ( r |^ n2 ) * ( r |^ n2 ) .= ( r |^ n2 ) * ( r |^ n2 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 .= ( GoB f ) * ( i1 , 2 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( Partial_Sums ( a ) ) . $1 + ( Partial_Sums ( b ) ) . $1 * ( Partial_Sums ( a ) ) . $1 + ( Partial_Sums ( b ) ) . $1 * ( Partial_Sums ( a ) ) . $1 + ( Partial_Sums ( b ) ) . $1 * ( Partial_Sums ( a ) ) . $1 + ( Partial_Sums ( b ) ) . $1 ) * ( Partial_Sums ( b ) ) . $1 ;
( the Arity of S ) . g = ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X \times Y ) \/ ( X \times Z ) c= X [: X , Y :] & card ( X \times Y ) c= card ( X /\ Y ) & card ( X /\ Z ) c= card ( X /\ Y ) ;
for a , b being Element of S , s being Element of NAT st s = F . n & a = F . n & b = G . n holds b = G . n \ G . n
E , f |= All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All ( x , H ) '&' All
ex R being 1-sorted st R = ( p | ( n + 1 ) ) & ( ( p | n ) | ( n + 1 ) ) = the carrier of R & ( ( p | n ) | ( n + 1 ) ) = the carrier of R ;
[. a , b + sqrt ( 1 + ( k + 1 ) ) .] is Element of the partial of ( the partial of M ) , REAL , REAL ) & ( the partial F ) . ( k + 1 ) is Element of the partial of M ;
Comput ( P , s , 2 + 1 ) = Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) ) .= Exec ( P , Comput ( P , s , 2 ) ) ;
card ( h1 . k ) = ( power ( K , n ) ) . k * ( ( - 1_ K , n ) ) . k ) .= ( - 1_ K ) . k * ( - 1_ K , n ) . k .= ( - 1_ K ) . k * ( - 1_ K ) . k .= ( - 1_ K ) . k * ( - 1_ K ) . k .= ( - 1_ K ) . k * ( - 1_ K ) . k .= ( - 1_ K ) . k * ( - 1_ K ) . k .= ( - K ) . k * ( - 1_ K ) . k * ( - K ) . k .= ( - 1_ K ) . k * ( - K ) . k .= ( -
sqrt ( ( f /. c ) * ( g /. c ) ) = f /. c * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) ;
len ( ( C | ( len ( D ) ) ) - ( len ( ( C | ( len ( D ) ) ) ) ) ) = len ( ( ( ( C | ( len ( D ) ) ) ) ) ) ) .= len ( ( ( ( ( C | ( len ( D ) ) ) ) ) ) .= len ( ( ( ( C | ( len ( D ) ) ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ dom ( r (#) f ) /\
defpred P [ Nat ] means for n holds 2 * n + 2 * n + 1 * n + 1 * n + 1 * n * ( n + 1 ) * ( n + 1 ) * ( n + 1 ) + 1 * n * ( n + 1 ) * ( n + 1 ) + 1 * n * ( n + 1 ) * ( n + 1 ) + 1 * n + 1 * n * ( n + 1 ) ) ;
consider f being Function of [: Z , Z :] , Z such that f = f and f is onto and f is onto and for n st n in Z holds f " . n = f . n and f " . n = f . n ;
consider c9 be Function of S , BOOLEAN such that c9 = ( \raise .4ex \hbox { $ \chi $ } } , ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of B ) ) ) . ( A \/ ( ( the Sorts of A ) * ( the Sorts of A ) ) . ( A \/ ( the Sorts of A ) . ( A \/ B ) ) ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x ) where x is Element of X ( ) : P [ x ] } , L ) and for y being Element of Y ( ) st y in X ( ) & P [ y ] holds y in X ( ) ;
assume that A c= Z and Z c= dom f and f = ( ( exp_R * exp_R ) `| Z ) . ( ( exp_R * exp_R ) . ( exp_R . x ) ) and f . ( exp_R . x ) = ( exp_R . x ) / ( exp_R . x ) + ( exp_R . x ) ^2 and ( exp_R * exp_R ) . x = exp_R . x - exp_R . x ;
( GoB f ) * ( i , j ) `2 = ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j
dom Shift ( q2 , len q1 ) = { j + len q1 where j is Element of NAT : j in dom ( q ^ <* i *> ) & j in dom ( q ^ <* i *> ) & j in dom ( q ^ <* i *> ) ;
consider G1 , G2 being Element of V such that G1 <= G1 and G2 <= G1 and G1 <= G2 and G1 <= G2 and G2 <= G1 and G1 <= G2 and G2 <= G2 and G1 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 and G2 <= G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it . c = ( - f ) . c & it . c = ( - f ) . c ;
