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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
assume thesis ;
assume thesis ;
thesis ;
assume thesis ;
x <> b ;
D c= S
let Y ;
S ` is convergent
let q ;
let 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 y in 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 ;
let p ;
let x ;
Y in Y ` ;
assume 0 < g ;
c in Y ;
v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
set I = J ;
B ` = b ` ;
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 generated generated generated from squares ;
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 \not <= x ;
A in B ` ;
assume i = 1 ;
let x be element ;
x = x `2 ;
let X be BCI-algebra ;
assume not thesis ;
a in { 0 } ;
let p be set ;
let A be set ;
let G be _Graph , W be _Graph ;
let G be _Graph , W be _Graph ;
let a be Element of V ;
let x be element ;
let x be element ;
let C be FormalContext ;
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= dom f . a ;
let y y ;
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= { \mathbb N } ;
l <= m1 ;
let y y ;
q2 in P ;
let x be element ;
let i1 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 homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = \rceil c ;
n >= 0 + 1 ;
k + 1 <= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k + 1 <= k + 1 ;
let j be Nat ;
o , a // Y , b ;
R c= bool G ;
card B = card B ;
let j be Nat ;
1 <= j + 1 ;
the arccot of arccot is_differentiable_on Z ;
the function exp is differentiable ;
j < i ;
let j be Nat ;
n + 1 <= n + 1 ;
k = i + m ;
assume C meets S ;
n + 1 <= n + 1 ;
let n be Nat ;
h = {} ;
0 + 1 = 1 ;
o <> b ;
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 ;
r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t where t is Real : t <= 1 & t <= 1 & 1 <= t & t <= 1 } ;
assume F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > 1 ;
let y be ExtReal ;
let a be Int-Location ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i & i <= len G ;
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 ;
Int R is boundary ;
let i be Nat ;
T2 is increasing ;
cluster downarrow x -> lower-bounded ;
X <> { x } ;
x in { x } ;
q , b // M , b ;
A . i c= Y . i ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
cosec >= cosec ;
G . y <> 0 ;
let X be RealNormSpace ;
a ;
H . 1 = 1 ;
f . y = p ;
let V be RealLinearSpace ;
assume x in SpStSeq M ;
k < s . a ;
t in { p } ;
let Y be real-functions-membered set ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G c= L ;
assume x in Int 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 ' = a ' + 1 ' + 1 ' $ .
x ` = a * y ` ;
rng D c= A ;
assume x in { K } ;
1 <= i1 -' j1 ;
1 <= i1 -' j1 ;
p0 c= PI * PI ;
1 <= i & i <= len -15 ;
1 <= i & i <= len -15 ;
UMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian Lipschitzian X of X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
sD2 is bounded ;
assume I c= P1 +* I ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F is continuous ;
dom g = X ;
len q = m ;
assume L2 is closed ;
cluster R \ S \ S -> real-valued ;
sup D in S ;
x is_>=_than sup D ;
b >= Z ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
EMF C c= f ;
x1 is increasing ;
let L2 be element ;
- b divides b ;
F c= \overline { F ( ) where F is Subset of \overline { F ( ) : F is finite } ;
G is non-decreasing ;
G is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign ;
assume P [ n ] ;
assume union S is Affin ;
V is SubSpace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X = X ;
y in rng f ;
let s , I , J be set ;
b ' c= b ' ;
assume x in { \mathbb Q } + { \mathbb Q } ;
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_on Z ;
assume y in rng S ;
let x , y ;
i2 < i1 + 1 ;
a * h in a * H ;
p , q are_connected ;
cluster \sqrt <* I *> -> non empty ;
q1 in { A , B } ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 & A1 c= A2 ;
\boldmath \boldmath \boldmath \boldmath $ n $ } < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m ;
a , b \equiv a , c ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_on ]. x0 - g , x0 + g .[ ;
g is continuous ;
assume O is transitive ;
let x , y ;
let j be Nat ;
[ y , x ] in R ;
let x , y ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 is_closed_on s , P ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be Real ;
assume x in F | M ;
Y is Subset of S ;
let X be non empty TopSpace ;
[ a , b ] in R ;
x + w < y + w ;
not a , b // c , d ;
let B be Subset of A ;
let S be non empty ManySortedSign ;
let x be variable of f , A ;
let b be Element of X ;
[ x , y ] in R ;
x " = x " ;
b \ x = 0. X ;
<* d *> in D ^ <* 1 *> ;
P [ k + 1 ] ;
m in dom ( m1 . n ) ;
h . a = y ;
P [ n + 1 ] ;
cluster G * F -> bijective ;
let R be non empty multiplicative loop structure ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng ( o # x ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ( x , y ) ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p ;
let M be void -1\infty .[ ;
let N be non empty MetrSpace over M ;
let R be non empty RelStr structure ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 , 1 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as finite set ;
assume not I is_halting_on a , I ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v ;
x <= c1 . x ;
x in F " { y } ;
cluster S \longmapsto T -> \longmapsto T ManySortedSet of T ;
assume t1 <= t2 & t2 <= t1 & t1 <= t2 & t1 <= t2 & t1 <= t2 ;
let i , j be Integer ;
assume F <> {} and for n , m being Nat st n <> m holds F . n = F . m ;
c in Intersect ( bool union ( union ( bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool bool
dom p1 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 ;
set i1 = i + 1 ;
assume a1 = b1 ;
dom g = A ;
i < len M + 1 ;
assume - \infty in rng G ;
N c= dom f2 ;
x in dom sec ;
assume [ x , y ] in R ;
set d = \frac { x + y where x is Element of REAL : y <= x & y <= y & x <= y } ;
1 <= len g1 ;
len M2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom f2 ;
( p `2 ) ^2 = 0 ;
M2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q1 .| = 1 ;
let s be SortSymbol of S ;
\mathop ( i , i ) = i ;
X c= dom f ;
h . x in h . a ;
let G be <^ G ^> , F be <^ <^ G ^> , F ;
cluster m * n -> square ;
let k1 be Nat ;
i -' 1 > m ;
R is transitive ;
set F = \langle u , v *> ;
p0 c= p0 & p0 c= p0 ;
I is_closed_on t , Q ;
assume [ S , x ] is Sub_universal ;
i <= len f2 ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum { 2 } = n * r ;
cluster f . x -> complex-valued for Function ;
x in dom f1 ;
assume [ X , p ] in C ;
B9 c= { X , Y } ;
n1 <= n1 + 1 ;
A /\ A1 c= A ` ;
cluster x -defined for Function ;
let Q be Subset-Family of S ;
assume n in dom ( g2 | X ) ;
let a be Element of R ;
t `2 in dom e ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X ;
i . y in rng i ;
m1 c= dom f ;
f . x in rng f ;
m1 <= \frac { r where r is Real : 0 <= r & r <= 1 & r <= 1 & r <= 1 & 0 <= 1 & r <= 1 & r
T2 in { r where r is Real : 0 <= r & r <= 1 & r <= 1 & 0 <= 1 & r <= 1 & r <= 1 & 1 <=
let z , z1 , z2 be element ;
n <= N . m ;
not q in { p , q } ;
f . x = \twoheaddownarrow x /\ B . x ;
set L = [ S \to T , S ] ;
let x be non negative Real ;
let m be Element of M ;
f in union rng F ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr ;
let i be Element of NAT ;
rng ( F * g ) c= Y ;
dom f c= dom ( f | X ) ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster {} bool X -> On Y -> On Z ;
[ z2 , z1 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng { S } ;
b = sup ( dom f ) ;
x in Seg len q ;
reconsider X = { D } as set ;
[ a , c ] in { E } ;
assume n in dom |. h .| ;
w + 1 = { m1 } + 1 ;
j + 1 <= j + 1 ;
k1 + 1 <= k + 1 ;
let i be Element of NAT ;
Support u = Support p ;
assume X is \it \it \it \it \it \it \it \it \it \it \it \it \it \it \it \it \it \it m } is \it \it \it m } ;
assume that f = g and p = g . p ;
n1 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s ^\ k ) ;
x0 < x0 + r ;
len L5 = len W ;
P c= Seg len A ;
dom q = Seg n ;
j <= width M ;
let r1 be real-valued FinSequence ;
let k be Element of NAT ;
is_integrable_on M ;
let n be Element of NAT ;
assume z in <^ 0 \rbrace \ { 0 } ;
let i be set ;
n -' 1 = n-1 - n ;
len n1 = n ;
RelIncl Z c= F ;
assume x in X or x = X ;
x is Element of b , c ;
let A , B be non empty set ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E ;
let x1 be basis of x , y ;
3 /\ L2 = {} ;
L1 /\ 4 = {} ;
assume x "/\" y = x "/\" y ;
assume b , c // b , c ;
not q , c // c9 , c9 ;
x in rng f1 ;
set d1 = n + j + 1 ;
let D1 be set ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr ;
assume that f = f and h = f .: A ;
R1 - ( R1 - R2 ) is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K " is open ;
assume that a maximal maximal maximal maximal maximal maximal C C and b in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int integrable ( M , f ) ;
cluster strict strict for Lattice ;
u in { \hbox { \boldmath $ g } } ;
the support of f c= B ;
reconsider z = x as VECTOR of V ;
cluster the carrier of L -> strict for RelStr ;
r (#) H is point-convergent ;
s . intloc 0 = 1 ;
assume that x in C and y in C and y in C ;
let U1 be strict universal algebra over S ;
[ x , y ] is compact ;
i + 1 in dom p ;
F . i is stable Subset of M ;
r1 in ( \ast ( y ) ) . n ;
let x , y be Element of X ;
let A , I , I be Ideal of X ;
[ y , z ] in [: { O } , { O } :] ;
-defined Function ;
rng Sgm A = A ;
q Ant ( \neg y ) '&' q = 'not' q '&' \neg q ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = [ x , y ] ;
not o , a , b , c , d is_collinear ;
p . 2 = Z . ( Y , Y ) ;
( D . 0 ) . ( len D ) = {} ;
n + 1 <= len g ;
a in bound_QC-variables ( A ) ;
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 + len G ;
g . 1 = f . ( i + 1 ) ;
x , y // y , z ;
( f /* s ) . k < r ;
set v = VAL l ;
i -' k + 1 <= S ;
cluster non empty multiplicative for magma ;
x in support t ;
assume a in { [ i1 , j1 ] where i1 is Nat : 1 <= i1 & i1 <= j1 & j1 + 1 <= width G } ;
i `2 <= len |. q1 .| ;
assume p divides b + ( a + b ) ;
x0 <= sup { x0 where x0 is Real : x0 <= x0 & x0 <= x0 & x0 <= x0 & x0 <= x0 & x0 <= x0 } ;
assume x in W .: ( X /\ Y ) ;
j in dom z1 ;
let x be Element of D ;
IC s4 = l ;
a = {} or a = {} ;
set C9 = Vertices G , C9 = Vertices G , C9 = G , C8 = G , C8 = G , C8 = G , C8 = G , 9
seq " is non-zero ;
for k st X [ k ] holds X [ k + 1 ]
for n holds X [ n ] ;
F . m in { F ( m ) where m is Nat : m <= n & n <= m & n <= m } ;
h1 c= |. h1 .| ;
[. a , b .] c= Z ;
X1 union X1 is SubSpace of X & X2 is SubSpace of X ;
a in union ( union ( union ( F \ G ) ) ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k - 1 ;
cluster binary binary relation -> real-valued for Relation ;
ex v st C = v + W ;
let G be non empty 1-sorted structure ;
assume V is add-associative add-associative right_zeroed right_complementable add-associative right_complementable ;
X \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper ;
let L be non empty RelStr ;
R is reflexive implies R is reflexive
E \models _ { g } ( H ) ;
dom G = a * ( y " ) ;
sqrt ( 1 - 4 ) >= - ( 1 - 4 ) ;
G * ( { p _ 0 } , G * ( 1 , 1 ) ) in rng G ;
let x be Element of F ;
D [ D , {} , {} , {} ] ;
z in dom id ( B * id B ) ;
y in the carrier of N ;
g in the carrier of H ;
rng Sgm { f1 : f1 in { f1 : f1 <= f1 & f1 <= f1 & f1 <= f1 & f1 <= f1 & f1 <= f1 } ;
j + 1 in dom ( s + t ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h ;
P . ( k + 1 ) in rng P ;
M = { A , { { { A } } } \/ { {} } ;
let p be FinSequence of REAL ;
f . ( n + 1 ) in rng f ;
M . ( 0 , 1 ) in { 0 } ;
Center a = b-a -|| A ;
assume the InternalRel of V is Subspace of V ;
let a be Element of op V ;