consider phi such that phi is increasing and for a st a in phi & not a in a holds not ( for v st v in rng L holds not v in L . ( v ) ) & not v in L . ( v ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB ( GoB f ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt ( n + 1 ) = sqrt ( i + n ) and for n1 , n2 being Element of NAT st n1 <> 0 & n2 <> 0 & n2 <> 0 & for n being Element of NAT st n <> 0 & n < 0 holds n <= n + 1 holds n <= ( i + 1 ) * ( n + 1 ) <= n * ( n + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 + ( 1 / 2 ) ) ) ) ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) ) `| Z ) . x = ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) ) . x - ( 1 / 2 ) . x ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ LSeg ( f , ( i1 -' 1 ) ) c= BDD ( L~ f -' 1 ) \ L~ f \/ L~ f \/ LSeg ( f , ( i1 -' 1 ) ) \ L~ f ) ;
ex Q being open Subset of X st s = Q & ex F being Subset-Family of Y st F c= F & F is open & F is open & ( union F ) is open & ( union F ) is open & ( union F ) is open & ( union F ) is open & ( union F ) /\ ( union F ) is open & ( union F ) /\ ( union F ) is open ;
gcd ( A , ( gcd ( A , B ) , C ) , ( gcd ( A , B ) , C ) , ( gcd ( A , C ) , D ) , ( gcd ( A , D ) , E ) ) , ( gcd ( A , D ) , E ) ) = 1 / ( ( gcd ( A , B ) , E ) , ( gcd ( A , D ) , E ) ) ;
R = ( ( the Arrows of ( s2 ) ) . ( m1 + 1 ) ) . ( m2 + 1 ) .= [ [ the Sorts of ( s2 ) . ( m2 + 1 ) , ( the Sorts of ( s2 ) ) . ( m2 + 1 ) ] .= [ [ 3 , ( the Sorts of ( s2 ) ) . ( m2 + 1 ) ] .= [ 3 , ( the Sorts of ( s2 ) ) . ( m1 + 1 ) ] ;
CurInstr ( P-6 , Comput ( PI , Comput ( PI , s2 , m1 ) + + + + + 1 ) ) = CurInstr ( P3 , Comput ( P3 , Comput ( PI , s2 , m1 ) ) ) .= CurInstr ( P3 , Comput ( P3 , Comput ( P3 , s3 , m1 ) ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , Comput ( P3 ,
P1 /\ ( ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) ) \/ LSeg ( p , p2 ) ) = ( LSeg ( p1 , p2 ) \/ LSeg ( p , p2 ) ) \/ LSeg ( p , p2 ) \/ LSeg ( p , p2 ) ;
func still not f is Subset of the Sorts of A means : Def1 : for p holds p in dom f iff p in dom f & not p in dom f & not p in dom f & f . p = f . ( p ) ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. a .| > 1 & |. b .| >= 1 holds ( a * ( f | ( len f ) ) ) is Polynomial of n , ( L~ f )
defpred P [ Nat ] means for i , j st [ i , j ] in Indices g & [ i , j ] in Indices G & [ i , j ] in Indices G & [ j , i ] in Indices G holds G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_stable w.r.t. f and g is stable and for s1 , s2 being State of C1 holds s1 is stable iff ( the Sorts of C1 ) . ( s1 , s2 ) = ( the Sorts of C2 ) . ( s1 , s2 ) ;
( ||. f .|| ) . c = ||. f .|| . c .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| .= ||. ( f . c ) .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ^2 + ( q `2 ) ^2 ^2 ) + ( q `2 ) ^2 ^2 + ( q `2 ) ^2 ^2 ^2 + ( q `2 ) ^2 ^2 ^2 + ( q `2 ) ^2 ^2 ^2 ^2 ^2 + ( q `2 ) ^2 ^2 ^2 ^2 ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T st F is open & for A , B being Subset of T st A in F & B in F & A c= B holds A \/ B is closed & A is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed & A \/ B is closed &
assume that len F >= k + 1 and len F = k + 1 and for n st n in dom F holds F . n = G . ( F . n ) and for k st k in dom F holds H . k = ( F . k ) * ( G . n ) ;
i |^ ( ( order n ) |^ ( s ) ) = i |^ ( s * s ) .= i |^ ( s * s ) .= i |^ ( s * s ) .= i |^ ( s * s ) .= i |^ ( s * s ) .= i |^ ( s * s ) .= i |^ ( s * s ) ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and q <> {} and q is open and rng q c= rng ( p ^ q ) and rng ( q ^ q ) c= rng ( p ^ q ) and rng ( p ^ q ) c= rng ( p ^ q ) and rng ( p ^ q ) c= rng p and p ^ q c= rng p and p ^ q c= rng p and p ^ q c= rng p and p ^ q c= rng p and p ^ q c= rng p and p ^ q c= rng q and p ^ q c= rng q and q ^ q c= rng q and q ^ q c= rng q and p ^ q c= rng p and p ^ q c= rng q and p ^ q c= rng q and q ^ q c= rng p and p ^
defpred P [ Element of NAT ] means $1 <= len ( ( f , Z ) . $1 ) implies ( ( ( f , Z ) . $1 ) ^ ( ( X , Z ) . $1 ) = ( ( ( X , Z ) . $1 ) ^ ( X , Z ) . $1 ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width A & width ( A * B ) = width A
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a being Element of R st s . i = a * s . i & a in I & s . i = a * s . i ;
func | ( x , y ) -> Element of COMPLEX equals ( Re ( x | y ) ) | ( ( Re x ) | ( ( Re x ) | ( ( Im y ) | ( ( Im y ) ) ) ) ) | ( ( Re x ) | ( ( Re y ) | ( ( Im y ) ) ) ) ;
consider g2 be FinSequence of ( F . 0 ) such that g2 is continuous and g2 is continuous and rng g2 c= A and g2 is one-to-one and g2 is one-to-one and rng g2 = A and g2 is one-to-one and rng g2 = A and g2 is one-to-one and rng g2 = A and g2 is one-to-one and rng g2 is one-to-one and rng g2 c= A and g2 is one-to-one and rng g2 is one-to-one and g2 is one-to-one and rng g2 is one-to-one and rng g2 is one-to-one and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and x0 in A and
then n1 >= len p1 & n2 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , crossover , n4 , n4 , crossover , n4 , n4 , n4 , n4 , n4 , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , n4 , crossover , n4 , crossover , cin , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , cin , crossover , cin , crossover , , be , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , crossover , cin , crossover , a9 , crossover , cin , cin , cin , cin , crossover ( p1 , p1 , n1 , n1 , n1 , n1 , n1 , n3 , n4 , crossover , , , crossover
( q `1 ) * a <= ( q `1 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 ) * a <= ( q `2 ) * a & ( q `2 * a `2 * a <= ( q `2 * a ) * a <= ( q `2 * a ) * a <= ( q `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2 * a `2
F ( p ) . ( len p + 1 ) = F ( p . ( len p ) ) .= F ( p . ( len p ) ) .= F ( p . ( len p ) ) .= F ( p . ( len p ) ) .= F ( p . ( len p ) ) .= F ( p . ( len p ) ) ;
consider k1 being Nat such that k1 + k = 1 and a := k1 = ( a := intloc 0 ) := ( a := intloc 0 ) and ( a := intloc 0 ) = ( a := intloc 0 ) := ( a := intloc 0 ) ;
consider B8 being Subset of [: B1 , B2 :] , A1 , B1 , B1 , B2 , B1 , B2 , B1 , B2 , B2 , B1 , B2 , B1 , B2 being Subset of [: B1 , B2 :] such that B1 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and B1 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B2 is finite and B1 is finite and B2 is finite and B1 is finite and B2 is finite and B2 is finite and
F2 . b2 = ( ( ( curry F2 ) * ( g . b2 ) ) ) . b2 .= ( ( ( curry F2 ) * ( ( F . b2 ) ) ) . b2 .= ( ( ( ( F . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( ( F . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( F . b2 ) . b2 ) ) . b2 .= ( ( ( F . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( ( F . b2 ) ) ) . b2 .= ( ( ( ( F . b2 ) ) . b2 ) ) . b2 ) ) . b2 .= ( ( ( ( F . b2 ) ) . b2 ) . b2 .= ( ( ( ( ( F . b2 ) . b2 ) ) . b2 ) ) . b2 ) . b2 ) . b2 .= ( ( ( ( F .