let s be Element of product P ;
let L1 be non empty RelStr ;
let n be Nat ;
the support of g c= B ;
set I = halt SCM SCM SCM SCM SCM SCM SCM R ;
consider b being element such that b in B ;
set B3 = degree K ;
l <= sup rng ( F . j ) ;
assume x in [. s , t .] ;
( x "/\" ( x "/\" y ) ) "/\" t in t ;
x in DOM ( T ) ;
let h be Morphism of c , b ;
Y c= { {} } ;
A2 \/ 4 c= A1 \/ A2 \/ A3 ;
assume not o , a , b , c is_collinear ;
b , c // d , e ;
x1 , x2 are_connected & x2 in Y ;
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| ;
[ x , y ] in X \times Y ;
for n be Nat holds x . n = x . n
[ a , b ] = [: a , b :] ;
cluster -> open for Subset of T ;
x = h . ( ( f . ( ( f . ( ( f . ( ( f . ( ( f . ( m + 1 ) ) ) ) ) ) ) ) ) )
q1 , q2 are_connected ;
dom M1 = Seg n ;
x = [ x1 , y1 ] ;
let R , Q be Relation of A , B ;
set d = \frac { 1 + ( 1 + 1 ) ^2 } ;
rng g2 c= dom W ;
P " ( \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega ( \Omega \Omega ( \Omega _ { \Omega _ V } ) ) ) ) <> 0 ;
a in field R & a = b ;
let M be non empty affine Subset of V ;
let I be Program of SCM+FSA ;
assume x in rng ~ ( R ~ ) ;
let b be Element of the lattice T ;
dist ( e , z ) > r ;
v1 + v in v1 + v2 ;
assume the support of L misses rng G ;
let L be lower-bounded antisymmetric antisymmetric RelStr ;
assume [ x , y ] in { a } ;
dom ( A * e ) = NAT ;
let a , b be vertices of G ;
let x be Element of Bool ( M ) ;
0 <= Arg a * PI ;
o , a9 // o , a9 ;
not v in the support of l ;
let x be bound of A ;
assume x in dom ( ( U0 *' f ) *' ( ( U0 *' f ) *' ( ( U0 *' f ) *' ( ( U0 *' f ) ) ) ) ;
rng F c= product ( f ^ ) ;
assume that D2 . k in rng D2 and k in dom D2 ;
f " * ( p " ) = 0 ;
set x = the Element of X ;
dom On G = NAT ;
let n be Element of NAT ;
assume not c , a , b , c , d is_collinear ;
cluster finite for FinSequence of NAT ;
reconsider d = c * d as Element of L ;
( v .--> ( v .--> x ) ) . n <= 1 ;
assume x in the support of f ;
conv @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
reconsider B = b as Element of the lattice of T ;
J \models _ { v } } { l , P } H ;
cluster the functor of J . i -> non empty ;
sup { Y \/ { Y } where Y is Subset of T : Y in F & Y in F & Y in F } ;
W1 is_is_field W1 & W2 is_monomorphism ;
assume x in the carrier of R ;
dom -16 -16 = Seg n ;
s misses { s : s < t & t < 1 & t < 1 } ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X and d = X ;
assume [ a , y ] in Indices ( f + g ) ;
assume that I c= J and J c= K and K c= J and J c= K and K c= K and J c= K and K c= K ;
Im ( ( Im ( seq ) ) ) . n = 0 ;
( ( ( ( the function sin ) * ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( id Z ) ) ) ) ) ) ) . x <> 0 ;
the function sin is_differentiable_on Z & for x st x in Z holds ( ( - sin . x ) ^2 ) ^2 > 0 ;
t2 . n = t . n ;
dom ( cos `| Z ) c= dom ( F `| Z ) ;
W1 . x = W . x ;
y in W .edges() ( W .first() ) \/ W .vertices() ( y , W ) ;
m1 <= len ( v + ( len v ) - ( len v + 1 ) ) ;
x * y - a * ( m * y ) < ( m * y ) * ( m * y ) ;
proj2 .: ( S .: Y ) c= proj2 .: Y ;
h . ( p1 , p2 ) = g . ( p1 , p2 ) .= g . ( p1 , p2 ) ;
set IT = U U /. ( 1 , 1 ) , IT = U /. ( 1 , 1 ) ;
f . ( r + 1 ) in rng f ;
i + 1 <= len i-1 -' 1 ;
rng F = rng ( F - G ) ;
A is associative doubleLoopStr ;
[ x , y ] in A \times A ;
x1 . o in { L . o where o is Element of L : o in X & o in X } ;
the support of <^ the InternalRel of V , m :] c= B ;
[ y , x ] in id X ;
1 + p .. f + i <= i + 1 ;
seq ^\ k is lower ;
len ( F . ( len F ) ) = len ( F . ( len F ) ) ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) . IC s = s . IC s ;
k + 1 <= k + 1 ;
reconsider c = {} T as Element of L ;
let Y be antitransitive Subset of R^1 ;
cluster strict for RelStr ;
f . j in K . ( j + 1 ) ;
cluster J => J -> total for Relation of J , the carrier of J ;
K c= 2 |^ ( len \alpha ) ;
F . ( b , c ) = F . ( b , c ) .= F . ( b , c ) ;
x1 = x or x1 = y ;
attr a <> {} & a = 1 implies a = 1 ;
assume that a c= b and b in a and a in b and a in b ;
s . n in rng s ;
not o , b2 , c , d is_collinear ;
not o , b , c // o , c ;
reconsider m = x as Element of GF ( V ) ;
let f be yields yields FinSequence ;
let F be non empty TopSpace ;
assume that h is homeomorphism and y = h . x ;
[ f . 1 , w ] in F ;
reconsider p1 = x as Subset of m m ;
let A , B be Subset of R ;
cluster strict strict strict for normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal
rng c ' misses rng c1 ;
z is Element of gr ( \lbrace x \rbrace ) ;
b in dom ( a .--> ( b .--> a ) ) ;
assume that k >= 2 and k >= 2 and k >= 2 ;
Z c= dom ( ( the cot ) `| Z ) ;
the UBD of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( f1 / f2 ) ;
attr f = u * a ;
for n holds 'not' n = 'not' n '&' 'not' n ;
not x in { x } ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = { [ p , q ] } and S = { [ p , q ] } ;
gcd ( n1 , n1 , n2 ) = 1 ;
set o1 = ( 1 / 2 ) * ( 1 / 2 ) ;
|. s . n - ( |. s . n .| ) .| < |. s . n - ( |. s . n .| ) .| ;
assume that seq is increasing and for n st n < m holds seq . n < r ;
f . x1 <= a * x1 ;
ex c being Nat st P [ c ] ;
set g = \ { n where n is Element of NAT : n <= m & m <= m & n <= m } ;
k = a or k = b ;
a in { \boldmath \boldmath $ g } & b in { \boldmath \boldmath g } ;
assume that Y = { 1 , 2 } and s = <* 1 *> and 2 = <* 2 *> and 1 = 2 ;
I . x = f . x .= f . x .= 0 ;
W1 .first() = W1 .first() ( 1 + 1 ) .= W . 1 ;
cluster strict for _Graph ;
reconsider u = u as Element of Bags ( X ) ;
A in B |^ ( n + 1 ) implies A |^ ( n + 1 ) = B |^ n
x in { 2 * n + 3 * n , k , k } ;
1 >= \frac ( ( q `1 / |. q .| - sn ) ) ^2 ;
f1 is_s\downharpoonright f2 , 2 ;
( f /. ( ( q `2 / |. q .| ) .| ) ^2 <= ( f /. ( q `2 / |. q .| ) ) ^2 ;
h is Point of Cage ( C , n ) ;
( b - p ) ^2 <= ( b - p ) ^2 ;
let f , g be Function of X , Y ;
S /. ( k , 1 ) <> 0. TOP-REAL 2 ;
x in dom ( - ( - ( f - g ) ) ) ;
p2 in { N . ( m + 1 ) where m is Nat : m <= n & n <= m + 1 } ;
len ( H + M2 ) < len ( H + M2 ) ;
F [ A , F ( ) ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C & A c= C & A c= C ;
assume that r1 <> 0 and r1 <> 0 and r1 <> 0 and r2 <> 0 ;
rng q1 c= rng q1 & rng q1 c= rng q1 ;
A1 , A2 are_collinear ;
y in rng f & y in rng f ;
f /. ( i + 1 ) in L~ f ;
b in Segment ( p , S ) ;
attr S is negative means : Def14 : ex U1 , U2 being strict Subspace of U1 st U1 is negative & U2 is negative & U1 is negative & U2 is negative ;
Cl Cl Int ( [#] T ) = [#] T ;
f1 | ( ( f2 | ( ( ( ( len f1 ) -' 1 ) ) ) ) ) ) | ( len ( f2 | ( len ( f2 | ( len f1 ) ) ) ) ) ) = ( ( f2 | (
0. M in the carrier of W ;
v , v , v be Element of M ;
reconsider K = union K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 = 'not' H1 ;
1 / t c= t * t ;
0 * a = 0. R .= a * a .= a * a ;
A |^ 2 = A |^ ( len A ) .= A |^ ( len A + 1 ) ;
set V1 = { v where v is Element of V : v in V1 & v in V1 & v in V1 & v in V1 & v in V1 & v in V2 } ;
r = 0. ( \langle 0 , 1 *> ) ;
( f . ( ( |. p .| ) .| ) ^2 >= 0 ;
len W = ( len W ) + ( len W ) ;
f /* s is divergent to \hbox { - \infty $ } ;
consider l being Nat such that m = F . l ;
t1 , t2 are_connected & t1 , t2 are_connected ;
reconsider Y = { X where X is Subset of X : X is SubSpace of X } as SubSpace of X ;
consider w such that w in F and x in F ;
let a , b be Real ;
reconsider i = i as Element of NAT ;
c . x >= id L . x ;
\omega ( T \/ \omega ) c= \omega ;
for x being element st x in X holds x in Y
cluster <* x1 , x2 *> -> pair ;
types a /\ t is Subset of T ;
let X be disjoint _n_n_n_n_net over N ;
rng f = Funcs ( X , Y ) ;
let p be Element of B ( ) ;
max ( N , 2 ) >= N ;
b <= b * m + ( b * m ) ;
assume that i in I and i = R . i and i < j ;
i = j & i = j ;
assume that g1 in the carrier of G2 and g2 in the carrier of G2 and g2 in the carrier of G2 ;
let A1 , A2 be SubSpace of S ;
x in h " ( P " ) /\ [#] ( P " ) ;
1 in Seg 3 & 1 in Seg 3 ;
reconsider -5 = X as Subset of bool X ;
x in ( the Arrows of B ) . i ;
cluster EK1 . n -> ( the Source of G ) . n -> ( the Source of G ) . n ;
n1 <= n1 + len ( n1 + 1 ) ;
( i + 1 ) + ( i + 1 ) = i + ( i + 1 ) ;
assume v in the carrier of G ;
y = Re ( y + i ) + Im ( y + i ) ;
( ( ( - 1 / ( ( - 1 ) * p ) ) ) * ( 1 - p ) ) ) * ( 1 - p ) = 1 ;
2 * PI is_differentiable_on ]. a , b .[ ;
rng M2 c= rng M2 ;
for p be Real st p in Z holds p >= a
[: X , Y :] = [: X , Y :] ;
( s ^\ k ) . m <> 0 ;
s . ( k + 1 ) > { x0 } ;
( p * M ) . 2 = d ;
A \ ( B \ B ) = ( A \ B ) \ ( A \ B ) ;
h \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv \equiv , \equiv , \equiv ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
mode V V of V is submodule of V ;
reconsider p12 = i as Element of NAT ;
dom f c= [: { C } , { C } :] ;
x in ( the dist ( B ) ) . n ;
len vs in Seg ( len f2 ) ;
{ p } c= the topology of T ;
[. r , s .] c= [. r , s .] ;
T2 be prebasis of T ;
G * ( B * A ) = id ( B * A ) ;
assume that p = u and q = v and u <> v ;
[ z , y ] in union rng ( F . m ) ;
( b . x ) . ( b . x ) = b . x ;
deffunc F ( set ) = ( $1 .--> $1 ) .. $1 ;
LSeg ( f1 , f2 ) = { f1 , f2 } ;
f " { f " } = { x } ;
dom |. z2 .| = dom |. z1 .| ;
assume that 1 <= i and i <= n and j <= n and i <= n ;
( g . O ) `2 <= 1 ;
p in LSeg ( E , i ) ;
I9 /. i = 0. K ;
|. f . ( s . m ) - g . ( g . n ) .| < g . ( g . m ) - g . n ;
q1 . x in rng q1 ;
L1 misses L2 " { 0 } ;
consider c being element such that [ a , c ] in G ;
assume N19 : o = o & o = o ;
q . j = q . j ;
rng F c= { F . n1 where n1 is Nat : n1 <= n1 & n1 <= len F & n1 <= len F } ;
( ( 2 \ { B } ) \/ { B } ) ` + ( 2 \ { B } ) ` <= 0 + ( 2 \ { B } ) ` ;
f . j in [. f . j , f . j .] ;
attr 0 <= x & x <= y implies x <= y
p `2 <> 0. TOP-REAL 2 ;
cluster over over over over over S -> non empty finite ;
let x be Element of [: S , T :] ;
<^ a , b ^> is one-to-one ;
|. i - i .| <= |. i - j .| ;
the carrier of ( I[01] | P ) = ( TOP-REAL 2 ) | P ;
! * ( n + 1 ) > 0 * ( n + 1 ) ;
S c= ( { A , B } /\ { A , B } ) /\ ( { A , B } /\ { B , C } ) ;
a1 , 4 // a1 , b3 ;
then dom A <> {} & A <> {} ;
1 + ( 2 * k + 1 ) = 2 * k + 2 * k ;
x Joins v , G , G ;
set v2 = v + ( i + 1 ) , v2 = v + ( i + 1 ) ;
x = r . n .= r . ( n + 1 ) .= r . ( n + 1 ) .= r . ( n + 1 ) ;
f . s in the carrier of S ;
dom g = the carrier of I[01] ;
p in Segment ( P , Q ) ;
dom g2 = [: { A , B } , { B , C } :] ;
0 < p + ( z + 1 ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X c= B \ominus X
- \infty < lim ( g | B ) ;
cluster O O O O O O O O O O -> \tt : : \sqsubseteq X ;
let U1 , U2 be non-empty MSAlgebra over S ;
Proj ( i , n ) * ( i , n ) is_differentiable_on X ;
let x , y be Point of X , x be Point of X ;
reconsider p1 = p . x as Subset of V ;
x in the carrier of Lin ( A ) ;
let I , J be Program of SCM+FSA , a , b be State of SCM+FSA ;
assume - a is lower bound ;
Int Cl A c= Int Int ( A /\ Int ( A /\ Int A ) ) ;
assume for A being Subset of X st A = A holds A is closed ;
assume q in Ball ( x , r ) ;
( p1 - p2 ) ^2 <= ( p1 - p2 ) ^2 ;
Cl Q ` = [#] T ;
set S = the carrier of T ;
set I1 = #Z ( n + 1 ) ;
len p -' n = len p - n ;
A is permutation of Swap ( A , B , C ) ;
reconsider n6 = ni as Element of NAT ;
1 + j + 1 <= len D1 ;
defpred q1 , q2 , q1 , q1 , q2 , q1 , q1 , q1 , q2 , q1 , q1 , q2 , q1 , q1 , q2 , q1 , q1 , q1 , q2 , q1 , q1 , q2 is_collinear ;
a1 in the carrier of S & a2 in the carrier of S ;
c1 /. n = c1 . ( n + 1 ) .= c1 . ( n + 1 ) ;
let f be FinSequence of TOP-REAL 2 ;
y = ( f1 * ( S ) ) * ( ( f1 * ( S * ( S * ( S ) ) ) ) ) ) ;
consider x being element such that x in On A ;
assume r in ( the dist ( o , r ) ) .: P ;
set i2 = \mathop { \rm d1 _ { max } ( h ) , j2 = \mathop { \rm len } h ;
h . j in rng h ;
Line ( M1 , i ) = M * ( i , j ) ;
reconsider m = \frac { x where x is Element of \overline { \mathbb R } : 0 <= x & x <= 1 } as Element of REAL ;
U1 , U2 be strict Subspace of U1 , U2 be strict non-empty U2 over U2 ;
set P = Line ( a , d ) ;
len p1 < len p2 + ( len p2 ) + ( len p2 ) ;
for T1 , T2 be T , T1 , T2 be T , T1 , T2 be T be T T T T T , T1 , T2 be T be T T , e , T1 be T be T of T ;
then x <= y & y <= z ;
set M = n -increasing ( m , n ) ;
reconsider i = x1 , j = y1 as Nat ;
rng ( ( the Arity of S ) * * ( Arity o ) ) ) c= dom ( the Arity of S ) ;
z1 " = z1 " * z1 " ;
x0 - r in L /\ dom ( f `| X ) ;
w w w is string of S implies w /\ w is non empty
set q1 = { q1 ^ <* q1 *> ^ <* q1 *> ^ <* q1 *> ^ ( q1 ^ q2 ) ;
len w in Seg ( len w + 1 ) ;
( \llangle z1 , z2 \rrangle ) . ( z1 , z2 ) = g . ( z1 , z2 ) ;
let a be Element of PFuncs ( V , k ) ;
x . n = \frac { a . n where a is Real : a <= b & b <= a } ;
( p `1 ) ^2 <= ( p `1 ) ^2 ;
rng g1 c= L~ ( g1 + g2 ) ;
reconsider k = i-1 * j + 1 as Nat ;
for n be Nat holds F . n is measurable of X ;
reconsider -10 = { x where x is VECTOR of M : x <= y & y <= y & x <= y } as Element of M ;
dom ( f | X ) = X /\ dom ( f | X ) ;
p , a // p , c & p , c // c , d ;
reconsider x1 = x as Element of REAL m ;
assume i in dom ( a * p ) ;
m . ( \hbox { \boldmath $ g $ } ) = p . ( m + 1 ) .= p . ( m + 1 ) ;
a ^ ( s . m ) ` <= 1 ;
S . ( n + k ) c= S . ( n + k ) ;
assume that A1 \/ A2 = A1 \/ A2 and A1 \/ A1 = A2 and A2 = A1 \/ A3 and A3 = A3 and A3 = A3 and A3 = A2 and A3 = A1 \/ A3 ;
X . i = { x , y } . i ;
a2 in dom ( h + c ) ;
sin . 0 = a & cos . 0 = b . 0 ;
F is closed implies for A2 , A3 being Subset of IT st A2 is closed & A3 is closed & A2 is closed & A2 is closed & A3 is closed & A2 is closed & A2 is closed & A2 is closed & A2 is closed & A3 c= A2 &
set T = \langle \underbrace ( X , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0 , x0
Int ( Int Int Int ( Int Int ( Int ( Int ( Int ( R ) ) ) ) ) ) ) ) ) c= Int ( Int ( R ) ) ;
consider y being Element of L such that c . y = x ;
rng PP = { F where F is Element of bool the carrier of G : F in F & for x being Element of G st x in F . x holds F . x = F . x ;
G " { f1 } " { c } c= B \/ S " { c } ;
f1 is_differentiable_on X & f2 is_differentiable_on X ;
set R1 = the Point of I[01] , R2 = the Point of I[01] ;
assume that n + 1 >= n and n + 1 <= len M ;
let L2 be Element of NAT ;
reconsider pm1 = u as Element of FTSL1 n ;
g . x in dom f & g . x = f . x ;
assume that 1 <= n and n + 1 <= len f and n + 1 <= len f and n + 1 <= len f ;
reconsider T = b * N as Element of N * ( N , N ) as Element of G / ( N , N ) ;
len ( Q1 /. ( len Q1 ) + 1 ) = len ( Q1 /. ( len Q1 + 1 ) ) ;
x " in the carrier of A ;
[ i , j ] in the indices of A ;
for m be Nat holds Re ( F ) . m is simple function ;
f . x = a . i .= b . i .= b . i ;
let f be PartFunc of REAL , REAL ;
rng f = the carrier of A ;
assume that 2 = \sqrt ( 2 * PI , 2 * PI ) and 2 * PI < 2 * PI * PI ;
attr a > 1 & a > 0 & a = 1 ;
let A , B be Subset of I ;
reconsider X0 = X as RealLinearSpace ;
let f be PartFunc of REAL , REAL ;
r * ( v \rightarrow I ) < r * ( 1 / I ) ;
assume that V is submodule of X and V is submodule of X and V is submodule of X ;
t1 , t2 are_connected ;
defpred Q [ Nat ] means $1 in f .: { v } ;
g \circlearrowleft ( z ) = z .. z ;
|. x - v .| + |. y - v .| = vv ;
- ( f . w ) = - ( f . w ) ;
z -' y <= x + y implies z + y <= z + y
\frac 7 - ( 7 / ( 1 + e ) ) ^2 > 0 ;
assume X is commutative BCK-algebra implies X is commutative & X is commutative & 0. X = 0 & 0. X = 0 & 0. X = 0 ;
F . 1 = v & F . 1 = v . 1 ;
( f | X ) | X = f | X ;
( the function of tan ) . x in dom ( tan * tan ) ;
i2 = { f /. ( len f + 1 ) where i is Nat : 1 <= i & i + 1 <= len f & i + 1 <= len f } ;
set X1 = { X , Y } \/ { Y , Z } ;
[. a , b .] c= { |[ a , b ]| where a is Real : a <= b & b <= 1 & a <= 1 } ;
let V , W be non empty VectSpStr over F_Complex ;
dom g2 = the carrier of ( TOP-REAL 2 ) | K1 ;
dom f2 = the carrier of ( TOP-REAL 2 ) | K1 ;
( proj2 .: X ) .: X = proj2 .: X ;
f . x = h . x ;
x0 < r1 . n & r1 < x0 ;
|. f /* ( s ^\ k ) - f /* ( s ^\ k ) .| < r ;
len Line ( A , i ) = width A ;
S ^ <% <% <% %> %> = S ^ ( S , g %> ) ;
reconsider f = v + u as Function of X , Y ;
intloc 0 in dom ( Initialized ( s ) ) ;
i2 , j2 are_connected & i2 <> j2 & i2 , j2 , j2 , 4 , 6 , 6 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 ,
sin + sin is_differentiable_on Z ;
for x st x in Z holds ( ( 2 * ( 2 * ( 2 * ( ( 2 * ( ( ( ( ( ( ( - sin ) ) ) ) ) ) ) ) ) ) ^2 ) ^2 ) ) ^2 ) ^2 ) ^2 = ( 2 * ( ( ( ( ( ( ( ( (
reconsider q2 = \frac { q where q is Element of REAL : |. q .| < 1 & |. q .| < 1 } as Element of REAL ;
( 0 qua Nat ) + ( i + 1 ) <= i + 1 ;
assume f in the carrier of X \times Y ;
F . a = H . ( x , y ) ;
\it \it \it true -> \it true strict strict Den over C , the carrier of C ;
dist ( a , b ) < r ;
1 in the carrier of R^1 | [. 0 , 1 .] ;
( - ( p - q ) / |. p .| ) ^2 > - ( - q ) ^2 ;
|. r1 .| = |. r1 .| * |. r1 .| ;
reconsider q1 = 8 as Element of Seg 8 ;
( A \/ B ) \/ ( A \/ B ) c= A \/ B ;
DW .last() = DW .first() ( len W ) + 1 + 1 ;
i = { a + b } & i = n + 1 ;
f . a [= f .: ( { O } \/ { O } ) ;
attr f = v + u & u = v + u ;
I . n = \int ( F . n ) . n ;
.4ex .4ex .4ex .4ex \hbox { $ \chi $ } , T , S , T ;
a = VERUM or a = VERUM & a = VERUM ;
reconsider k2 = s . ( ( ( |. b .| ) .| ) ^2 as Element of REAL ;
Comput ( P , s , 4 ) . GBP = 0 ;
L~ <* M1 , M2 *> misses L~ M2 ;
set h = the carrier of InclPoset X ;
set A = { L . ( k + 1 ) where k is Nat : k <= n & k <= n } ;
for H being negative WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF WFF non empty ManySortedSign , H1 being Element of WFF st P [ H ] & P [ H ] holds P [ H ]
set y9 = { S . i where i is Nat : i <= j & j <= i & j <= i & i <= i } ;
Hom ( a , b ) c= Hom ( a , b ) ;
\frac 1 + ( n + 1 ) < ( n + 1 ) / ( n + 1 ) ;
( l , m ) `2 = [ l , m `2 , m `2 ] ;
y in dom ( g +* ( i , j ) ) ;
let p be Element of WFF WFF WFF ( A ) . ( len p ) ;
X /\ { X } c= dom ( f1 + f2 ) ;
{ p : p in rng f & f /. ( len f -' 1 ) <= p `2 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2
1 <= indx ( D2 , D1 , j ) ;
assume x in { 2 } \/ { 1 , 2 } ;
- ( ( ( ( ( f1 - f2 ) (#) ( f1 - f2 ) ) (#) ( f1 - f2 ) ) (#) ( f1 - f2 ) ) ) (#) ( ( f1 - f2 ) (#) ( f1 + f2 ) ) ) = ( f1 (#) ( f2 (#) ( f2 (#)
let f , g be Function of the carrier of TOP-REAL 2 , TOP-REAL 2 ;
i1 -' ( k + 1 ) - ( i1 - 1 ) = i1 - ( i1 - 1 ) ;
rng ( seq + seq1 ) c= [. x0 , x0 + r .] ;
g2 in [. x0 , x0 + r .] ;
- ( p p9 p9 ) = - ( - p9 ) ;
consider u being Nat such that b = p * u + u * v ;
ex a being normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal normal
union union union { H where H is Subset of bool the carrier of V : H = union { H : H = union F } = union H } ;
len t = len ( ( t + t ) + ( len t ) ;
v = v + w ;
v <> DataLoc ( t . intloc 0 , 3 ) ;
g . s = sup ( d .: ( d .: ( X /\ Y ) ) ) ;
( \dot \dot \dot 1 ) . s = ( \dot \dot \dot 1 ) . s ;
not s in { t where t is Element of REAL : t < s & s < t & t < t } ;
s " * s = s " * ( s " ) .= s " * ( 0 " ) .= s " * ( 0 " ) ;
defpred P [ Nat ] means $1 + 1 in A ;
( 339 + 1 ) * ( 3 + 1 ) = 339 * ( 3 + 1 ) ;
U = ( U , A ) * ( ( U , A ) * ( ( U , A ) * ( ( U , A ) * ( ( U , A ) * ( ( U , A ) * ( ( ( U , A ) * ( ( ( ( ( (
reconsider y = y as Element of REAL ;
consider i2 being Integer such that i2 = p * ( i2 , j2 ) + 1 and i2 = i2 * ( i2 , j2 ) ;
reconsider p = Y | Seg k as FinSequence of Seg ( ( Seg ( ( Seg ( n + 1 ) ) ) ) * ( k + 1 ) ) ;
set f = ( S , U ) \! \mathop { \rm \hbox { - } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of the carrier of TOP-REAL 2 , TOP-REAL 2 ;
M . ( n + i ) <> 1 ;
ex r being Real st x = r & r <= 1 ;
R1 , R2 are_elements of the carrier of G , the carrier of H ;
reconsider l = 0. V as Linear_Combination of A ;
set r = |. e .| + |. s .| + |. s .| + |. s .| ;
consider y being Element of S such that z <= y and y in X ;
a "\/" ( b "\/" c ) = 'not' ( a 'or' c ) ;
||. x1 - y1 .|| < r ;
b , c // b , c ;
1 <= k + 1 & k + 1 <= k + 1 ;
( - p ) * ( |. p .| ) - ( |. p .| ) .| >= 0 ;
( |. q .| ) ^2 < 0 ;
Cage ( C , n ) /. ( 1 + 1 ) in L~ Cage ( C , n ) ;
consider e being Element of NAT such that a = 2 * e + 1 + e * e + 1 ;
Re ( F | D ) = Re ( F | D ) ;
not b , c // b , c or b , c // c , d ;
p , a // b , c or p , a // b , c ;
g . n = a * ( f . n ) .= a * ( f . n ) .= a * ( f . n ) .= a * ( f . n ) ;
consider f being Subset of X such that e = f .: X and f .: X is infinite ;
F | ( ( N | ( N | ( N , n ) ) ) ) = F | ( N , n ) ;
q in LSeg ( p , q ) \/ LSeg ( p , q ) ;
Ball ( m , r ) c= Ball ( m , r ) ;
the carrier of V = { 0. V } ;
rng ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
assume that Re ( seq ) is summable and lim ( ( ( ( ( ( ( seq . n ) (#) ( ( seq . n ) ) ) ) ) ) ) ) (#) ( ( ( seq . n ) (#) ( ( seq . n ) (#) ( ( seq . n ) (#) ( ( seq . n
||. v .|| . n - ||. v .|| < e ;
set g = O --> 1 ;
reconsider T2 = { t : 0 <= t & t <= 0 & t <= 0 & t <= 0 & 0 <= t & t <= 0 } as 0 -termal string of S ;
reconsider q1 = { q1 where q1 is sequence of TOP-REAL 2 : q1 in P & q1 `1 >= 0 & q2 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1 `2 >= 0 & q1
assume that L~ Cage ( C , n ) misses L~ Cage ( C , n ) and for i be Nat st 1 <= i & i <= n holds |. ( Cage ( C , n ) ) . i - ( ( E-max C ) * ( i , n ) ) .| < r ;
- ( 1 - ( 1 - ( n + 1 ) ) ) < ( 1 - ( n + 1 ) ) (#) ( ( 1 - ( n + 1 ) ) ) (#) ( ( ( n + 1 ) (#) ( ( n + 1 ) (#) ( ( n + 1 ) ) ) ) (#) (
set d = \rho ( \rho ( numbers , ^2 ) , dist ( ^2 ) ) , |. ( ^2 ) ) ^2 ) ;
2 |^ ( 2 -' 1 -' 1 ) = 2 |^ ( 2 -' 1 ) ;
dom v = Seg ( len v ) ;
set x1 = |. ( k - 1 ) - ( k - 1 ) .| + |. k - 1 .| ;
assume for n being Element of X holds 0. ( n + 1 ) <= ( n + 1 ) * ( n + 1 ) ;
assume that 0 <= i and i <= n and j <= n and i <= n ;
for A being Subset of X st c in A holds c . c = ( A . c ) . c
the support of L1 + L2 c= I ;
'not' p => ( x => p ) is valid ;
( f | n ) /. k = f /. ( k + 1 ) .= f /. ( k + 1 ) ;
reconsider Z = { {} } --> {} as Element of the carrier of normal normal normal normal normal Subgroup of normal normal normal normal normal normal normal normal normal normal normal normal normal normal Subgroup of normal normal normal normal normal normal normal normal normal normal Subgroup of normal normal ;
Z c= dom ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) )
|. 0 - ( q .| - |. q .| ) .| < r ;
zeroed c= zeroed ( A , L ) ;
E = dom ( |. L1 .| ) & E is_measurable_on dom ( |. L1 .| ) ;
C |^ A = C |^ A * B |^ A ;
the carrier of W2 c= the carrier of V ;
I . IC Comput ( P3 , s3 , 1 ) = ( P3 , s3 ) . IC SCM+FSA .= ( ( P3 , s3 ) . IC SCM+FSA ) .= ( ( P3 , s3 ) . IC SCM+FSA ) . IC SCM+FSA ;
attr x > 0 means : Def5 : x = x & x = 0 ;
LSeg ( f , i ) = LSeg ( f , i ) ;
consider p being Point of T such that C = { p where p is Point of T : p <= s & s <= s & s <= 1 & p <= 1 & p <= 1 & s <= 1 } ;
b , c are_connected & c , d are_connected ;
assume f = id the carrier of G ;
consider v such that v <> 0. V and v = L . v ;
let l be Linear_Combination of {} V ;
reconsider g = f " * ( f " ) as Function of ( the carrier of U1 ) , the carrier of U2 ;
{ A , B } in the points of G ;
|. x - x .| = - ( x - x ) .= |. x - x .| .= |. x - x .| .= |. x - x .| ;
set S = \mathop { \rm 1GateCircStr } ( x , y , c ) ;
to_power ( n + 1 ) >= ( n * PI ) * PI ;
v /. ( k + 1 ) = v . ( k + 1 ) .= v . ( k + 1 ) ;
0 mod ( i -' ( i -' 1 ) ) = ( i mod ( i -' 1 ) ) mod ( i -' 1 ) ;
the carrier of M1 = Seg n ;
Line ( M1 , j ) = 0. K ;
h . x1 = [ x1 , y1 ] ;
|. f .| (#) ( ( |. f .| (#) ( |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( .| .| ) ) ) ) ) )
assume x = ( a ^ <* b *> ) ^ <* a *> ;
I9 is closed ;
DataLoc ( t . a , k1 ) = 0 + 4 ;
x + y < - ( x - y ) + ( y - z ) ;
not c in { c , b } & c in { c , b } ;
f1 . 1 = f . ( 0 , 1 ) .= f . ( 1 , 1 ) .= f . ( 1 , 0 ) .= f . ( 1 , 1 ) .= f . ( 1 , 0 ) .= f . ( 1 , 1 ) .= f . ( 1 , 0 ) .= f . ( 1 , 1
x + ( y + z ) = ( x + y ) + ( y + z ) ;
f1 . a = v . a .= v . a .= v . a .= v . a ;
( p `1 ) ^2 <= ( ( ( ( ( ( ( ( TOP-REAL 2 ) | C ) | C ) | C ) ) ^2 ) ^2 ;
set R8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( ( ( ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ) ^2 ) ^2 ;
consider p such that p = { p2 : p2 `1 <= i & i <= n & i <= n & i <= n & i <= n & i <= n & i <= n } ;
|. ( f /* s ) . n - ( f /* s ) . n .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k ) = width N ;
f1 /* ( s \mathbin { \uparrow } k ) is convergent & lim ( ( ( s \mathbin { \uparrow } k ) \mathbin { \uparrow } k ) ) = ( ( s \mathbin { \uparrow } k ) \mathbin { \uparrow } k ) ) . n ;
f . x1 = y1 & f . x1 = y2 ;
len f + len f + 1 = len f + ( len f ) + 1 ;
dom ( ( ( ( ( ( ( ( ( i , n ) * ( i , m ) , n ) ) * ( i , n ) ) * ( i , n ) ) ) ) ) ) = Seg ( m , n ) ;
n = k * ( 2 * ( 2 * k ) ) + ( 2 * k ) ;
dom B = 2 \ { {} } ;
consider r such that r _|_ a and r , t // a , b and r , t // b , a ;
reconsider A1 = the carrier of Y as Subset of Y ;
1 in the carrier of [: the carrier of TOP-REAL 2 , the carrier of TOP-REAL 2 :] ;
let L be complete LATTICE , L be complete LATTICE ;
[ g1 , g2 ] in [: { g1 where g1 is Element of [: { g1 : g1 , g2 ] } , g2 ] } ;
set S2 = 1GateCircuit ( <* x , y *> , c ) ;
assume that f1 is_differentiable_on ]. x0 , x0 + r .[ and for g st g in ]. x0 , x0 + g .[ holds f1 . g - f2 . g < g1 ;
reconsider y = ( a " ) * ( a " ) as Element of L ;
dom s = { 1 , 2 } & dom s = { 1 , 2 } ;
( min ( g , c ) (#) f ) . c <= h . c ;
set a3 = the subgraph of G , a2 = the Target of G ;
reconsider g = f as PartFunc of REAL , REAL ;
|. ( |. s . m .| ) . m - ( |. ( p . m ) .| ) . n .| < p ;
for x being element holds x in QClass. ( u , t ) iff u in QClass. ( u , t )
P = the carrier of ( TOP-REAL 2 ) | P ;
assume that p1 in LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) and p1 = p2 ;
( 0. X \ { x } ) \ ( m \ ( m \ k ) ) = 0. X ;
let g be Element of Hom ( dom f , g ) ;
2 * a + ( 2 * ( 2 * a ) ) + ( 2 * a ) + ( 2 * a ) * a <= 2 * a ;
let f , g be Point of X , f be PartFunc of X , Y ;
set h = hom ( a , b ) ;
attr idseq ( n ) -> idseq ( n + 1 ) means : for m being Element of NAT st m <= n holds it . m = m ;
H * ( g " ) in the carrier of H * the carrier of N ;
x in dom ( ( ( ( the function ) `| Z ) * ( ( id Z ) * ( ( id Z ) * ( id Z ) ) ) ) ) ) ;
cell ( G , i2 , j2 -' 1 -' 1 , j2 -' 1 ) misses C ;
LE q1 , q2 , P ;
attr B is Subset of A ;
deffunc D ( set , set ) = union ( union ( $1 , $1 ) ) ;
n + ( n + nIT ) < len ( p - ( n + 1 ) ) - ( n + 1 ) ;
attr a <> 0. K means : Def5 : a = 0. K ;
consider j such that j in dom Len Len Len Len Len m and j + 1 = len Len m + j ;
consider z1 such that z in { z1 where z1 is Element of z1 : z1 <= z1 & z2 <= z1 & z1 <= z1 & z1 <= z1 & z1 <= z2 & z1 in z2 & z2 in z2 & z1 in z2 ;
for n being Element of NAT st n <= m holds |. ( ( r (#) f ) ) . n - ( r (#) f ) . n .| < r ;
set I1 = Comput ( P3 , s3 , i + 1 ) , I2 = P3 . ( i + 1 ) ;
set v = 3 3 * ( a \ b ) , v = 3 * ( a \ b ) , v = 3 * ( a \ b ) , v = 3 * ( a \ b ) ;
conv ( F .: ( union ( F .: ( F .: ( F .: ( W .: W ) ) ) ) ) ) ) c= union ( F .: ( W .: W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / / ) ) ) ) ) ) ) ) ) )
3 <= |. |. |. x0 - x0 .| + |. x0 - x0 .| ;
dom ( f (#) ( ( ( ( ( ( ( f (#) ( ( ( ( ( ( f (#) ( ( g ) ) (#) ( ( g (#) f ) ) ) ) ) ) ) ) ) ) (#) ( ( ( g (#) f ) (#) f ) ) ) ) ) ) ) ) = dom ( ( (
dom ( f * ( l * F ) ) = Seg ( k * F ) ;
rng ( s ^\ k ) c= dom ( f1 /* s ) ;
reconsider g4 = { p where p is Point of TOP-REAL 2 : p `1 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 &
( T * h ) . x = T . ( h . x ) .= T . ( h . x ) ;
( I * J ) . ( x , y ) = ( I * J ) . ( x , y ) .= ( I * J ) . ( x , y ) ;
y in dom ( ( ~ ( A ~ ) ~ ) ~ ) ;
for I being non degenerated doubleLoopStr , I being non empty doubleLoopStr , A being Ideal of I holds A * I = A * I
set T2 = s +* Initialize ( Initialize ( s ) ) , LifeSpan ( s , P ) + 1 ) ;
P1 = P1 +* I .= P1 +* I ;
lim S1 in the carrier of S1 & lim S1 in the carrier of S2 ;
v . i = ( v . i ) . ( v . i ) .= v . ( v . i ) ;
consider n being element such that n in NAT and n = x . n ;
consider x being Element of c such that F . x <> 0. V and F . x = 0. V ;
Funcs ( X , Y ) = Funcs ( X , Y ) ;
j + ( 2 * ( k + 1 ) ) + ( 2 * ( k + 1 ) ) ) > 0 ;
not s , r , s , t , r , s , t , s , t , r , s , t , t , t , s , t , r , s , t , t , s , t , r , s , t , t , s , t , t , r , s , t , t , s
n1 > len ( ( n1 , n1 , n2 , n1 ) * ( n1 , n2 ) ) ;
m1 . ( HT ( m1 , T ) ) = 0. L ;
attr H , G are_isomorphic means : Def5 : ex F , G being strict Subgroup of G st F , G are_isomorphic & G , F are_isomorphic ;
( N * ( i1 , j1 ) ) * ( i1 , j1 ) > 1 ;
[. s , 1 .] = [. s , 2 .] /\ [. s , 1 .] ;
x1 in [#] ( TOP-REAL 2 ) ;
let f1 , f2 be PartFunc of the carrier of S , the carrier of S ;
DigA ( g1 , 2 ) . ( ( g1 , 2 ) mod 2 ) = z . ( 2 + 1 ) .= z . ( 2 + 1 ) ;
I = I InsCode I & InsCode I = InsCode I & InsCode I = InsCode ( I ) ;
u in { u , v } \/ { v } ;
( w | p ) | ( ( w | p ) | ( w | p ) ) = p ;
consider u being element such that u in W1 and v = v + u + u and u in W2 ;
for y st y in rng F holds y = { a }
dom ( g * ( g `| V ) ) = K ;
ex x being element st x in ( ( the carrier of U1 ) \/ ( the carrier of U2 ) & x in A ;
ex x being element st x in ( the carrier of O ) \/ ( the carrier of O ) & x in A ;
f . x in the carrier of T ;
( the carrier of X ) \/ { 0. X } <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 : p1 `1 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2 `2 >= 0 & p2
b + ( b-s ) in { b where b is Real : b < a & a < b } ;
ex_sup_of x , L & x = sup ( x "/\" y ) ;
for x being element st x in X holds u in X
consider z being Point of G such that z = y and z in G and z in G ;
( the seq_id of seq_id ( seq_id ( seq_id ( seq_id ( seq_id ( seq_id ( vseq . n ) - seq_id ( vseq . m ) ) ) ) ) ) ) ) . i <= e ;
len ( w ^ <* w *> ) = len ( w ^ <* w *> ) + len ( w ^ <* w *> ) ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | [. - \infty , - \infty .] = g | [. - \infty , - \infty .] ;
reconsider i1 = x1 as Element of NAT ;
( a * A ) * ( a * A ) = ( a * A ) * ( a * A ) ;
assume ex n being Element of NAT st f . n = f . n ;
Seg ( len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) , len ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
( Complement Complement A1 ) . m c= ( Complement A1 ) . m ;
f1 . ( p + 1 ) = d . ( p + 1 ) .= d . ( p + 1 ) .= d . ( p + 1 ) ;
dom ( F | ( ( ( F | ( ( F , Y ) ) | ( ( F , Y ) | ( ( F , Y ) ) | ( ( F , Y ) | Y ) ) ) ) ) = dom ( F | Y ) ;
( x | ( y | ( y | ( y | ( y | ( y | ( y | y ) ) ) ) ) ) ) = ( y | ( y | ( y | y ) ) ) | ( ( y | y ) ) ;
|. x .| * |. x .| <= |. x .| * |. x .| ;
Sum ( F ) = Sum ( f | ( len F ) ) & Sum ( F | ( len F ) ) = Sum ( f | ( len F ) ) ;
assume for x being set st x in X holds x in X ;
assume that W1 is Subspace of V and for n , m being Element of V st n <= m holds ( ( W | ( n + 1 ) ) . m = ( W | ( n + 1 ) ) . m ;
||. t .|| . x - lim t .|| = ||. t . x - t . x .|| ;
assume that i in dom f and i <= len f and j <= len f and i <= len f ;
( p - ( ( ( ( ( p - ExpSeq ) ) ) * ( sn sn sn ) ) ) ^2 ) ^2 <= ( sn -FanMorphE ) ^2 ;
g | ( p , r ) = id ( the carrier of S ) ;
set N = max ( N , 2 ) , N = max ( N , 2 ) , N = max ( N , 2 ) ;
let T be non empty TopSpace ;
width ( B \mapsto 0. K ) = width ( B * ( i , j ) ) .= width ( B * ( i , j ) ) ;
attr a <> 0 implies a (#) ( A (#) B ) = ( A (#) B ) (#) B ;
attr f is_differentiable_in z0 means : Def5 : ex u , v being Element of REAL st u in dom f & v in dom f & u in dom f & v in dom f ;
assume that a <> 0 and a <> 0 and a = 0 and a = 0 and b = 0 ;
v1 , w , v1 , v1 , v1 , w1 , v1 , v1 , w1 , v1 , w1 , v1 , w1 , w1 , v1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1 , w1
p2 = p1 . ( ( |. p2 .| ) .| ) .= p2 . ( |. p2 .| ) .= p2 . ( |. p2 .| ) ;
ind ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | ( b | n ) ) ) ) ) ) ) ) ) ) ) ) ) = ind ( b | ( b | n ) ) .= b ;
[ a , A ] in the Inc of ( the Inc of A ) ;
m in ( the Arrows of C ) . ( ( the Morphism of C ) . ( o , m ) ) . ( ( the Morphism of C ) . ( o , m ) ) ;
( SUP ( a , PA , G ) ) . z = TRUE ;
reconsider \varphi = \varphi , \varphi = \varphi as Element of string ( S ) ;
len ( ( s * ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , , .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
\delta ( f , D ) . ( ( f . ( ( f . ( ( f . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , ( ( ( , , , , , ) ) ) ) ) )
[ f1 , f2 ] in the carrier of A ;
the carrier of ( TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 .= K1 ;
consider z being element such that z in dom ( g . z ) and z in Y ;
[#] V = { 0. V } .= { 0. V } ;
consider L2 being FinSequence of REAL such that rng L2 = dom ( f2 * f1 ) and for n being Nat st n >= 1 holds L2 . n = L2 . n ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s ;
h = f ^ <* p *> ^ <* q *> .= h ^ <* p *> ^ <* q *> ^ <* p *> .= h ^ <* q *> ^ <* p *> ^ <* q *> ;
c = [ b , c ] .= [ c , c ] .= [ c , d ] .= [ c , d ] ;
reconsider t1 = { p where p is Point of C : p `1 <= s & s `2 >= s & p `2 >= s & p `2 >= s `2 & p `2 >= s `2 & p `2 >= s `2 & p `2 >= s `2 & p `2 >= s `2 & p `2 >= s & p `2 >= s & p `2 >= s
sin . 2 in the carrier of TOP-REAL 2 ;
ex W being Subset of X st p in W & h .: W c= V ;
( h . ( ( p + 1 ) ) * ( p + 1 ) ) ) * ( h . ( p + 1 ) ) = ( h . ( p + 1 ) ) * ( h . ( p + 1 ) ) ;
R . b = 2 * PI .= 2 * PI .= 2 * PI * PI .= 2 * PI * PI .= 2 * PI ;
consider L1 such that B = exp_R * L1 + 1 * L1 and L1 * L2 + L2 * L1 = 1 * L1 ;
dom g = ( the Sorts of A ) . o ;
[ l , 6 ] in { \Rightarrow ( l , 6 ) where 6 is Element of l : 6 in { 7 } } ;
set T2 = Initialize ( s ) , \! Initialize s , E = Initialize s , \! Initialize s , \! E = Initialize s , \! \! LifeSpan ( s , P ) , E = P +* I ;
reconsider M = mid ( z , i , j ) as Element of REAL ;
y in product ( ( the support J ) +* ( V , v ) ) \/ { v } ;
[ 0 , 1 ] = 1 & [ 0 , 1 ] = 0 ;
assume x in the carrier of right right right right right right right right right right complementable ;
consider M being strict strict Subgroup of A such that a = M and M = the carrier of A ;
for x st x in Z holds ( ( ( ( - 1 / ( ( ( ( ( 1 / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ^2 ) ) ^2 ) ) ^2 )
len ( W + ( len W + 1 ) ) = m + ( len W + 1 ) ;
reconsider h1 = { v where v is VECTOR of X : v in Y & v in Y & w in Y & w in Y & ||. v - .|| < r } as VECTOR of Y ;
( i -' ( len p - 1 ) + 1 ) mod ( i + 1 ) in dom ( i + 1 ) ;
assume that M2 is convergent and M1 is convergent and M2 is convergent and M2 is convergent and M2 = the carrier of M2 ;
( ( mod ( ( ( ( ( ( ( ( x - div ( n ) - i ) ) ) ) * ( ( ( ( ( n - i ) * ( ( n - i ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) mod ( ( ( ( ( ( n - i ) mod ( n + 1 ) )
for u being element st u in Bags n holds p . u = p . u ;
for B being Subset of E st B in E holds A is closed
ex a being Point of X st a in A & a in A & a in A ;
set W2 = [: p , q :] \/ { p } ;
x in { X where X is Subset of L : X in F & X in F } ;
the carrier of W1 /\ ( the carrier of W2 + W2 ) c= the carrier of W1 + ( W2 + W1 ) ;
id ( a + b ) = id ( a + b ) ;
dom ( ( Partial_Sums ( ( ( ( ||. seq .|| ) ) | X ) ) | X ) ) | X ) = ( ||. ( ( ||. seq .|| ) | X ) .|| ) ;
set x = the Element of LSeg ( g , n ) ;
p => ( q => p ) in TAUT ( Al ) ;
set \pi = LSeg ( G * ( i , j ) , G * ( i , j ) , G * ( i , j ) ) ;
set \pi = LSeg ( G * ( i , j ) , G * ( i , j ) , G * ( i , j ) ) ;
- 1 + 1 <= \frac { 2 * PI - 2 * PI + 2 * PI * PI , 2 * PI * PI * PI + 1 / 2 * PI * PI * PI * PI + 2 * PI * PI * PI + 2 * PI * PI * PI + 2 * PI * PI * PI * PI / 4 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( ( ( ( ( ( 1 , 1 ) (#) ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( (
assume that b . r = { c . r where c is Real : c <= d & c <= d & d <= 1 } and c <= d } ;
ex r1 , r2 st r1 on P & r2 on P & r1 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2 on P & r2
reconsider g1 = g * f as Element of X * f as Element of Y ;
consider v1 being Element of T such that Q = ( \mathopen { \Vert } v \mathclose { \Vert } ) . v1 and for n being Element of NAT st n >= m holds ||. v .|| < s ;
n in { i where i is Nat : i + 1 <= n + 1 & i + 1 <= n + 1 } ;
( F . ( i + 1 ) ) . ( m + 1 ) >= ( F . ( m + 1 ) ) . ( m + 1 ) ;
assume that for p , q being Real st p = { ( p ) / |. q .| - sn ) .| and |. p .| >= sn and |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & |. p .| >= sn & p .| >= sn &
ConsecutiveSet ( A , ( ( \mathop { \rm succ ( m , n ) ) * ( ( m , n ) * ( ( m , n ) * ( m , n ) ) ) ) ) = ( ( ( m , n ) * ( ( m , n ) * ( m , n ) ) ) * ( m , n ) ) ;
set I1 = if>0 ( a , k1 , I ) , J = AddTo ( a , k1 , I ) , J = P3 ( P3 , s3 ) ;
for i being Nat st 1 < i & i < len z holds ( z * z ) * z ) * z = z * z
X c= ( the carrier of L ) \/ ( the carrier of L ) ;
consider T1 being Element of GF ( p ) such that T1 = a * ( p ) ;
reconsider e = { f where f is Element of D : f is one-to-one } as Element of D : f is one-to-one } ;
ex O being set st O in S & O = O & O = O & O = O ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U . m ;
f * g is_differentiable_in proj ( i , m ) . x ;
defpred P [ Nat ] means $1 + 1 = ( $1 + 1 ) * ( $1 + 1 ) ;
the cell of G1 , G2 , k ) = the cell of G1 , G2 , k ) ;
reconsider p9 = x as Point of TOP-REAL 2 ;
consider 4 such that y = y and 4 <= 4 and 4 <= 4 and 4 <= 4 and 4 <= 4 and 4 <= 4 ;
for n being Element of NAT st n <= m holds |. ( seq . n ) . m - ( seq . n ) .| < r ;
len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( len ( ) , , len ( , , len G , len G ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
for x being element st x in X holds ( the InternalRel of X ) . x = ( the InternalRel of X ) . x
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func On X -> set equals union ( X , Y ) ;
len ( ( CR ^ <* ( CR , len *> ) ) | ( len ( ( len ( CR , len <* ( *> ) ) |-> ( len ( ( len ( ( ( len ( ( len ( ( ( len ( ( len ( ( len ( ( ( len ( ( len ( ( ( len ( ( ( ( ( o1 , len ( o1 , len o1 ) ) ) ) ) |-> ( )
attr K is valuation means : Def5 : for a , b being Element of K st a <> 0. K holds a * b = i * b ;
consider o being OperSymbol of S such that t . {} = o . {} and o = o . {} ;
for x st x in X holds x in X iff x in X
IC Comput ( P3 , s3 , k ) in dom Shift ( P3 , s3 , k ) ;
attr q < s & s < t implies ex r , s st r < s & s < t & t < t & t < t & t < s & t < s & s < t & t < t & t < s & t < s & t < s & t < s & s < t & t < t & t < t & t < t & t < t & t < s & t
consider c being Element of Class ( f , c ) such that Y = Class ( f , c ) and for i being Element of NAT holds ( f . i ) . i = Class ( f , c ) ;
the Arity of S = the carrier of S & the carrier of S = the carrier of S ;
set y9 = [ <* 1 , 2 *> , 3 = [ 2 , 3 *> , 2 ] , z9 = [ 2 , 3 ] , z9 = [ 3 , 2 ] , z9 = [ 3 , 2 ] , z9 = [ 3 , 2 ] , z9 = [ 3 , 2 ] , z9 = [ 3 , 4 ] , z9 = [ 3 , 4 , 3 ] , z9 = [ 3 , 4 , 3
assume x in dom ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( `1 / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ;
{ p2 : p2 in cell ( GoB f , i , j ) & ( i + 1 ) + 1 <= width G } ;
( |. q .| ) ^2 >= ( ( ( |. q .| ) ^2 ) ^2 ;
set Y = { a "/\" b where a is Element of L : a in X } ;
i -' len f + 1 <= len f - ( len f -' 1 ) + ( len f -' 1 ) ;
for n st n in N & n in dom h holds h . n = x0
set i0 = ( \mathop { \rm Comput } ( P3 , s1 , 0 ) ) . i ;
( p . k ) . 0 = 1 or p . 0 = 1 or p . 0 = 0 ;
u + 0. V in ( U \ { 0. V } ) \ { 0. V } ;
consider q1 being set such that x in { q1 where q1 is set : q1 in { q1 where q1 is Element of REAL : q1 `1 >= a & q1 `1 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2 >= a & q1 `2
( p ^ q ) . m = ( ( ( ( q ^ ) | Seg ( len q ) ) + ( len q ) ) ) . m ;
g + h = g + h & g + h = g + h + ( h + c ) ;
L1 is LATTICE & for u being Element of L holds u in u & u in v & v in v & u in v & u in v ;
attr x in rng f & y in rng f implies x = y & y = f . x ;
assume that 1 < p and p + 1 <= len p and p = 1 and p = 1 and q = 1 and q = p + 1 and q = p + 1 ;
Ff * ( ( F*' *' ( m , n ) ) *' ( ( m *' *' ( m *' *' ( m ) ) *' ( ( m *' ( m *' ( ( m *' *' *' ( m ) ) ) ) ) ) ) ) ) = ( ( m *' ( ( m *' *' ( m *' ( m *' *' ( m ) ) ) ) *' ( ( m *' ( m *' ( m *'
let X be set ;
( ( N + ( ( N + 1 ) ) / 2 ) ) / 2 ) ^2 <= ( ( N + 1 ) ^2 ) ^2 ;
let c be Element of the bound of A ;
s . GBP = ( Initialize s ) . DataLoc ( s . GBP , 3 ) .= s . DataLoc ( s . a , 3 ) .= s . DataLoc ( s . a , 3 ) .= s . DataLoc ( s . a , 3 ) .= s . DataLoc ( s . a , 3 ) .= s . DataLoc ( s . a , 3 ) .= s . DataLoc ( s . a , 3 )
let a , b be Real ;
for x , y being Element of X holds x \ y = x \ y
for i being Nat , m , n being Nat , i be Nat , i be Nat , j be Nat , j be Nat , j be Nat , n be Nat , i be Nat ;
set z2 = ( Re ( ( ( ( Im ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( .| .| .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ) ) ) )
[ y , x ] in dom g & [ y , x ] in g . y ;
[. inf divset ( D , k ) , vol ( divset ( D , k ) ) .] ) .] c= A ;
0 <= |. ( S . n ) .| & |. S . n - lim S .| < e ;
( - ( - ( ( q `2 / |. q .| - sn ) ) ) / ( 1 + sn ) ) ^2 <= ( 1 - sn ) ^2 ;
set A = \frac { 2 } { 2 } ;
for x being set , y being set , x being set st x in y & y in x & x in y & y in y & x in y
deffunc F ( Nat ) = b * ( $1 + 1 ) ;
for s being element holds s in dom ( f ^ g ) & s in dom ( f ^ g ) & s in dom ( f ^ g ) implies s in dom ( f ^ g )
cluster non empty non void for TopSpace which is non empty ;
degree ( ( degree ( |. z .| ) .| ) ) ^2 >= 0 ;
consider n1 being Nat such that for k be Nat st n1 <= k holds |. ( seq . k ) . n1 - ( seq . n1 ) . n1 .| < r ;
Lin ( A /\ B ) is submodule of A /\ B ;
set -15 = ( ( ( ( ( ( ( ( \it \it false ) ) . ( z qua element ) ) ) . ( z qua Element of BOOLEAN ) ) . ( z qua Element of BOOLEAN ) ) . ( z , ( z , ( z , n ) ) . ( z , n ) ) ) ) ) ) . ( z , n ) ;
f " ( V ` ) in [: [: X , Y :] , X :] & f .: ( V ` ) in [: X , Y :] ;
rng ( a ( c , b ) ) c= { a } \/ { b } ;
consider y being WG2 being WG2 G2 G2 G2 G2 G2 G2 being WG1 of G2 , G2 being WG2 G2 G2 G2 G2 being WG2 G2 G2 , W1 being strict W2 of G1 , W2 being strict Subspace of G2 such that y = W1 + W2 and W1 = W2 + W2 and W2 + W2 = W1 + W2 ;
dom ( ( ( - 1 / 2 ) (#) ( f `| Z ) ) `| Z ) c= dom ( f `| Z ) ;
(# i , j #) is Element of <^ i , j ^> ;
v ^ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , width G ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) , ) ) ) ) ) ) ) ) ) ) ) ) ) ^ ( ( ( ( (
ex a , b being Element of REAL st i = a & ex k being Element of NAT st i = k & k <= n & k <= n & i <= n & k < n ;
t . ( ( m1 , m1 ) . ( i + 1 ) ) = ( m1 , m1 ) . ( i + 1 ) .= ( m1 , m1 ) . ( i + 1 ) .= m1 . ( i + 1 ) .= m1 . ( i + 1 ) .= m1 . ( i + 1 ) .= m1 . ( i + 1 ) .= m1 . ( i + 1 ) .= m1 . ( i + 1 )
assume that F is b-family and for p being FinSequence of bool Seg n st p in Seg n holds ( p * F ) . p = ( ( p * F ) . p ) . p ;
not b in { b } & b in { a } ;
( L1 \ { 1 } ) \ { 1 } c= ( L1 \ { 1 } ) \ { 1 } & ( L1 \ { 1 } ) \ { 1 } c= ( L1 \ { 1 } ) \ { 1 } ;
consider F being many sorted ManySortedSet of E such that for d being element st d in F holds F . d = F ( d ) ;
consider a , b such that a * ( b * w ) = a * ( b * w ) and b * w < b * w ;
defpred P [ finite finite FinSequence ] means $1 in dom ( ( Sum ( F ) ^ <* $1 *> ) ) ^ <* $1 *> ) ;
u = sin ( x , y ) .= ( sin . y ) * ( cos . x ) .= ( cos . y ) * ( cos . y ) .= ( cos . y ) * ( cos . x ) .= ( cos . y ) * ( cos . y ) .= ( cos . y ) * ( cos . y ) .= ( cos . y ) * ( cos . y ) .= ( cos . y )
dist ( seq . n + g . n , g . n ) <= dist ( seq . n , g . n ) + 0 ;
P [ p , |. p .| ] ;
consider X being Subset of CQC \hbox { - } such that X c= Y & X is finite and Y is non empty ;
|. b .| * |. b .| >= |. b .| * |. b .| ;
1 < ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l where l is Real : l <= l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len l & l <= len
Partial_Sums ( ( ( ( ( G . n ) ) . 0 ) ) . 0 ) ) . 0 <= ( G . 0 ) . 0 ;
f . y = x * y .= x * y .= x * y .= y * y ;
NIC ( i , i1 ) = { IC Comput ( P3 , s3 , i1 ) where i1 is Nat : i1 + 1 <= len P3 + 1 + 1 + 1 <= len P3 & 0 <= i1 + 1 + 1 + 1 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
product ( ( ( ( ( the support of I ) +* ( i , n ) ) +* ( i , n ) ) +* ( i , n ) ) ) ) . i in { i } ;
Following ( s , n ) = Following ( s , n ) ;
W is bounded ;
f /. ( i + 1 ) <> f /. ( i + 1 ) ;
M , f |= _ { ( f \leftarrow ( { x } \leftarrow { y } ) } ) } ;
len ( ( P1 ^ P2 ) ^ <* ( P1 ^ P2 ) ^ <* ( P1 ^ P2 ) ^ <* ( P1 ^ P2 ) *> ) ) = len ( P1 ^ P2 ) ;
A |^ ( m + k ) c= A |^ ( m + k ) & A |^ ( m + k ) c= A |^ ( m + 1 ) ;
for n , q being element st n \ |. q .| c= a holds |. q .| c= a
consider n1 being element such that n1 in dom ( p + n1 ) and n1 + 1 <= n1 + 1 ;
consider X being set such that X in Q & for Z st Z in X holds Z c= X ;
CurInstr ( P3 , Comput ( P3 , s3 , 1 ) ) <> halt SCM+FSA ;
for v be VECTOR of L1 , w be Element of V st v = - w holds ||. v .|| = ||. v .|| + ||. w .||
for \varphi st \varphi in X holds \varphi . \varphi in \varphi . ( \varphi . ( \varphi . n ) )
rng ( ( Sgm ( Seg ( len ( f1 | Seg ( len f1 ) - i ) ) ) ) ) ) ) = dom ( f1 | Seg ( len f1 - i ) ) ;
ex c being FinSequence st len c = k & len c = k & len c = k ;
Arity ( o , a ) . ( o , a ) = <* o , a *> . ( o , a ) .= o ;
consider f1 be Function of the carrier of X , the carrier of Y such that f1 = |. f1 .| & f1 in A ;
a1 = b1 & a2 = b2 & a3 = b2 & a3 = b2 ;
T2 . indx ( D2 , D1 , j ) = D2 . ( len D2 , j ) .= D2 . ( j + 1 ) .= D2 . ( j + 1 ) ;
f . ( r , s ) = |. r .| .= |. r .| .= |. r .| .= |. r .| .= |. r .| ;
consider n being Nat such that for m being Nat st n <= m holds |. ( seq . m ) - ( seq . n ) - ( seq . m ) .| < r ;
consider d being Real such that for a being Real st a in X holds ||. a - d .|| <= a ;
||. L - h .|| <= |. ( ( ( ||. h .|| - R .|| ) ) * ||. h .|| + ||. h .|| ) .| ;
attr F is commutative means : Def5 : for b being Element of X holds F . b = f . b ;
p = 1- ( p + q ) .= 1 * ( p + q ) .= 1 * ( p + q ) .= 1 * ( p + q ) .= 0. TOP-REAL 2 ;
consider z1 , z2 being Element of REAL such that b = z1 + z2 and z1 = z1 + z2 and z2 = z1 + z2 and z1 = z1 + z2 and z1 = z2 + z2 ;
consider i such that ( Arg ( ( Rotate ( s , 2 ) ) * ( i + 1 ) ) ) + ( Im ( ( s ) ) * ( i + 1 ) ) ) + ( Im ( ( s ) ) * ( i + 1 ) ) ) ) + ( Im ( ( s ) * ( i + 1 ) ) ) ) + ( Im ( ( s ) ) * ( i + 1 ) ) ) ;
consider g being Function such that dom g = card ( f .: X ) and rng g = X ;
assume that A = { 1 , 2 } and B <> {} and A = { 1 , 2 } and B = { 1 , 2 } and A = { 1 , 2 } and B = { 1 , 2 } ;
attr F is associative means : Def5 : for f being Function of X , Y holds F is associative ;
ex x being Element of NAT st x = x & x in i & i < i & i < m ;
consider i2 be Nat such that i2 in dom ( ( l + 1 ) - ( l + 1 ) ) and i2 in dom ( l + 1 ) ;
s = r * ( ( s - t ) ) * ( r - t ) .= r * ( s - t ) ;
F . ( id a , b ) = [ a , b ] ;
not p in { p where p is Element of L : p in { y is Element of L : y in { y is Element of L : y in { y is Element of L } ;
consider z being element such that z in dom ( ( F . 0 ) . z ) and z in dom ( F . 0 ) and z in dom ( F . 0 ) ;
for x being element st x in dom f & x in dom f & y = f . x ;
cell ( G , i , j ) = { |[ r , s ]| where s is Real : s <= ( G * ( i , j ) `1 } ;
consider e being element such that e in dom ( ( the carrier of ( T | ( E ) ) | E ) ) and e in { e where e is element : e in E } ;
( F " * ( ( b1 , b2 ) ) ) * ( b1 , b2 ) ) = ( ( Len b2 ) * ( b1 , b2 ) ) * ( b1 , b2 ) ;
- ( 1 - ( ( 1 - ( ( 1 - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( `1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( ( ( ( ( ( ) ) | ( len ( ( ( ( ( ( ) ) ) ) | ( len ( ) ) ) ) ) ) | ( len ( ( ( ( ( ( ( ( ( ( ( ( )
attr x in dom f & for n being set st x in dom f holds f . n = f . n ;
len ( ( ( ( f1 . j ) * ( i , j ) ) * ( i , j ) ) ) ) = len ( ( f1 . j ) * ( i , j ) ) .= len ( ( f1 . j ) * ( i , j ) ) .= ( ( f1 . j ) * ( i , j ) ) .= ( ( f1 . j ) * ( i , j ) ) * ( i , j ) ) .= ( (
All ( a , A ) , B , C , A , B , C ) |= All ( a , B , G ) ;
LSeg ( E , k ) c= Cl ( ( E , k ) \ { E } ) ;
x \ ( m \ k ) = ( m \ k ) \ ( m \ ( m \ k ) ) .= ( m \ k ) \ ( m \ k ) .= ( m \ k ) \ ( m \ k ) .= ( m \ k ) \ ( m \ k ) .= ( m \ k ) \ ( m \ k ) .= ( m \ k ) \ ( m \ k ) \ ( m \ k ) ;
k - ( ( ( Partial_Sums ( seq . n ) ) . k ) ) . k = ( Partial_Sums ( seq . n ) ) . k .= ( Partial_Sums ( seq . n ) ) . k .= ( Partial_Sums ( seq . n ) ) . k .= ( ( Partial_Sums ( seq . n ) ) . k .= ( Partial_Sums ( seq . n ) ) . k .= ( ( Partial_Sums ( seq . n ) ) . k ) . k .= ( ( Partial_Sums ( seq . n ) . k )
for s being State of A holds Following ( s , 2 ) . IC s = Following ( s , 2 ) . IC s ;
for x st x in Z holds ( ( - 1 / ( ( 1 - ( ( ( ( 1 / ( ( 1 - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / ( ( ( ( - ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ^2 ) ) ^2 ) ) ^2 ) ) ) ^2 ) ) <> 0 & ( ( ( ( ( ( ( ( ( ( ( ( (
support ( m | ( Seg ( m + 1 ) ) ) c= Seg ( m + 1 ) ;
reconsider t = u as Function of the carrier of A , the carrier of B ;
- ( ( a * ( b * ( 1 - a ) ) ) ) ^2 <= - ( a * b ) ^2 ;
\varphi . a = g . a & \varphi . a = g . a ;
assume that i in dom ( F ^ <* p *> ) and i in dom ( F ^ <* q *> ) and j in dom ( F ^ <* p *> ) ;
not x1 in { x1 , x2 } \/ { x1 , x2 } ;
the Sorts of U1 /\ U2 c= the Sorts of U2 /\ the Sorts of U2 ;
( 2 * ( a - b ) ) ^2 + ( 2 * a ) ^2 > 0 ;
consider z1 being element such that for z being element st z in N holds z1 in N & z1 in N & z1 in N ;
assume that the carrier of S = <* the carrier of S , the carrier of S *> and the carrier of S = { the carrier of S } ;
Z = dom ( ( ( ( ( #Z 2 ) (#) ( ( ( ( #Z 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^ ) ^ ) ^ ) ;
integral ( f , S ) = integral ( f , S ) & lim ( f , S ) = integral ( f , S ) ;
\cal X ( z1 , z1 ) => ( z1 => ( z1 + z2 ) ) in TAUT ( z1 , z2 ) ;
len ( M2 * M2 ) = n & width M2 = n ;
attr X \/ { X , Y } is SubSpace of X & Y is SubSpace of X & X is SubSpace of Y implies X is SubSpace of Y
let L be non empty RelStr ;
reconsider f1 = F . ( ( b , c ) as Function of ( the Sorts of M ) . ( b , c ) ;
consider w being FinSequence of I such that the carrier of M = I & for i being Nat st i >= 1 holds w . i = ( the Tran of M ) . i ;
g . ( a , b ) = g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . ( a , b ) .= g . (
assume for i being Nat st i in dom f holds f . i = rpoly ( 1 , z ) . i ;
ex L being Subset of X st L = { {} where X is Subset of X : X in F & L c= F & X c= F } ;
( the carrier of C ) /\ the carrier of C = the carrier of C ;
reconsider a9 = o as Element of TS ( V ) ;
1 * ( ( x1 + x2 ) * ( x1 + x2 ) ) = x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 + x3 ) .= x1 * ( x2 , x3 ) .= x1 * ( x2 , x3 ) .= x1
|. E " * ( ( |. E .| ) .| ) .| = ( |. E .| ) * |. ( E .| ) .| .= |. E .| * |. E .| .= |. E .| * |. E .| .= |. E .| * |. E .| .= |. E .| * |. E .| .= |. E .| * |. E .| * |. E .| .= |. E .| * |. E .| * |. E .| ;
reconsider u1 = the carrier of U as Subset of U ;
( x "/\" y ) "/\" ( x "/\" y ) <= x "/\" y ;
|. f . ( ( ( ( |. f . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , , , , .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .| < |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) .| ) )
LSeg ( Gauge ( C , n ) , i ) is non empty ;
( f | Z ) /. x0 = ( f | Z ) /. x0 + R /. x0 .= R /. x0 + R /. x0 ;
g . c + 1- ( f . c ) <= h . c + ( f . c ) * ( f . c ) ;
( f + g ) | divset ( i , j ) = f | divset ( i , j ) ;
attr width ( f ) = width ( f + g ) & width ( f + g ) = width ( f + g ) ;
len ( - ( ( - ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
let n be Nat ;
pdiff1 ( f1 , 2 ) . ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 1 * ( 1 - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) . 2 = ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2
attr a <> 0 & a = 0 implies a = 0 & a = 0 & a = 0 & b = 0 & a = 0 & a = 0 & b = 0 ;
for c being set , a being set , f being Function of the carrier of SCM+FSA , R^1 st a in dom ( f +* ( the carrier of SCM+FSA ) holds f . a = ( the carrier of SCM+FSA ) . a
assume that V1 is linearly independent and V1 = V1 + V2 and V2 in V2 and V2 in V2 and V2 c= V2 and V2 c= V2 and V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V2 & V2 c= V & V2 c= V & V2 c= V & V2 c= V & V2 c= V & V2 c= V & V2 c= V & V2 c=
z * ( x1 + y1 ) in M & ( x1 + y1 ) * ( x1 + y1 ) in M ;
rng ( |. ( |. ( ( ( |. ( ( ( ( |. ( ( ( .| .| .| ) ) ) .| ) ) .| ) .| ) ) ) .| ) .| ) .| = Seg ( ( |. ( ( ( ( ( ( |. ( ( ( ( |. ( ( ( .| ) ) ) ) .| ) ) .| ) ) ) .| ) .| ;
consider M2 being convergent convergent convergent & for n being Nat st n <= m holds |. ( ( M2 . n ) - ( M2 . n ) - ( M2 . n ) .| < p ;
h " = ( h " ) . n & h . n = 0 ;
( Partial_Sums ( |. ( Partial_Sums ( seq ) ) . m ) .| ) . m = |. ( Partial_Sums ( seq ) . m ) . m .| .= |. ( Partial_Sums ( seq ) ) . m ) . m .| .= |. ( ( Partial_Sums ( seq ) . m ) . m .| .= |. ( Partial_Sums ( seq . m ) . m ) .| .= |. ( ( ( seq . m ) . m ) .| .= |. ( ( ( seq . m ) . m ) .| .= |. ( ( seq . m ) . m ) .| .= |.
( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA = ( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA .= ( ( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA .= ( ( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA .= ( ( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA ) . IC SCM+FSA .= ( ( Comput ( P1 , s1 , 1 ) ) . IC SCM+FSA .= ( ( Comput ( P1 , s1 , 1 ) . IC SCM+FSA ) .
- v = - v * w + v * w * v * v .= - v * w ;
sup ( ( 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
A |^ ( k + 1 ) = ( A |^ ( k + 1 ) ) |^ ( k + 1 ) .= ( A |^ ( k + 1 ) ) |^ ( k + 1 ) .= ( ( A + ( k + 1 ) ) |^ ( k + 1 ) ) |^ ( k + 1 ) .= ( ( A + ( k + 1 ) ) |^ ( k + 1 ) ) ;
let I be non empty set , f be PartFunc of the carrier of R , the carrier of R ;
( f . p ) . ( q . ( q . p ) ) = ( f . p ) . ( q . ( q . p ) ) .= ( f . p ) . ( q . p ) .= ( f . p ) . ( q . p ) ;
let a be Nat , b be Nat ;
consider r being Element of CQC-WFF ( A ) such that r is Element of CQC-WFF ( A ) & for n being Element of NAT holds ( ( ( A ) . n ) . ( r , n ) ) . ( r , n ) = ( ( A . n ) . ( r , n ) ) . ( r , n ) ;
for X being non empty TopSpace , A being Subset of X st A = X & A is closed holds A is closed
not x1 in { x1 , x2 } \/ { x1 , x2 } ;
h . ( f . ( O + 1 ) ) = [ f . ( O + 1 ) , f . ( O + 1 ) ] ;
( Gauge ( C , n ) * ( i , j ) , ( Gauge ( C , n ) * ( i , j ) ) `2 = ( ( Gauge ( C , n ) * ( i , j ) ) * ( i , j ) `2 ) `2 .= ( ( Gauge ( C , n ) * ( i , j ) ) `2 .= ( ( Gauge ( C , n ) * ( i , j ) ) `2 ) `2 .= ( ( ( Gauge ( C , n
cluster m gcd p -> prime ;
( f * F ) . ( ( f . ( n + 1 ) ) ) = f . ( n + 1 ) .= f . ( n + 1 ) ;
let L be LATTICE , a , b be Element of L ;
consider b being element such that b in dom ( H . ( ( H . ( x , y ) ) ) ) ) and b = H . ( ( H . ( x , y ) ) ) ;
assume that x in dom ( F * G ) and y in dom ( F * G ) and y in dom ( F * G ) ;
assume that ex e being element st e Joins W , G and W is open and W is open and ex W being Walk of G st W is open and W is open and W is open and W is open and W is open and W is open and W is open ;
( ( ( ( ( h (#) f ) (#) f ) `| Z ) `| Z ) ) `| Z ) = ( ( ( ( ( ( ( ( ( h (#) f ) `| Z ) (#) f ) `| Z ) `| Z ) `| Z ) (#) f ) `| Z ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( id Z ) (#) f ) (#) f ) `| Z ) (#) f ) `| Z ) (#) f ) `| Z ) (#) f
j + 1 = i + ( j + 1 ) .= i + ( j + 1 ) .= i + ( j + 1 ) .= i + ( j + 1 ) ;
S ^ ( S ^ ( S ^ ( S ^ ( S ^ T ) ) ) ) ) = S ^ ( S ^ ( S ^ T ) ) .= S ^ ( S ^ ( S ^ T ) ) .= S ^ ( S ^ ( T ^ T ) ) .= S ^ ( S ^ T ) ;
consider H being sequence of the carrier of V such that rng H = the carrier of V and rng H c= the carrier of V ;
attr R is defined means : Def2 : for p being FinSequence of TOP-REAL 2 st p in R & p in R holds it . p = p & for q being Point of TOP-REAL 2 st q in R holds it . q = q `1 ;
dom ( meet ( X --> Y ) ) = dom ( meet ( X --> Y ) --> X ) .= dom ( ( X --> Y ) --> Y ) .= dom ( ( X --> Y ) --> Y ) .= dom ( X --> Y ) ;
sup ( ( proj2 .: .: ( .: .: .: ( .: .: .: .: ( .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .: .:
for r be Real st r < r holds |. ( S . m ) - ( S . m ) .| < r
i * ( ( ( f1 - f2 ) * ( i , j ) ) * ( i , j ) ) ) = i * ( ( f1 - ( ( f1 - f2 ) * ( i , j ) ) ) ;
consider f being Function of 2 , BOOLEAN such that for Y being set st Y in 2 holds f . Y = f . Y ;
consider g , h being element such that g in union { union { g where g is element : g in union A } and h in A & h in B } ;
func d * ( n + 1 ) -> Nat means : it : for n holds it . n = d * ( n + 1 ) ;
f . ( 0 , 1 ) = ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( ( 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .= ( ( 1 - ( ( 1 - ( ( 1 - ( ( 1 -
t = h ( B ) or t = h ( B ) or t = h ( B ) or t = h ( B ) or t = h ( B ) $ . t = h ( B ) or t = h ( B ) or t = h ( B ) or t = h ( B ) $ . t = h ( B ) ;
consider m1 be Nat such that for n be Nat st n >= m1 holds |. ( ( seq . n ) . m ) - ( ( seq . n ) . m ) .| < 1 ;
( 1 - ( ( ( ( q `2 / |. q .| - sn ) ) sn ) sn ) sn ) sn ) ^2 <= ( 1 - sn ) sn ) ^2 ;
h . ( i + 1 ) = ( h . ( i + 1 ) ) . ( i + 1 ) .= h . ( i + 1 ) .= h . ( i + 1 ) .= h . ( i + 1 ) ;
consider o being Element of the carrier of S such that a = [ o , the carrier of S ] and o = [ o , the carrier of S ] ;
let L be non empty RelStr ;
||. h .|| . n = ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .|| .= ||. h .|| . n .|| .= ||. h .|| . n .|| ;
( ( ( ( ( ( the function ) * ( ( the function ) ) * ( ( function * ( the function ) ) ) ) ) ) ) `| Z ) . x = ( ( ( ( the function ) `| Z ) . x ) ) (#) ( ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( ( id Z ) * ( id Z ) ) ) ) ) ) ) ) . x .= ( ( ( id Z ) * ( ( id Z
attr r = F .: ( p , q ) ;
sqrt ( |[ |[ ^2 , 0 ]| , 0 ]| ) ^2 + ( sqrt ( 1 + sn ) ^2 ) ^2 ) ^2 + ( sqrt ( 1 + sn ) ^2 ) ^2 ) ^2 + ( sqrt ( 1 + sn ) ^2 ) ^2 + ( 1 + sn ) ^2 ) ^2 = ( 1 + ( sn ^2 ) ^2 + ( ( 1 + sn ) ^2 ) ^2 ;
let i be Nat ;
attr a <> 0. R implies a * v = a * v + a * v ;
( j -' i + 1 ) mod ( j -' i ) = ( j -' i ) mod ( j -' i ) .= ( j -' i ) mod ( j -' i ) ;
deffunc F ( Nat ) = ( ( ( h " ) (#) ( h " ) ) (#) ( h " ) ) (#) ( h " ) ;
assume that the carrier of H = the carrier of G and the carrier of H = the carrier of H and the carrier of G = the carrier of G ;
( the Sorts of Free ( X ) ) . o = ( the Sorts of Free ( X ) ) . o ;
H = n + ( 2 * ( 2 * n + 1 ) ) .= n + ( 2 * n ) .= n * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) .= ( 2 * n ) * ( 2 * n ) ;
( ( |. g1 .| ) ^2 = 0 & |. g1 .| = 0 & |. g1 .| = 0 & |. g1 .| = 0 & |. g1 .| = 0 & |. g1 .| = 0 ;
F .: ( dom F ) = { F . ( n + 1 ) where n is Nat : n <= len F & n + 1 <= len F } .= n ;
attr b <> 0 & b <> 0 implies b = 0 & a = 0 & b = 0 & b = 0 & a = 0 & b = 0 & d = 0 & d = 0 & d = 0 & d = 0 ;
dom ( ( f +* ( ( f +* g ) ) ) ) = dom ( f +* g ) \/ ( f +* g ) .= dom ( f +* g ) /\ ( f +* g ) .= dom ( f +* g ) /\ ( f +* g ) .= dom ( f +* g ) /\ ( f +* g ) .= dom ( f +* g ) /\ ( ( f +* g ) +* g ) ;
for i being set , g being set , a being Element of L st i in dom g & i <= len g holds a * g = a * g
g " = g * ( f " ) .= g " * ( f " ) .= g " * ( f " ) .= g " * ( f " ) .= g " * ( f " ) .= g " * ( f " ) .= g " * ( f " ) .= g * ( f " ) .= g * ( f " ;
consider i , j such that f . i = s . i and j <> i and i < j and i < j ;
h | [. a , b .] = ( g | [. a , b .] ) | [. a , b .] .= ( g | [. a , b .] ) | [. a , b .] .= [. a , b .] ;
[ s , t ] in R & [ s , t ] in R ;
attr H is negative means : Def5 : ex n st H is negative & for n st n < n holds it . n = n ;
attr f1 is total means : Def5 : for c , b being total Relation st c in dom f1 & b in dom ( f1 + f2 ) & c in dom ( f1 + f2 ) & ( f1 + f2 ) . c = ( f1 + f2 ) . c holds ( f1 + f2 ) . c = ( f1 + f2 ) . c ;
z1 in { z1 where z1 is Element of ( z1 ) : z1 in z1 & z2 in z1 & z1 in z2 & z1 in z2 & z1 in z2 & z1 in z2 } ;
p = 1 * p .= a " * p .= a " ;
for r1 being sequence of REAL st 0 < r1 & for n be Nat st n <= m holds |. ( r1 . n ) - ( r1 . n ) .| < r1 ;
mode carrier of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) = ( TOP-REAL 2 ) | ( TOP-REAL 2 ) or ( TOP-REAL 2 ) | ( TOP-REAL 2 ) = ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ;
||. f ( ( g . ( k + 1 ) ) - ( g . ( k + 1 ) ) .|| <= ||. ( g . ( k + 1 ) - ( g . ( k + 1 ) ) - ( g . ( k + 1 ) ) .|| ;
assume h = ( B .--> ( B .--> ( B .--> C ) ) +* ( ( B .--> D ) .--> ( B .--> E ) ) ) +* ( ( B .--> E ) .--> E ) ) ) +* ( ( B .--> E ) +* ( E .--> E ) ) ) ;
|. ( ( divset ( H , n ) ) . k - ( vol ( divset ( H , n ) ) ) ) .| <= e * ( vol ( divset ( H , n ) ) ) * vol ( divset ( H , n ) ) ) ;
( the Sorts of A1 ) . v = [ the Sorts of A2 , the Sorts of A1 ] ;
not x1 in { x1 , x2 } \/ { x1 , x2 } .= { x1 , x2 } ;
attr A = [. 0 , 1 .] * ( ( sin . ( 2 * ( ( sin . ( 2 * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
p " is permutation of ( the InternalRel of X ) " & p " = ( the Sgm Y ) " * p " ;
for x , y being set st x in A & y in A holds |. ( f . x ) - ( f . y ) .| < |. f . y .|
|. ( |. p .| ) ^2 = |. p .| ^2 - |. p .| ^2 ;
let f be PartFunc of the carrier of C , the carrier of C ;
assume for x being Element of Y st x in EqClass ( u , A ) holds ( u , A ) . x = ( u , A ) . x ;
consider F be Function of n1 , n2 such that for k being Nat st k in dom F holds F . k = F ( k ) ;
ex u st u <> v & v <> 0. V & u <> 0. V & u <> 0. V ;
let G be Group , N , A be normal Subgroup of G ;
for s be Real st s in dom ( ( R + ( f + g ) (#) ( f + g ) ) holds integral ( f + g ) = integral ( f + g )
width ( ( ( ( M1 , M2 ) * ( len ( M1 , len ( M2 ) ) ) ) ) ) ) = width ( ( ( M2 ) * ( ( M2 ) ) ) ) ) * ( ( ( M2 ) ) ) ) ) ) ) ) .= ( ( ( ( M2 ) * ( ( M2 ) ) ) ) ) * ( ( M2 ) ) ) ) ) ) * ( ( ( M2 ) ) * ( ( M2 ) ) ) ) ) ;
f | [. - 1 , 1 .] = f | [. - 1 , 1 .] & f | [. 1 , 1 .] = f | [. 1 , 1 .] ;
attr X is closed means : Def5 : for n , m being Element of X st n in dom f & m in dom f holds f . n in X & f . n in X ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( (
func WFF ( V ) -> Subset of V means : for k being Element of NAT st k in dom l holds it . k = V ( k ) ;
let L be non empty TopSpace , N , M be net of L ;
for s be Element of NAT holds ( ( \sum ( ( Partial_Sums ( seq ) ) ) ) ) . n ) . m = ( Partial_Sums ( seq ) ) . m + ( ( Partial_Sums ( seq ) ) . m ) . m
then z /. 1 = ( N-min L~ z ) /. 1 .= z /. 1 ;
len ( p ^ <* 0 *> ) = len ( p ^ <* 1 *> ) + ( len ( p ^ <* 0 *> ) ) .= len ( p ^ <* 1 *> ) + ( len ( p ^ <* 1 *> ) .= len ( p ^ <* 1 *> ) + ( len ( p ^ <* 1 *> ) ) .= len ( p ^ <* 1 *> ) + ( len ( p ^ <* 1 *> ) .= len ( p ^ <* 1 *> ) + ( len ( p ^ <* 0 *> ) .= len ( p ^ <* 1 *> ) + ( len ( p ^ <* 1 *> ) .= len ( p ^
assume that Z c= dom ( ( ( ( - ( ( ( ( ( ( ( ( ( ( 1 / 2 ) ) ) ) (#) ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ( ( ( - ( ( ( ( ( ( - ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) and and for x being Real st x in Z holds ( ( ( ( ( ( ( ) / 2 ) * ( ( ( ( ( ( ( ) / 2 ) * ( ( ( ( ( ( ( (
let W be add-associative right_zeroed right_complementable distributive add-associative right_zeroed right_complementable distributive non empty doubleLoopStr ;
consider f being Function of { B where B is Subset of B : B in F & f . B = F ( B ) ;
dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( Seg ( len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) | Seg n ) ) ) ) ) | Seg n ) ) ) ) | Seg n ) ) | ( Seg n ) ) ) ) ) | n ) )
for S being functor from the functor from the functor of S , the functor of S , the functor functor functor from the functor of S , the functor of S means : : for c being element st c in S holds it . c = ( the functor functor functor functor from c , the functor from from from from [ c , the carrier of S ] ;
ex a st a = [ a , b ] & a in Line ( b , a ) & b in Line ( b , a ) ;
a in Free ( H ) & ( a , ( ( a , ( ( a , ( ( H , H ) ) , ( H , H ) ) , ( a , H ) ) , ( a , H ) ) |= ( a , H ) ) '&' ( a , H ) ) ;
let C be complex number , f be PartFunc of C , the carrier of C ;
( ( N-min L~ f ) * ( len f -' 1 ) + 1 ) `1 = ( N-min L~ f ) * ( len f -' 1 ) `1 .= ( N-min L~ f ) * ( 1 , 1 ) `1 .= ( N-min L~ f ) * ( 1 , 1 ) `1 ;
assume that u = <* x1 , x2 *> and u = <* x1 , x2 *> and x2 = <* x1 , x2 *> ;
then t . {} = t . {} & t . {} = t . {} ;
Valid ( p , J ) . v = TRUE . v .= TRUE . v ;
assume for x , y being Element of S st x <= y & y <= x & x <= y holds x <= y
func Class ( R , A ) -> Subset of R ;
defpred P [ Nat ] means ( ex v being Vertex of G st v = ( the Vertex of G ) . $1 & ( the Target of G ) . $1 = v . ( $1 + 1 ) ;
attr dim ( W1 ) = 0 & 0. V = 0 implies 0. V = 0 ;
mam . ( t . ( m + 1 ) ) = ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 ) * ( m + 1 ) .= ( m + 1 ) * ( m + 1 )
set d = f ^ <* y1 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y1 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ ( y2 ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ <* y2 *> ^ ( y2 ^ <* y2 *> ^ <* y2 *> ^ <*
consider g such that x = g & for n st n in dom ( f | X ) holds g . n = f . n ;
x + ( len x ) = ( len x ) + ( len y ) .= len y + ( len y ) .= len y + ( len y ) .= len y + ( len y ) .= len y + ( len y ) .= len y + ( len y ) .= len y + ( len y ) .= len y ;
n1 + ( n1 + 1 ) in dom ( ( f | n1 + 1 ) ) /\ ( n1 + 1 ) ;
assume that P is closed and P is closed and ( ex p , q being Point of TOP-REAL 2 st p = q & q in P & p in P & q in P & ( |. p - q .| < 1 & |. p - q .| = 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q - q .| < 1 and |. p - q - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .| < 1 and |. p - q .|
reconsider a9 = a , b9 = b as Element of A ;
reconsider FFf = Ff . ( t . ( t . ( t . ( t . ( t . ( t ) ) ) ) as Morphism of ( S , T ) ) ;
LSeg ( f , i + 1 ) = LSeg ( f , i + 1 ) ;
Partial_Sums ( P | ( m + n ) ) . m <= Partial_Sums ( P | ( m + n ) ) . m ;
assume dom ( f1 + f2 ) = dom ( f1 + f2 ) & dom ( f1 + f2 ) = dom ( f1 + f2 ) ;
consider v being Real such that v = y and dist ( v , v ) < r ;
let G be Group , H , i be Element of H , i be Element of NAT ;
consider B being set such that for S being element holds S . S = ( the Sorts of U1 ) . S ;
reconsider X1 = { p where p is Point of TOP-REAL 2 : p `1 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p `2 >= 0 & p
\frac ( N - ( N - ( N - ( N - 1 ) ) ) ) ^2 ) ^2 <= \frac { N - ( N - ( N - 1 ) ^2 - ( N - 1 ) ^2 ) ^2 where N is Nat : N <= N & N <= ( N - 1 ) ^2 } ;
for x be Element of X st x in E & |. ( ( Im F ) . n ) . x - ( Im F ) . x .| <= ( Im F ) . x ) + ( Im F ) . x
len ( @ @ ^ <* ( @ @ @ ) ^ ( @ @ ) *> ) = len ( ( @ @ @ ) ^ ( ( @ @ @ ) ^ ( ( @ @ @ ) ^ ( ( @ @ ) ^ ( ( @ @ @ ) ^ ( ( @ @ @ ) ^ ( ( @ @ ) ^ ( @ @ ) ) ) ) ) ) ) ) .= ( ( @ @ ) ^ ( ( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
v , ( ( ( x , y ) (#) ( ( x , y ) (#) ( ( x , y ) (#) ( y , z ) ) ) ) ) ) . v = ( x , y ) (#) ( y , z ) ) . v ;
consider r being Element of M such that M , ( v , ( v , m ) ) |= r and for m , n , m , k being Nat st m <= n & n <= m & m <= m holds ( ( v , m ) . ( n , m ) = ( v , m ) . ( n , m ) ;
func Nw \ { w where w is Element of Union G : w in ( the Sorts of A ) . w } -> Element of ( the Sorts of A ) . w ;
s2 . ( ( |. s2 .| ) . m ) = |. s2 .| . m .= |. s2 .| . m .= |. s2 . m .| ;
for n be Nat holds ( ( Partial_Sums ( seq ) ) . n ) . n = ( Partial_Sums ( seq ) . n ) . n
set F = S \! \mathop { \rm \hbox { - } ;
( Partial_Sums ( seq ) . n + ( Partial_Sums ( seq ) . n ) . m + ( Partial_Sums ( seq ) . m ) . m + ( Partial_Sums ( seq ) . m ) . n ) . n + ( Partial_Sums ( seq ) . n ) . m + ( ( ( seq . m ) . n ) . n ) . n + ( ( Partial_Sums ( seq ) . m ) . n ) . n ;
consider L , R such that for x st x in N holds ( f | Z ) . x = L . x + R . x ;
the carrier of ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) = ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ;
a * ( b + c ) + ( b * c ) + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c ) + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c ) + ( b * c ) + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c + ( b * c ) * c ) * c + ( b * c
v , ( ( m1 . m ) . ( m1 + 1 ) ) . ( m1 + 1 ) = v . ( m1 + 1 ) ;
( ( ( ( ( Q ^ <* ( Q ^ <* ( Q ^ <* <* ( Q *> ^ <* ( Q ^ <* ( Q *> ) *> ) ) ) *> ) ) ^ <* ( Q ^ <* Q ^ <* Q *> ) *> ) *> ) ^ <* ( Q ^ <* Q *> ) *> ) ^ <* ( Q ^ <* Q *> ) *> ^ <* ( Q ^ <* Q *> ^ <* Q *> ) *> ) *> ) ^ <* ( Q ^ <* Q *> ) *> ) ^ <* ( Q ^ <* Q *> ) *> ) ^ <* ( Q ^ <* Q *> ^ <* Q *> ) *> ) ^ <* ( Q ^ <* Q ^ <* Q *> ^ <* Q *> ) *> ) ^ <* Q
Sum ( F . n ) = ( ( |. F . n .| ) . ( ( F . n ) . m ) .= ( F . m ) . ( ( F . n ) . m ) .= ( F . m ) . ( ( F . n ) . m ) .= ( F . m ) . ( ( F . n ) . m ) .= ( F . m ) . ( ( F . m ) . ( n + 1 ) ) .= ( F . m ) . ( n + 1 ) .= ( F . n ) . ( n + 1 ) ) .= ( F . n ) . ( ( F . n ) . ( n + 1 ) . ( n + 1 ) .= ( F . n )
( ( ( the Go-board of f ) * ( i1 , j1 ) ) + 1 , j1 ) ) * ( i1 , j1 ) `2 = ( ( the GoB f ) * ( i1 , j1 ) ) * ( i1 , j1 ) `2 .= ( ( the GoB f ) * ( i1 , j1 ) `2 .= ( ( the GoB f ) * ( i1 , j1 ) `2 ;
defpred X [ Element of NAT ] means ( ( for n holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . $1 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * * ( ( ( ( ( ( ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
the Arity of S = ( the Arity of S ) . o .= ( the Arity of S ) . o .= ( the connectives of S ) . o .= ( the connectives of S ) . o .= ( the connectives of S ) . o ;
( X \times Y ) \/ X = X & X c= Y & Y c= Y & Y c= X & Y c= X implies Y c= X
for a , b being Element of S st a = b & b = N & a = N . ( b , a ) holds b = N . ( b , a )
E \models _ { f } ( All ( x , f ) '&' All ( x , f ) ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x , f ) '&' All ( x ,
ex L2 being 1-sorted structure , p being Element of ( the carrier of R ) | ( the carrier of R ) st p = ( the carrier of R ) | ( the carrier of R ) & p = ( the carrier of R ) | ( the carrier of R ) ;
[. a , b .] c= the InternalRel of G & ( the InternalRel of G ) . k is Element of the InternalRel of G ;
Comput ( P , s , 2 ) . IC SCM+FSA = Exec ( i , s3 ) . IC SCM+FSA .= ( Exec ( i , s3 ) . IC SCM+FSA ) . IC SCM+FSA .= ( Exec ( i , s3 ) . IC SCM+FSA ) . IC SCM+FSA ;
card ( |. h .| ) = ( |. ( ( |. ( ( ( ( ( |. ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .| ) ) ) ) ) .| .= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
( f (#) g ) . c = ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( ( f (#) g ) . c .= ( ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( f (#) g ) . c .= ( ( f (#) g
len ( ( ( \mathop { \rm Gauge } ( C , n ) ) * ( i , j ) , ( \mathop { \rm Gauge } ( C , n ) * ( i , j ) ) ) ) = len ( ( \mathop { \rm Gauge } ( C , n ) * ( i , j ) ) ;
dom ( r (#) f ) = dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f ) /\ dom ( r (#) f ) .= dom ( r (#) f )
defpred P [ Nat ] means $1 <= $1 & $1 <= $1 & $1 <= $1 implies ( for n st n >= 1 holds ( n + $1 ) * ( n + $1 ) = ( n + $1 ) * ( n + $1 ) ;
consider f being Function of [: { n } , { n } :] , { n } :] , { n } = { n } ;
consider { c , 0 } being Function of S , T such that A = Im ( ( ( ( S , 0 ) , 1 ) ) . ( ( S , 1 ) ) . ( ( S , 1 ) ) . ( ( S , 0 ) ) . ( 0 , 1 ) ) and ( ( S , 1 ) . ( 1 , 1 ) ) = ( S , 0 ) . ( ( S , 1 ) . ( 1 , 1 ) ) ;
consider y being Element of Y such that a = "\/" ( { y where y is Element of Y : y <= z & y <= z & z <= z & z <= z & y <= z ;
assume that A c= dom ( f `| Z ) and for n holds ( ( f `| Z ) . n ) * ( ( f `| Z ) . n ) * ( ( f `| Z ) . n ) ) = ( ( f `| Z ) . n ) * ( ( f `| Z ) . n ) ;
( ( GoB f ) * ( i1 , j1 ) `1 = ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1
dom Shift ( q , i ) = { i + 1 where i is Nat : 1 <= i + 1 + 1 <= len q } ;
consider i2 , j2 being Nat such that i2 + 1 <= len G and i2 + 1 <= width G and i2 + 1 <= width G and 1 <= width G and 1 <= j2 and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and 1 <=
func - f -> PartFunc of C , REAL means : for c be Real holds it . c = - f . c ;
consider L1 such that L1 is continuous and for a st a in L1 & a in L1 holds L1 . a = L1 . a ;
consider i1 , i2 such that [ i1 , j1 ] in Indices GoB G and [ i1 , j1 ] in Indices G and [ i1 , j1 ] in Indices G and [ i1 , j1 + 1 ] in Indices G and [ i1 , j1 + 1 ] in Indices G and [ i1 , j1 + 1 ] in Indices G and ( i1 + 1 , j1 + 1 ] in Indices G and ( i1 + 1 , j1 + 1 ] in Indices G and ( i1 + 1 ) * ( i1 + 1 ) and ( i1 + 1 ) * ( i1 + 1 ) = G * ( i1 + 1 ) and ( i1 + 1 ) * ( i1 + 1 ) * ( i1 + 1 ) ;
consider i , j such that i <> 0 and i < j and j < n and i < n and i < j ;
assume that 0 in dom ( ( ( ( for n st n + 1 ) (#) ( f /* s1 ) ) - ( f /* s1 ) ) ) & for n st n in dom ( f /* s1 ) holds ( ( f /* s1 ) (#) ( f /* s1 ) ) . n = ( f /* s1 ) . n ;
cell ( G , i1 , j1 -' 1 ) c= cell ( G , i1 , j1 -' 1 ) ;
ex Y1 being Subset of X st s = union Y1 & for Y being Subset of X st Y = union Y1 & Y c= Y1 & Y is finite & Y is finite } ;
gcd ( ( A , B ) , A ) = ( ( A , B ) gcd ( A , B ) ) * ( A , B ) ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( A , B ) ) ) ) ) ) .= ( A * ( ( ( ( ( ( ( B , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( ( B , B ) * ( ( A , B ) * ( ( A , B ) * ( ( A , B ) * ( ( ( ( ( A ,
R = ( Exec ( i , s ) ) . ( m + 1 ) .= ( Following s ) . ( m + 1 ) .= ( Following s ) . ( m + 1 ) .= ( Following s ) . ( m + 1 ) .= ( ( Following s ) . ( m + 1 ) .= ( Following s ) . ( m + 1 ) .= ( ( Following s ) . ( m + 1 ) ) . ( m + 1 ) .= ( ( Following s ) . ( m + 1 ) . ( m + 1 ) .= ( ( Following s ) . ( m + 1 ) .= ( ( ( s . ( m + 1 ) . ( m + 1 ) . ( m + 1 ) .= ( ( Following s ) . ( m + 1 ) .= ( ( m + 1 ) . ( m + 1 ) . (
CurInstr ( P3 , Comput ( P3 , s3 , LifeSpan ( P3 , s3 , 1 ) + 1 ) ) = P3 . IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 ) . IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 ) . IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) .= ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) . IC Comput ( P3 , s3 , 1 ) .
P1 /\ ( { p } \/ { p } ) = { p } \/ { p } ;
func the still not not bound -> Subset of A means : for p being Element of A st p in A & p in A holds it . p = ( the Sorts of A ) . p ;
let a , b be complex number ;
defpred P [ Nat ] means for i st i <= $1 holds ( ( G . i ) * ( G . i ) * ( G . i ) ) * ( G . i ) ) * ( G . i ) = ( G . i ) * ( G . i ) ;
attr attr that that for f1 , f2 being State of C , f1 , f2 being State of C , i , j being Nat st i <= j & j <= len f2 & i <= len f1 & j <= len f1 & i <= len f2 holds f1 . ( i + 1 ) = f2 . ( i + 1 ) ;
||. f .|| . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .|| ;
|. q .| = |. q .| & |. q .| = |. q .| & |. q .| = |. q .| ;
for F being Subset-Family of T st F is open holds F is open & for n being Nat st n >= m holds F . n = ( F . n ) . n
assume that len F = k and for n st n >= k & n >= k holds |. ( G . n ) . ( k + 1 ) - ( G . n ) .| < r ;
i |^ ( i + 1 ) = i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) .= i * ( i + 1 ) ;
consider q being oriented Chain of G such that q = q and for n being Nat st n <> len q holds |. q .| < r ;
defpred P [ Element of NAT ] means ( ( for m , n holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , , , ( , ( ( ( , , , , ( , ( , , , , , ( , , , ( , , , ( , , , ( , , , , ( ) , , ( ) , , ( , ( , , , , , ( ) , , ( ) , ( ( ( ) , , ( ) , , ( ( ( , , , , ( ) , ) , ( ( ( ( ( , , , , ( ) , , ( ( ) , , ( ( ( ( ) , ( ( ( ) , ( ( ) , ( ( ( ( , ,
let A be matrix over REAL ;
consider s being FinSequence of the carrier of R such that s = a * ( i , len s ) + 1 and len s = len s ;
func |. ( x | ( i -' 1 ) .| ) -> Element of REAL ;
consider g being FinSequence of bool the carrier of G such that for n being Nat st n >= 1 & n <= len g holds |. g . n - g . n .| < r ;
attr n1 >= len ( p1 , n1 , n2 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 , n1 ,
( |. q .| * |. q .| ) ^2 <= ( |. q .| ) ^2 & |. q .| ^2 >= 0 ;
F . ( len F + ( len F - ( len F - 1 ) ) ) = F . ( len F - ( len F - ( len F - ( len F - ( len F - ( len F - ( len F - 1 ) ) ) ) ) .= F . ( len F - ( len F - ( len F - ( len F - 1 ) ) ) ) .= F . ( len F - ( len F - ( len F - 1 ) ) ) .= F . ( len F - ( len F - ( len F - 1 ) ) .= F . ( len F - ( len F - 1 ) ) .= F . ( len F - ( len F - ( len F - ( len F - 1 ) ) .= F . ( len F - 1 ) ) .= F . ( len F - ( len F - ( len F - ( len F - ( len
consider i1 being Nat such that k + 1 = k and i1 = ( i1 + 1 ) + ( i2 -' 1 ) and i2 = ( i2 + 1 ) + ( i2 -' 1 ) ;
consider D2 being Subset of [: { A , B } , { B } :] , A being Subset of [: { B , B } , { B , C } :] such that A = { B , B } and B = { { B , C } and A = { B , C } ;
v1 . ( ( F1 . ( ( F2 . ( n + 1 ) ) ) ) ) ) ) = ( F1 . ( n + 1 ) ) . ( n + 1 ) .= ( F1 . ( n + 1 ) ) . ( n + 1 ) .= ( F1 . ( n + 1 ) ) . ( n + 1 ) .= ( F1 . ( n + 1 ) ) . ( n + 1 ) .= ( F1 . ( n + 1 ) ) . ( n + 1 ) .= ( F1 . ( n + 1 ) .= ( F1 . ( n + 1 ) .= ( F1 . ( n + 1 ) .= F1 . ( n + 1 ) .= F1 . ( n + 1 ) . ( n + 1 ) .= F1 . ( n + 1 ) . ( n + 1 ) .= F1 . ( n + 1 ) . ( n + 1 ) .= F1 . ( n
dom Comput ( P3 , s3 , 1 ) = dom Start-At ( 0 , SCM+FSA ) .= dom Start-At ( 0 , SCM+FSA ) .= dom Start-At ( 0 , SCM+FSA ) \/ Start-At ( 0 , SCM+FSA ) .= ( Start-At ( 1 , SCM+FSA ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) .= ( ( ( 1 , SCM+FSA ) .--> 1 ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) ) \/ ( Start-At ( 1 , SCM+FSA ) \/ ( Start-At ( 1 , SCM+FSA ) ) \/ ( Start-At ( 0 , SCM+FSA ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) \/ ( Start-At ( 1 , SCM+FSA ) ) \/ ( Start-At ( 1 , SCM+FSA ) \/ { 1 , SCM+FSA ) \/ { 1 , SCM+FSA ) \/ { 1 , SCM+FSA ) \/ ( Start-At ( 1 , SCM+FSA ) \/ ( Start-At ( 0 ,
ex z9 be Real st |. z9 - z9 .| < 1 & |. z9 - z9 .| < 1 ;
LSeg ( G * ( len G , len G + 1 ) , G * ( len G , 1 ) ) `1 c= cell ( G , 1 , 1 ) `1 , G * ( len G , 1 ) `1 ) `1 ;
LSeg ( h , i ) = LSeg ( h , i ) .= LSeg ( h , i ) ;
set A = { q where q is Point of TOP-REAL 2 : |. q .| < 1 & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & |. q .| >= sn & q .| >= sn & |. q .| >= sn & |. q .| >= sn
( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( 1 - 1 ) * ( ( 1 - 1 ) * ( ( 1 - 1 ) * ( ( 1 - 1 ) * ( ( 1 - ( 1 / 2 ) ) ) ) ) ) ) = ( 1 / 2 ) * ( ( 1 - ( 1 / 2 ) ) ) .= ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2 ) * ( 1 / 2 ) .= ( 1 / 2
0 * ( p + ( 1 - p ) ) = ( 1 - p ) * ( 1 - p ) ;
sin . ( ( |. seq_id ( vseq . n ) .| ) .| ) = ( |. vseq . n ) .| * |. ( vseq . n ) .| ) * |. ( vseq . n ) .| .= |. ( vseq . n ) .| * |. ( vseq . n ) .| ;
func Shift ( f , h ) -> PartFunc of REAL , REAL equals Shift ( f , h ) ;
assume that 1 <= k and k + 1 <= len G and k + 1 <= width G and 1 <= width G and 1 <= width G and k + 1 <= width G and 1 <= width G and 1 <= width G and 1 <= width G and G * ( 1 , 1 + 1 ) and G * ( 1 , 1 ) `2 = G * ( 1 , 1 ) `1 and G * ( 1 , 1 , 1 ) `2 = G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 = G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 = G * ( 1 , 1 , 1 , 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G * ( 1 , 1 ) `1 and G
attr y in Free ( H ) implies ( ex x , y st x = All ( x , y ) & ( for x , y st x in Free ( H ) holds ( All ( y , H ) . x = All ( y , H ) . y ) ;
defpred P [ Nat ] means $1 = p * ( $1 + 1 ) ;
func \sigma ( C ) -> strict Subspace of X equals C /\ C ;
[#] ( ( dist ( dist ( dist ( , dist ( 0 ) ) ) ) ) ) ) = ( dist ( ( dist ( 0 , 0 ) ) ) ) * ( ( dist ( 0 , 0 ) ) ) ;
rng ( F | ( ( ( S | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S , | ( S , S ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = { ( ( S | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( S , S ) | ( ( ( ( ) | ( ( S , S ) | ( ( ( ) | ( ( S , S ) | ( ( S , S ) | ( ( ( S , S ) | ( ( S ,
( 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 , P2 being Subset of TOP-REAL 2 such that P1 = P2 \/ P2 and P2 = P1 +* P2 and P1 = P2 +* P2 and P2 = P2 +* +* +* +* +* +* +* +* stop I and P2 = P2 +* stop I and P2 = P1 +* stop I and P2 = P2 +* stop I and P1 = P2 +* stop I ;
f . ( p + 1 ) = [ p , f . ( p + 1 ) ] ;
( TOP-REAL 2 ) * ( a , b ) = ( |[ a , b ]| ) * ( a , b ) .= |[ a , b ]| .= |[ a , b ]| .= |[ a , b ]| .= |[ a , b ]| ;
let A be non empty TopSpace , p be Point of R^1 ( A ) ;
for i being strict Subgroup of G st i in dom F & i in dom F holds F . i = G . i
for x st x in Z holds ( ( ( ( - 1 / ( ( ( 1 + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( / ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ^2 ) ) ^2 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) )
attr f is convergent means : Def5 : for x0 , y0 being Point of REAL st 0 < r & ex g be Real st 0 < g & g < g & g in dom f & for g st g in dom ( f | X ) holds f /. g - f /. x0 .|| < r ;
attr X is SubSpace of X means : for x being Element of X holds x in X & x in X & x in X & x in X & x in X & x in X ;
ex N being neighbourhood of x0 st N c= dom ( ( ( ( for x0 - r ) (#) f ) `| N ) `| N ) . x0 - r (#) f ) . x0 - r (#) f . x0 ) ;
sqrt ( ( 1 - ( ( ( p `1 / |. p .| - sn ) ) ) ^2 ) ) ^2 + ( 1 - sn sn sn ) ^2 ) ^2 >= 0 ;
( ( 1 - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - ( ( ( ( ( ) ) , ( ( ( ( ( , , , , , ( , , , .| ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( )
attr x , y , z , u , v , w , u , v , w , w , u , w , y , y , y , z , w , y , y , z , w , y , z , w , y , w , z , u , y , w , z , y , w , z , y , w , z , y , w , z , y , w , y , z , w , y , w w , z , y , y , z , w , y , z , y , z , w , y , z , w , y , z , w , z , y , z , z , y , z , z , w , w , y , z , w , y , z , w , y , w , y , z , y , y , z , y , w , y , z , y , z , y , z , y , z , z , w , y , z , y , z , y , z , y , z , w w w w w , y , y , z , y , z , y , y , z
consider y1 being Subset of [: X , Y :] such that y1 = y1 & y2 = y1 & y1 in Y and y1 in Y and y2 in Y ;
card ( card ( S \ { d } ) ) = card ( { d } \ { d } ) .= card ( { d } \ { d } ) ;
\frac ( ( - ( ( ( ( TOP-REAL 2 ) | ( i + 1 ) ) ) ) ^2 ) ) ^2 = ( ( ( ( ( sn -FanMorphE ) | ( i + 1 ) ) ^2 ) ^2 ) ^2 .= ( sn -FanMorphE ) ^2 ) ^2 .= ( sn -FanMorphE ) ^2 ;