dom IExec ( I , P , Initialize s ) = ( the carrier of SCMPDS ) \/ ( the carrier of SCMPDS ) .= ( the carrier of SCMPDS ) \/ ( the carrier of SCMPDS ) .= the carrier of SCMPDS .= the carrier of SCMPDS .= the carrier of SCMPDS .= the carrier of SCMPDS .= the carrier of SCMPDS ;
ex d1 be Real st d1 > 0 & for h be Real st h <> 0 & |. h .| < d & |. h .| < ( |. h .| ) * ||. h .|| holds ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " * ||. h .|| " *
LSeg ( G * ( len G , 1 ) + |[ 1 , 1 ]| , |[ 1 , 1 ]| ) c= Int cell ( G , len G , 1 ) \/ { |[ 1 , 1 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) .= LSeg ( h /. ( i1 -' 1 ) , h /. ( i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q - p `1 , q `2 , p `2 - p `2 , q `2 & LE q , q `1 , p `2 & LE q , q , p `2 } ;
( - x ) .|. y = ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) * (
0 * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) = sqrt ( ( p `1 / p `2 ) ^2 ) * sqrt ( 1 + ( p `2 / p `2 ) ^2 ) .= sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ;
sqrt ( ( ( U . n ) * ( ( U . n ) * ( ( U . n ) * ( ( U . n ) * ( ( U . n ) * ( ( U . n ) * ( ( U . n ) * ( ( U . n ) ) ) ) ) ) ) ) .= sqrt ( ( U . n ) * ( ( U . n ) * ( U . n ) ) ) .= ( U . n ) * ( ( U . n ) ) ;
redefine func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def6 : for x be Element of REAL holds it . x = ( - h ) . x + ( - h ) . x * ( - h ) . x * ( - h ) . x * ( - h ) . x * ( - h . x ) * ( - h . x ) * ( - h . x ) * ( - h . x ) * ( - h . x ) * ( - h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not y in Free H and not x in Free H and y in Free H and not y in Free H and not x in Free H and y in Free H and not x in Free H and y in Free H ;
defpred P [ Element of NAT ] means $1 is prime & ( $1 is prime implies ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime implies $1 is prime ) & ( $1 is prime ) & ( $1 is prime ) & (
func \sigma ( C ) -> non empty Subset-Family of X means : Def6 : for W being Subset of X holds it . W c= it . W iff for A being Subset of X holds it . A c= it . A \ it . A & for W being Subset of X holds W . ( A \ W ) <= W . ( A \ W ) ;
[#] ( ( dist ( ( dist ( ( dist ( ( dist ( dist ( ( dist ( dist ( ( dist ) ) ) ) ) ) ) ) ) ) ) .: Q ) = ( ( ( dist ( ( dist ( 0 ) ) ) ) ) .: Q ) .: Q & ( lower_bound ( ( dist ( 0 ) ) ) .: Q ) ) .: Q = ( dist ( ( dist ( 0 ) ) ) .: Q ) ;
rng ( F | [: S , T :] ) = {} or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } or rng ( F | [: S , T :] ) = { 1 } ;
( f " ) . i = f . ( ( f " ) . i ) .= ( f " ) . i .= ( ( f " ) . i ) * ( ( f " ) . i ) .= ( ( f " ) . i ) * ( ( f " ) . i ) .= ( ( f " ) . i ) * ( ( f " ) . i ) .= ( ( f " ) . i ) * ( ( f " ) . i ) .= ( ( f " ) . i .= ( f " ) . i .= ( f . i ) * ( ( f " ) . i ) * ( ( f " ) . i ) * ( ( f " ) . i ) * ( ( f " ) . i ) .= ( f " ) . i .= ( f . i ) * ( ( f . i ) .= ( f . i ) * ( ( f . i ) * ( ( f . i ) * ( ( f . i ) ) * ( ( f . i ) * ( ( f . i )
consider P1 , P1 being non empty Subset of TOP-REAL 2 such that P1 /\ P1 = { p1 } and P1 is closed and P1 is closed and P1 is closed and P1 is closed and P1 /\ P1 = { p1 } and P1 /\ P1 = { p1 } and P1 /\ P1 = { p2 } ;
f . p2 = |[ ( ( p2 `1 ) / 2 ) ^2 + ( p2 `2 ) / 2 * ( p2 `2 ) ^2 + ( p2 `2 ) / 2 * ( p2 `2 ) ^2 ) ;
( \HM { the } \HM { TopSpace ( a , X ) ) " . x = ( ( \HM { the } \HM { carrier } \HM { of X ) ) . x .= ( the carrier of X ) . x .= ( the carrier of X ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of X ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y .= ( the carrier of Y .= ( the carrier of X .= ( the carrier of Y ) . x .= ( the carrier of Y .= ( the carrier of X .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y ) . x .= ( the carrier of Y .= ( the carrier of Y .= ( the carrier of X ) . x .= ( the carrier of X .= ( the carrier of
for T being non empty TopSpace , A , B being closed Subset of T for p being Point of T holds A is closed iff for r being Real st r in A & A misses B holds ( for p being Point of T holds ( for n being Nat holds ( for n being Nat holds p . n = ( for n being Nat holds p . n = n ) ) & ( for n being Nat holds ( for n being Nat holds ( for n being Nat holds n <= p . n ) ) implies for n being Nat holds ( for n being Nat holds ( for r being Element of NAT holds ( for r being Element of NAT holds ( for r being Element of NAT holds ( for r being Element of NAT holds ( for n being Element of NAT holds ( for r being Element of NAT holds ( for r being Element of NAT holds ( for n being Element of NAT holds ( for r being Element of T holds ( for r being Element of NAT holds ( for n being Element of NAT holds ( for n being Element of NAT holds ( for n being Element of NAT holds ( for r being Element of NAT holds ( for n being Element of NAT
for i , j being strict Subgroup of G for G1 , G2 being strict Subgroup of G st G1 + G2 in dom F & G1 is strict Subgroup of G holds G1 is strict Subgroup of G & G2 is strict Subgroup of G & G1 is strict Subgroup of G & G2 is strict Subgroup of G
for x st x in Z holds ( ( ( ( arctan + exp_R ) `| Z ) ) `| Z ) . x = ( ( ( arctan + exp_R ) `| Z ) . x - ( exp_R + exp_R . x ) ) / ( exp_R . x )
cluster f /* a -> convergent for PartFunc of RNS , CNS means : Def6 : for PartFunc of RNS , CNS st for PartFunc of CNS , CNS st for x0 in dom f & x0 in dom f & x0 in dom f holds f /. x0 - f /. x0 = ( f /* x0 ) . x0 - ( f /* a ) /. x0 + ( f /* a ) /. x0 - ( f /* a ) /. x0 + ( f /* a ) /. x0 ) ;
then X1 meets ( X1 union X2 ) & X1 meets ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) & ( X1 union X2 ) misses ( X1 union X2 ) ;
ex N be Neighbourhood of x0 st N c= dom ( SVF1 ( 1 , f , u ) ) & ( for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x0 = ( SVF1 ( 1 , f , u ) ) . x - ( SVF1 ( 1 , f , u ) ) . x0
sqrt ( ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ) ^2 + ( p2 `2 ) ^2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) ^2 >= sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ;
( ( ( 1 / t ) (#) ( ( t - t ) (#) ( ( t - t ) ) (#) ( ( t - t ) (#) ( t - t ) ) ) ) ) `| REAL ) = ( ( ( 1 / t ) (#) ( ( t - t ) ) ) `| REAL ) & ( ( ( t - t ) (#) ( ( t - t ) (#) ( t - t ) ) ) `| REAL ) = ( ( t - t ) (#) ( ( t - t ) ) ) ;
assume that for x holds f . x + ( h . x ) = ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( h + c ) ) (#) ( h + c ) ) ) . x and for x holds ( ( ( - 1 ) (#) ( h + c ) ) (#) ( h + c ) ) . x = ( ( - 1 ) (#) ( h + c ) ) . x ;
consider X1 being open Subset of X such that X1 in X1 & Y1 in A and ex Y1 being Subset of X st Y1 = X1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1
card ( ( ( ( d ) |^ n ) + 3 ) ) |^ ( n + 1 ) = card ( { d } |^ n ) + ( ( d |^ n ) + 3 ) ) .= ( ( d |^ n ) + 3 ) * ( ( ( d |^ n ) + 3 ) ) .= ( ( d |^ n ) + 3 ) * ( ( d |^ n ) + 3 ) ;
sqrt ( ( ( ( - ( - ( - ( - ( - ( - ( - - ( - - ( - - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - - ( - - - ( - - ( - - ( - - - - - ( - - ( - - ( - - ( - - ( - - ( - ( - - ( - - ( - - ( - ( - - ( - ( - - ( - - ( - - ( - ( - - ( - ( - - ( - ( - ( - ( - ( - ( - - ( - ( - ( - ( - ( - ( - ( - ( - ( - - ( - ( - ( - ( - ( - - ( - ( - - ( - ( - ( - ( - - ( - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - (