thesis ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
thesis ;
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contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
B ;
a <> c
T c= S
D c= B
c ;
b ;
X ;
b in D ;
x = e ;
let m ;
h is onto ;
N in K ;
let i ;
j = 1 ;
x = u ;
let n ;
let k ;
y in A ;
let x ;
let x ;
m c= y ;
F is onto ;
let q ;
m = 1 ;
1 < k ;
G is prime ;
b in A ;
d divides a ;
i < n ;
s <= b ;
b in B ;
let r ;
B is one-to-one ;
R is total ;
x = 2 ;
d in D ;
let c ;
let c ;
b = Y ;
0 < k ;
let b ;
let n ;
r <= b ;
x in X ;
i >= 8 ;
let n ;
let n ;
y in f ;
let n ;
1 < j ;
a in L ;
C is boundary ;
a in A ;
1 < x ;
S is finite ;
u in I ;
z << z ;
x in V ;
r < t ;
let t ;
x c= y ;
a <= b ;
m in NAT ;
assume f is a3 ;
not x in Y ;
z = +infty ;
k be Nat ;
K `2 is being_line ;
assume n >= N ;
assume n >= N ;
assume X is almost BCK-algebra ;
assume x in I ;
q is Matrix by 0 ;
assume c in x ;
exp > 0 ;
assume x in Z ;
assume x in Z ;
1 <= k12 ;
assume m <= i ;
assume G is prime ;
assume a divides b ;
assume P is closed ;
O > 0 ;
assume q in A ;
W is not bounded ;
f is IC ;
assume A is boundary ;
g is special ;
assume i > j ;
assume t in X ;
assume n <= m ;
assume x in W ;
assume r in X ;
assume x in A ;
assume b is even ;
assume i in I ;
assume 1 <= k ;
X is non empty ;
assume x in X ;
assume n in M ;
assume b in X ;
assume x in A ;
assume T c= W ;
assume s is negative ;
b `2 <= c `2 ;
A meets W ;
i `2 <= j `2 ;
assume H is universal ;
assume x in X ;
let X be set ;
let T be Tree ;
let d be element ;
let t be element ;
let x be element ;
let x be element ;
let s be element ;
k <= 5 ;
let X be set ;
let X be set ;
let y be element ;
let x be element ;
P [ 0 ]
let E be set , F be FinSequence of E ;
let C be category ;
let x be element ;
k be Nat ;
let x be element ;
let x be element ;
let e be element ;
let x be element ;
P [ 0 ]
let c be element ;
let y be element ;
let x be element ;
a be Real ;
let x be element ;
let X be element ;
P [ 0 ]
let x be element ;
let x be element ;
let y be element ;
r in REAL ;
let e be element ;
n1 is retraction ;
Q is_closed_on s , P ;
x in *> ;
M < m + 1 ;
T2 is open ;
z in b "\/" a ;
R2 is well-ordering ;
1 <= k + 1 ;
i > n + 1 ;
q1 is one-to-one ;
let x be trivial set ;
PP is one-to-one ;
n <= n + 2 ;
1 <= k + 1 ;
1 <= k + 1 ;
let e be Real ;
i < i + 1 ;
p3 in P ;
p1 in K ;
y in C1 ;
k + 1 <= n ;
let a be Real , x be Real ;
X |- r => p ;
x in { A } ;
let n be Nat ;
let k be Nat ;
let k be Nat ;
let m be Nat ;
0 < 0 + k ;
f is_differentiable_in x ;
let x0 ;
let E be Ordinal ;
o is_a_unity_wrt o1 ;
O <> O ;
let r be Real ;
let f be FinSeq-Location ;
let i be Nat ;
let n be Nat ;
Cl A = A ;
L c= Cl L ;
A /\ M = B ;
let V be complex RealUnitarySpace , W be Subspace of V , x be VECTOR of W ;
not s in Y |^ 0 ;
rng f <= w ;
b "/\" e = b ;
m = m2 ;
t in h . D ;
P [ 0 ] ;
assume z = x * y ;
S . n is bounded ;
let V be RealUnitarySpace , W be Subspace of V , x be VECTOR of W ;
P [ 1 ] ;
P [ {} ] ;
C1 is component ;
H = G . i ;
1 <= i `1 + 1 ;
F . m in A ;
f . o = o ;
P [ 0 ] ;
a! <= r ;
R [ 0 ] ;
b in f .: X ;
assume q = q2 ;
x in [#] V ;
f . u = 0 ;
assume e1 > 0 ;
let V be RealUnitarySpace , W be Subspace of V , x be VECTOR of W ;
s is trivial & s is trivial ;
dom c = Q ;
P [ 0 ] ;
f . n in T ;
N . j in S ;
let T be complete LATTICE , f be Function of T , T ;
the object of F is one-to-one ;
sgn x = 1 ;
k in support a ;
1 in Seg 1 ;
rng f = X ;
len T in X ;
vO < n ;
SMaxADSet is bounded ;
assume p = p2 ;
len f = n ;
assume x in P1 ;
i in dom q ;
let U1 , U2 ;
p-25 = c ;
j in dom h ;
let k ;
f | Z is continuous ;
k in dom G ;
UBD C = B ;
1 <= len M ;
p in dist x ;
1 <= jj & jj <= 1 ;
set A = Fint ;
card a [= c ;
e in rng f ;
cluster B \oplus A -> empty ;
H has no that G ;
assume that n1 <= m ;
T is increasing ;
e2 <> e1 ;
Z c= dom g ;
dom p = X ;
H is proper subformula of G ;
i + 1 <= n ;
v <> 0. V ;
A c= Affin A ;
S c= dom F ;
m in dom f ;
let X0 be set ;
c = sup N ;
R is connected ;
assume not x in REAL ;
Im f is complete ;
x in Int y ;
dom F = M ;
a in On W ;
assume e in [: A , B :] ;
C c= Cmax ( C , n ) ;
m1 <> {} ;
let x be Element of Y ;
let f be \pi Chain , g be FinSequence ;
not n in Seg 3 ;
assume X in f .: A ;
assume that p <= n and p <= m ;
assume not u in { v } ;
d is Element of A ;
A |^ b misses B ;
e in v .reachableFrom ( X ) ;
- y in I ;
let A be non empty set , B be set ;
P0 = 1 ;
assume r in F . k ;
assume f is simple ;
let A be exists countable set ;
rng f c= NAT ;
assume P [ k ] ;
f6 <> {} ;
let o be Ordinal ;
assume x is sum of squares ;
assume not v in { 1 } ;
let I1 , I2 ;
assume that 1 <= j and j < l ;
v = - u ;
assume s . b > 0 ;
d1 in d2 ;
assume t . 1 in A ;
let Y be non empty TopSpace , X be non empty TopSpace ;
assume a in ]. s , t .[ ;
let S be non empty Poset ;
a , b // b , a ;
a * b = p * q ;
assume x , y are_.--> the space of V ;
assume x in [#] ( f ) ;
[ a , c ] in X ;
m-14 <> {} ;
M + N c= M + N ;
assume M is connected ;
assume f is with_with_with_vrrr-rrron ;
let x , y be element ;
let T be non empty TopSpace ;
b , a // b , c ;
k in dom Sum p ;
let v be Element of V ;
[ x , y ] in T ;
assume len p = 0 ;
assume C in rng f ;
k1 = k2 & k2 = k2 ;
m + 1 < n + 1 ;
s in S \/ { s } ;
n + i >= n + 1 ;
assume Re ( y ) = 0 ;
k1 <= j1 & j1 <= j2 ;
f | A is constant ;
f . x > b ;
assume y in dom h ;
x * y in B1 ;
set X = Seg n ;
1 <= i2 + 1 ;
k + 0 <= k + 1 ;
p ^ q ^ q ^ p ^ q ^ p ^ q ^ q ^ p ^ q ^ q ^ p ^ q ^ q ^ q
j / y divides m ;
set m = max A ;
[ x , x ] in R ;
assume x in succ 0 ;
a in sup phi ;
Cj in Cj ;
q2 c= C1 & q2 c= C2 ;
a2 < c2 & c2 < c2 ;
s2 is 0 -started ;
IC s = 0 ;
s2 = s3 * ( 1 , j ) ;
let V ;
let x , y be element ;
let x be Element of T ;
assume a in rng F ;
x in dom T ` ;
let S be MSAlgebra over L ;
y " <> 0 ;
y " <> 0 ;
0. V = u-] ;
y2 , y is_collinear ;
R1 , R2 are_fiberwise_equipotent ;
let a , b be Real , r be Real ;
let a be object of C ;
let x be Vertex of G ;
let o be object of C , a , b be object of C ;
r '&' q = P \lbrack l \rbrack ;
let i , j ;
let s be State of A , v be Element of S , w be Element of A , f be FinSequence of A , g be FinSequence of B
s3 . n = N ;
set y = ( x `1 ) ^2 ;
NAT in dom g ;
l . 2 = y1 ;
|. g . y .| <= r ;
f . x in C1 ;
V is non empty ;
let x be Element of X ;
0 <> f . g2 ;
f2 /* q is convergent ;
f . i is_measurable_on E ;
assume \xi in N-22 ;
reconsider i = i as Ordinal ;
r * v = 0. X ;
rng f c= INT & rng f c= INT ;
G = 0 .--> goto 0 ;
let A be Subset of X ;
assume that A is dense and B is dense ;
|. f . x .| <= r ;
let x be Element of R ;
let b be Element of L ;
assume x in W & y in W ;
P [ k , a ] ;
let X be Subset of L ;
let b be object of B ;
let A , B be category ;
set X = Vars ( C , n ) ;
let o be OperSymbol of S ;
let R be connected non empty Poset ;
n + 1 = succ n ;
x9 c= Z & y9 c= Z ;
dom f = C1 & dom g = C1 ;
assume [ a , y ] in X ;
Re ( seq ) is convergent ;
assume a1 = b1 & a2 = b2 ;
A = sInt A ;
a <= b or b <= a ;
n + 1 in dom f ;
let F be Instruction of S , f be FinSequence of S ;
assume r2 > x0 ;
let Y be non empty set , f be Function of Y , BOOLEAN ;
2 * x in dom W ;
m in dom ( g2 | m ) ;
n in dom ( g1 | n ) ;
k + 1 in dom f ;
the still not bound in { s } ;
assume x1 <> x2 & x2 <> x3 ;
v1 in V & v2 in V ;
not [ b `1 , b `2 ] in T ;
ii + 1 = i ;
T c= s2 ( T ) ;
( l `1 ) ^2 = 0 ;
n be Nat ;
( t `2 ) ^2 = r ^2 ;
AK is integrable ;
set t = Bottom t ;
let A , B be real-membered set ;
k <= len G + 1 ;
[: C , D :] misses [: D , D :] ;
Product ( seq ) is non empty ;
e <= f or f <= e ;
cluster SCM -> non empty for set ;
assume c2 = b2 & c2 = b1 ;
assume h in [. q , p .] ;
1 + 1 <= len C ;
not c in B . m1 ;
cluster R .: X -> non empty ;
p . n = H . n ;
assume that v-4 is Cauchy and for n be Nat holds vK . n is convergent ;
IC s3 = 0 ;
k in N or k in K ;
F1 \/ F2 c= F ;
Int ( G1 ) <> {} ;
( z `2 ) ^2 = 0 ;
p01 <> p1 & p1 <> p2 ;
assume z in { y , w } ;
MaxADSet ( a ) c= F ;
ex_sup_of { s } , S & s in S ;
f . x <= f . y ;
let T be up-complete non empty reflexive antisymmetric antisymmetric RelStr ;
q / ( m + 1 ) >= 1 ;
a >= X & b >= Y ;
assume <* a , c *> <> {} ;
F . c = g . c ;
G is one-to-one ;
A \/ { a } c= B ;
0. V = 0. V & V is Subspace of Y ;
let I be db Instruction of S , s be State of S ;
f-24 . x = 1 ;
assume z \ x = 0. X ;
C4 = 2 / ( n + 1 ) ;
let B be sequence of \Sigma ;
assume X1 = p .: D ;
n + l in NAT ;
f " P is compact ;
assume x1 in REAL + 1 ;
p1 = K & p2 = K ;
M . k = <*> REAL ;
phi . 0 in rng phi ;
MMO is closed ;
assume z0 <> 0. L ;
n < N . k ;
0 <= seq . 0 ;
- q + p = v ;
{ v } is Subset of B ;
set g = f /. 1 , h = f /. 2 ;
[: R , S :] is stable ;
set \cal R = Vertices R , S = G \ { x } ;
p0 c= P4 & P4 c= P4 ;
x in [. 0 , 1 .] ;
f . y in dom F ;
let T be Scott TopAugmentation of S ;
inf the carrier of S in S ;
downarrow a = downarrow b ;
P , C , K is_collinear ;
assume x in LSeg ( s , r ) ;
2 / i < 2 / m ;
x + z = x + z + q ;
x \ ( a \ x ) = x ;
||. \mathopen { \Vert - x .|| <= r ;
assume that Y c= field Q and Y <> {} ;
a \times b , b :] , a are_equipotent ;
assume a in [: A ( ) , A ( ) :] ;
k in dom ( q | k ) ;
p is FinSequence of S ;
i -' 1 = i-1 - 1 ;
f | A is one-to-one ;
assume x in f .: [: X , Y :] ;
i2 - 1 = 0 ;
j2 + 1 <= i2 ;
g " * a in N ;
K <> { [ {} , {} ] } ;
cluster strict product eeed_Ring for Group ;
|. q .| ^2 > 0 ;
|. p3 .| = |. p .| ;
s2 - s1 > 0 ;
assume x in { G * ( len G , j ) } ;
W-min C in C & W-min C in C ;
assume x in { G * ( len G , j ) } ;
assume i + 1 = len G ;
assume i + 1 = len G ;
dom I = Seg n & dom I = Seg n ;
assume that k in dom C and k <> i ;
1 + 1-1 <= i + 1-1 ;
dom S = dom F & rng F c= dom G ;
let s be Element of NAT ;
let R be ManySortedSet of A ;
let n be Element of NAT ;
let S be non empty non void non empty non void ManySortedSign ;
let f be ManySortedSet of I ;
let z be Element of COMPLEX , v be Element of COMPLEX ;
u in { \hbox { \boldmath $ g } } ;
2 * n < ( 2 * n ) ;
let x , y be set ;
B-11 c= V & V is open ;
assume I is_closed_on s , P ;
U1 = U1 & U2 = U2 & U1 = U2 ;
M /. 1 = z /. 1 .= z /. 1 ;
x9 = x9 & y9 = y9 & x9 = y9 ;
i + 1 < n + 1 ;
x in { {} , <* 0 *> } ;
f7 <= f6 & f7 <= len f ;
let l be Element of L ;
x in dom ( F . n ) ;
let i be Element of NAT ;
r8 is ( len p ) -valued ;
assume <* o2 , o1 *> <> {} ;
s . x / ( 0 / 2 ) = 1 ;
card ( K + 1 ) in M ;
assume that X in U and Y in U ;
let D be Subset-Family of Omega ;
set r = { q + 1 } ;
y = W . ( 2 * x ) ;
assume dom g = cod f & dom g = cod f ;
let X , Y be non empty TopSpace , f be Function of X , Y , g be Function of X , Y , h be Function of Y , Y ;
x \oplus A is interval ;
|. <*> A .| . a = 0 ;
cluster strict Sublattice of L -> strict for Subspace of L ;
a1 in B . s1 & a2 in B . s1 ;
let V be finite .[ over F , v be Vector of V , w be Vector of V ;
A * B on B & A on B ;
f-3 = NAT --> 0 .= 1 ;
A , B be Subset of V ;
z1 = P1 . j .= P1 . j ;
assume f " P is closed ;
reconsider j = i as Element of M ;
a , b be Element of L ;
assume q in A \/ ( B "\/" C ) ;
dom ( F * C ) = o ;
set S = INT , T = 2 -tuples_on the carrier of R ;
z in dom ( A --> y ) ;
P [ y , h . y ] ;
{ x0 } c= dom f & { x0 } c= dom f ;
let B be non-empty ManySortedSet of I , A be non-empty ManySortedSet of I , B be ManySortedSet of B ;
sqrt ( PI / 2 ) < Arg z ;
reconsider z9 = 0 as Nat ;
LIN a , d , c ;
[ y , x ] in [: I , I :] ;
( Q ) `1 = 0 & ( Q `1 ) = 0 ;
set j = x0 div m , i = m mod m ;
assume a in { x , y , c } ;
j2 - ( j - 1 ) > 0 ;
I \! \mathop { \rm \hbox { - } \varphi = 1 ;
[ y , d ] in ( F . y ) ;
let f be Function of X , Y ;
set A2 = sqrt ( B / C ) ;
s1 , s2 be Element of L ;
j1 -' 1 = 0 + 1 - 1 .= 0 ;
set m2 = 2 * n + j ;
reconsider t = t as bag of n ;
I2 . j = m . j ;
i |^ s , n are_relative_prime ;
set g = f | divset ( D , j ) ;
assume that X is lower and 0 <= r ;
( 1 - r ) * ( 1 - r ) = 1 ;
a < ( p3 `1 ) ^2 / ( p3 `2 ) ^2 ;
L \ { m } c= UBD C ;
x in Ball ( x , 10 ) ;
not a in LSeg ( c , m ) ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
1 <= i1 -' 1 & i1 -' 1 <= len f ;
i + i2 <= len h - 1 ;
x = W-min ( P ) & x = W-min ( P ) ;
[ x , z ] in X ~ Z ;
assume y in [. x0 , x .] ;
assume p = <* 1 , 2 , 3 *> ;
len <* A1 *> = 1 ;
set H = h . ( g1 . ( g . ( h . ( g . ( h . ( h . ( g . ( h . ( g . ( h . ( h .
card b * a = |. a .| ;
Shift ( w , 0 ) |= v ;
set h = h2 ** h1 , h1 = h1 ** h2 , h2 = h2 ** h1 , h2 = h ** h1 , h2 = h ** h2 , h2 = h ** c ;
assume x in X0 /\ ( X1 \/ X2 ) ;
||. h .|| < d1 / ( 1 - e ) ;
not x in the support of f & not x in the carrier of f ;
f . y = F ( y ) ;
for n holds X [ n ] ;
k - l = kk - l ;
<* p , q *> /. 2 = q ;
let S be Subset of the lattice of Y ;
P , Q be Initialize s of s ;
Q /\ M c= union ( F | M )
f = b * canFS ( S ) ;
let a , b be Element of G ;
f .: X <= f . sup X ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L , x be Element of L ;
S-20 is x -to_power i -to_power i ;
let r be non positive Real ;
M , v |= x \hbox { x } y ;
v + w = 0. Z + w .= v ;
P [ len F ] & P [ len F ] ;
assume InsCode i = 8 & InsCode i = 8 ;
the non zero Element of M = 0 & the carrier of M = { 0 } ;
cluster z * seq -> summable ;
let O be Subset of the carrier of C ;
|. f .| " X is continuous ;
x2 = g . ( j + 1 ) ;
cluster -> AllTermsOf S for Element of S ;
reconsider l1 = lj as Nat ;
v is Vertex of r2 & v is Vertex of G ;
T3 is SubSpace of T2 & T3 is SubSpace of T2 ;
Q1 /\ Q19 <> {} ;
k be Nat ;
q " is Element of X ;
F ( t ) is set with non empty ;
assume that n <> 0 and n <> 1 ;
set e = EmptyBag n , f = EmptyBag n , g = EmptyBag n , h = EmptyBag n , e = EmptyBag n , h = EmptyBag n , e = EmptyBag n , e = EmptyBag n ,
let b be Element of Bags n ;
assume for i holds b . i is commutative ;
x is root of ( p ) . ( x , y ) ;
not r in ]. p , q .[ ;
let R be FinSequence of REAL ;
S7 does not empty \vert b1 & not b2 not empty ex b st b on b1 & not b on b2 , T ;
IC SCM R <> a & IC SCM R <> a ;
|. |[ x , y ]| .| >= r ;
1 * ( s - t ) = s - t ;
let x be FinSequence of NAT , y be Element of NAT ;
let f be Function of C , D , g be Function of C , D ;
for a holds 0. L + a = a
IC s = s . NAT .= ( the Sorts of A ) . NAT ;
H + G = F- ( G-] ) ;
C1 . x = x2 . x ;
f1 = f . x .= f2 . x .= f2 . x ;
Sum <* p . 0 *> = p . 0 ;
assume v + W = v + u + W ;
{ a1 } = { a2 } ;
a1 , b1 _|_ b , a ;
d1 , o _|_ o , a3 & d1 , d2 _|_ o , a4 ;
IX is reflexive & IX is reflexive implies X is reflexive
IX is antisymmetric & CX is antisymmetric implies [: X , Y :] is antisymmetric
upper_bound rng ( rng ( H1 | n ) ) = e ;
x = a1 * ( a2 * a3 ) ;
|. p1 .| ^2 >= 1 ^2 ;
assume j2 -' 1 < j - 1 ;
rng s c= dom f1 & rng s c= dom f2 & rng s c= dom f1 ;
assume support a misses support b & not b in support b ;
let L be associative non empty doubleLoopStr , F be FinSequence of L , G be FinSequence of F ;
s " + 0 < n + 1 ;
p . c = ( f " ) . 1 .= f . ( 1 / c ) ;
R . n <= R . ( n + 1 ) ;
Directed ( I1 ) = I1 +* ( I2 , I2 ) ;
set f = + ( x , y , r ) ;
cluster Ball ( x , r ) -> bounded ;
consider r being Real such that r in A ;
cluster non empty NAT -defined for Function ;
let X be non empty directed Subset of S ;
let S be non empty full SubRelStr of L ;
cluster <* L1 . N , \subseteq \rangle -> complete for non trivial TopSpace ;
sqrt ( 1 / a ^2 ) = a ;
( q . {} ) `1 = o ;
n - ( i -' 1 ) > 0 ;
assume sqrt ( 1 - 2 ) <= t `1 ;
card B = k + 1 - 1 ;
x in union rng ( f | n ) ;
assume x in the carrier of R & y in the carrier of R ;
d ;
f . 1 = L ( F . 1 ) ;
the vertices of G = { v } \/ { v } ;
let G be ee] ;
e , v6 , v6 , v6 , V ;
c . i0 in rng c & c . i0 in rng c ;
f2 /* q is divergent_to+infty & f2 /* q is divergent_to+infty ;
set z1 = - ( z2 - z1 ) , z2 = - ( z2 - z2 ) , z2 = - ( z2 - z1 ) , z2 = - ( z2 - z2 ) , z2 = - ( z2
assume w is llof S , G ;
set f = p \! \mathop { t } , g = p , h = q ;
let c be object of C ;
assume ex a st P [ a ] ;
let x be Element of REAL m , y be Element of REAL m ;
let I1 be Subset-Family of X ;
reconsider p = p as Element of NAT ;
v , w be Point of X ;
let s be State of SCM+FSA , p be FinSequence of SCM+FSA ;
p is FinSequence of SCM R & q is FinSequence of NAT ;
stop I c= P & stop I c= P & card I = card I + 2 ;
set ci = fi /. i , cj = f /. i , cj = f /. j , cj = f /. j , cj = f /. j , cj = f /.
w ^ t ^ t ^ s ^ t ^ w ^ t ^ w ^ t ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ t ^ w ^ w ^ w ^ t ^
W1 /\ W = W1 /\ W2 & W2 /\ W = W2 /\ W ;
f . j is Element of J . j ;
let x , y be Element of T2 ;
ex d st a , b // b , d & a , b // d , c
a <> 0 & b <> 0 & c <> 0 ;
ord ( x ) = 1 & x is positive ;
set g2 = lim ( s ) , g1 = lim ( s ) ;
2 * x >= 2 * sqrt ( 1 + ( 2 * x ) ^2 ) ;
assume ( a 'or' c ) . z <> TRUE ;
f \circ g in Hom ( c , c ) ;
Hom ( c , c + d ) <> {} ;
assume 2 * Sum ( q | m ) > m ;
L1 . ( F . k ) = 0 ;
R1 \/ R2 = R1 & R1 \/ R2 = R2 ;
( ( - 1 ) (#) sin ) . x <> 0 ;
( ( for x st x in Z holds exp_R . x = 1 / x ) ) & ( exp_R . x = 1 / x ) implies ( exp_R . x = 1 / ( exp_R . x ) ^2
o1 in [: X , Y :] /\ [: X , Y :] ;
e , v6 , v6 , v6 , V ;
r3 > sqrt ( 1 - 0 ) * sqrt ( 1 - 0 ) ;
x in P .: ( F .: ( B ) ) ;
J be closed closed Program of R , I be Subset of R ;
h . p1 = f2 . O .= f2 . O ;
Index ( p , f ) + 1 <= j ;
len ( q | i ) = width M & width ( q | i ) = width M ;
the support of K c= A & the support of K c= A ;
dom f c= union rng ( F | X ) ;
k + 1 in support ( n ) ;
let X be ManySortedSet of the carrier of S ;
[ x `1 , y `2 ] in InnerVertices ( R ) ;
i = D1 or i = D2 or i = D1 . i ;
assume a mod n = b mod n ;
h . x2 = g . x1 .= f . x2 ;
F c= 2 -tuples_on X & F is one-to-one ;
reconsider w = |. s1 .| as Real_Sequence ;
sqrt ( 1 / m * r + r ) < p ;
dom f = dom ( I . i ) .= dom ( I . i ) ;
[#] ( P-17 ) = [#] ( ( TOP-REAL 2 ) | K1 ) .= K1 ;
cluster - x -> ExtReal for ExtReal ;
then { d1 } c= A ;
cluster [: TOP-REAL n , { p } :] -> finite-ind ;
let w1 be Element of M ;
let x be Element of dyadic ( n ) ;
u in W1 & v in W2 & u in W2 ;
reconsider y = y as Element of L2 ( ) ;
N is full SubRelStr of T |^ the carrier of S ;
sup { x , y } = c "\/" c ;
g . n = n / 1 .= n ;
h . J = EqClass ( u , J ) ;
let seq be summable sequence of X ;
dist ( x , y ) < sqrt ( r / 2 ) ;
reconsider mm = m - 1 as Element of NAT ;
x- x0 < r1 - x0 & r1 < x0 + r2 ;
reconsider P = P ` as strict Subgroup of N ;
set g1 = p * idseq ( q `2 ) , g2 = q `2 ;
let n , m , k be non zero Nat ;
assume that 0 < e and f | A is bounded ;
D2 . [: x , y :] in { x , y } ;
cluster -> subcondensed for Subset of T ;
P be compact non empty Subset of TOP-REAL 2 , p1 , p2 be Point of TOP-REAL 2 ;
G * ( len G , 1 ) in LSeg ( \pi , 1 ) ;
n be Element of NAT , x be Element of NAT ;
reconsider S8 = S as Subset of T ;
dom ( i .--> X ) = { i } ;
let X be non-empty ManySortedSet of S ;
let X be non-empty ManySortedSet of S ;
op ( { {} } ) c= { [ {} , {} ] }
reconsider m = m2 as Element of NAT ;
reconsider d = x as Element of COMPLEX ;
let s be 0 -started State of SCMPDS , P be s of SCMPDS ;
let t be 0 -started State of SCMPDS , Q ;
b , b , x is_collinear & x , y \neq b ;
assume that i = n \/ { n } and j = k \/ { n } ;
let f be PartFunc of X , Y ;
N >= sqrt ( ( sqrt ( c ^2 + d ^2 ) / 2 ) ) ;
reconsider t9 = T" as Point of TOP-REAL 2 ;
set q = h * p ^ <* d *> ;
z2 in U ( y2 ) /\ Q ( y1 , y2 ) ;
A |^ 0 = { <* \rangle *> , A |^ 0 } ;
len W2 = len W2 + len W1 .= len W2 + len W1 ;
len h2 in dom h2 & len h2 in dom h2 & len h2 = len h2 ;
i + 1 in Seg len s2 & i + 1 in dom s2 ;
z in dom g1 /\ dom f & z in dom g ;
assume p2 = W-min ( K ) & p3 = W-min ( K ) ;
len G + 1 <= i1 + 1 ;
f1 (#) f2 is convergent & f2 (#) g2 is convergent & lim ( f1 (#) f2 ) = x0 ;
cluster seq + ( s + |. -> summable .| ) -> summable ;
assume j in dom M1 & i in dom M1 & j in dom M1 ;
let A , B be Subset of X ;
x , y , z be Point of X , x be Point of X , y be Point of X , z be Point of X , x be Point of X , y be Point of X , z be Point of X , x
b / ( 4 * a ) - ( 4 * c ) >= 0 ;
<* xy *> ^ <* y *> <=' x ;
a , b in { a , b } ;
len p2 is Element of NAT & len p1 = len p2 & len p2 = len p1 ;
ex x being element st x in dom R & x in X ;
len q = len ( K * G ) ;
s1 = Initialize ( ( Initialized s ) . f ) .= ( Initialized s ) . f ;
consider w being Nat such that q = z + w ;
x ` ` ` is \ of x ` ;
k = 0 & n <> k or k > n & n > 0 ;
then X is discrete for A being Subset of X ;
for x st x in L holds x is FinSequence
||. f /. c - f /. c .|| <= r1 ;
c in ]. p , q .[ & not c in { p } ;
reconsider V = V as Subset of the carrier of \hbox { n } ;
N , M be \langle L , M *> , L ;
then z >= \twoheaddownarrow x & z >= compactbelow x ;
M | M = f & M = g & M = g | M ;
( to_power ( 1 / 2 ) ) /. 1 = TRUE ;
dom g = dom f /\ X .= dom f /\ X .= dom g ;
mode \upupharpoons of G , the carrier of G ;
[ i , j ] in Indices M & [ i , j ] in Indices M implies i = j
reconsider s = x " as Element of H ;
let f be Element of dom ( Subformulae p ) ;
F1 ( a1 , - a2 ) = G1 ( a1 , a2 ) ;
cluster rectangle ( a , b , r ) -> compact ;
let a , b , c be Real ;
rng s c= dom ( 1 / 2 ) & rng ( 1 / 2 ) c= dom ( 1 / 2 ) ;
curry ' ( F , k ) is additive ;
set k2 = card ( dom B ) , k1 = card ( dom B ) , k2 = card ( dom B ) ;
set G = ( the Sorts of A ) . o ;
reconsider a = [ x , s ] as Object of G ;
let a , b be Element of M , x be Element of M ;
reconsider s1 = s as Element of [: S , T :] ;
rng p c= the carrier of L & rng p c= the carrier of L ;
let d be Subset of the Sorts of A ;
( x | x ) = 0 iff x = 0. W & x = 0. W
I-21 in dom stop I & card I = card J + 3 ;
let g be continuous Function of X , Y ;
reconsider D = Y as Subset of TOP-REAL n ;
reconsider i0 = len p1 - 1 as Integer ;
dom f = the carrier of S & rng f = the carrier of S ;
rng h c= union ( ( the support of J ) * the Arity of S ) ;
cluster All ( x , H ) -> reconsider All ( x , H ) ;
d * N1 / ( 1 / 2 ) > N1 * ( 1 / 2 ) ;
]. a , b .[ c= [. a , b .] ;
set g = f " ( D1 , D2 ) , h = f " ( D1 , D2 ) ;
dom ( p | ( m + 1 ) ) = [: Seg m , Seg m :] ;
3 + 2 - 1 <= k + - 1 ;
tan is_differentiable_in ( ( - 1 ) (#) ( arccot ) ) . x ;
x in rng ( f /^ p ) ;
let f , g be FinSequence of D ;
[: p , q :] in the carrier of [: S1 , S2 :] & [: p , q :] in the InternalRel of S1 ;
rng f " { 0 } = dom f /\ dom g .= dom g ;
( the Target of G ) . e = v ;
width G -' 1 < width G - 1 ;
assume v in rng ( S | E1 ) ;
assume x is root or x is root ;
assume 0 in rng ( ( g2 | A ) ^ ( g2 | A ) ) ;
let q be Point of TOP-REAL 2 , a , b be Real ;
let p be Point of TOP-REAL 2 , a , b be Real ;
dist ( O , u ) <= |. p2 .| + 1 ;
assume dist ( x , b ) < dist ( a , b ) ;
<* S *> is Element of the carrier of C & <* S , T *> is Element of the carrier of C ;
i <= len ( G * ( i , 1 ) ) -' 1 ;
let p be Point of TOP-REAL 2 , a , b be Real ;
x1 in the carrier of I[01] & x2 in the carrier of I[01] & x1 in the carrier of I[01] ;
set p1 = f /. i , p2 = f /. j ;
g in { g2 : r < g2 & g2 < x0 } ;
Q = [: S , T :] " ( Q /\ R ) .= [: S , T :] ;
( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 )
- p + I c= - p + A ;
n < LifeSpan ( P1 , s1 ) ;
CurInstr ( p1 , s1 ) = i .= ( CurInstr ( p1 , s1 ) ) ;
A /\ Cl { x } <> {} ;
rng f c= ]. r , s + 1 .[ ;
let g be Function of S , V ;
let f be Function of L1 , L2 , g be Function of L1 , L2 ;
reconsider z = z as Element of ( .| L ) . s ;
let f be Function of S , T ;
reconsider g = g as Morphism of c ' , b ' ;
[ s , I ] in S [: A , B :] ;
len ( the connectives of C ) = 4 & len ( the connectives of C ) = 2 ;
let C1 , C2 be subFunctor of C , D ;
reconsider V1 = V as Subset of X | B ;
attr p is valid means : Def1 : All ( x , p ) is valid ;
assume that X c= dom f and f .: X c= dom g and f .: X c= dom g ;
H |^ a " is Subgroup of H |^ a ;
let A1 be |^ of O , B1 , B2 be Element of E ;
p2 , p3 , q2 is_collinear & q2 , q2 , q3 is_collinear ;
consider x being element such that x in v ^ K ;
not x in { 0. TOP-REAL 2 } ;
p in [#] ( I[01] | B ) & q in [#] ( I[01] | B ) ;
0 in M . ( E , F ) ;
^ ( c , c ) / ( c , d ) = c ;
consider c being element such that [ a , c ] in G ;
a1 in dom ( F . ( s2 . m ) ) ;
cluster pre-RSLattice -lattice L -> distributive for distributive LATTICE of L ;
set i1 = the Nat , i2 = the Element of NAT , n = the Element of NAT ;
let s be 0 -started State of SCM+FSA , p be s -started FinSequence of SCM+FSA ;
assume y in ( f1 \/ f2 ) .: A ;
f . len f = f /. len f .= f /. len f ;
x , f . x '||' f . x , f . y ;
attr X c= Y means : Def1 : cos ( X ) c= cos ( Y ) ;
let y be upper Subset of Y , x be Element of X ;
cluster ( x `1 ) ^2 -> J -element for Relation ;
set S = <* Bags n , i *> , T = <* i , j *> ;
set T = [. 0 , PI / 2 .] ;
1 in dom mid ( f , 1 , 1 ) ;
sqrt ( 4 * PI / 2 ) < sqrt ( 2 * PI / 2 ) ;
x2 in dom f1 /\ dom f2 & x1 in dom f1 /\ dom f2 ;
O c= dom I & { {} } = { {} } ;
( the Target of G ) . x = v ;
{ HT ( f , T ) } c= Support f ;
reconsider h = R . k as Polynomial of n , L ;
ex b being Element of G st y = b * H ;
let x , y , z be Element of G ` ;
h19 . i = f . ( h . i ) ;
( p `1 ) ^2 = ( ( p `1 ) ^2 ) ^2 .= ( p `2 ) ^2 ;
i + 1 <= len Cage ( C , n ) ;
len <* P *> = len P & len <* P *> = len P ;
set NN = the InitS of N , NN = the InternalRel of N , NN = the InternalRel of N ;
len g-2 + ( x - 1 ) <= x - 1 ;
a on B & b on B implies b on B
reconsider r-12 = r * I . v as FinSequence ;
consider d such that x = d and a [= d and a [= c ;
given u such that u in W and x = v + u ;
len f /. n = len f -' n .= len f ;
set q2 = W-min ( C , n ) , q1 = W-min ( C , n ) ;
set S = LSeg ( S1 , S2 ) , T = LSeg ( S2 , T1 ) ;
MaxADSet ( b ) c= MaxADSet ( P /\ Q ) ;
Cl ( G . q1 ) c= F . r2 ;
f " D meets h " ( V ) ;
reconsider D = E as non empty directed Subset of L1 ;
H = ( the_left_argument_of H ) '&' ( the_right_argument_of H ) ;
assume t is Element of ( \mathfrak S ) . X ;
rng f c= the carrier of S2 & rng f c= the carrier of S2 ;
consider y being Element of X such that x = { y } ;
f1 . ( a1 , b1 ) = b1 . ( a1 , b1 ) ;
the carrier' of G = E \/ { E } ;
reconsider m = len \langle k *> as Element of NAT ;
set S1 = LSeg ( n , UMP C ) , S2 = LSeg ( n , UMP C ) ;
[ i , j ] in Indices M1 & [ i , j ] in Indices M1 & [ i , j ] in Indices M1 ;
assume that P c= Seg m and M is Matrix and P is Matrix of m , D ;
for k st m <= k holds z in K . k
consider a being set such that p in a and a in G ;
L1 . p = p * ( 1 / p ) ;
p-7 . i = pp . i .= pp . i ;
let PP , P2 be a_partition of Y , BOOLEAN , f be Function of Y , BOOLEAN ;
attr 0 < r & 1 < 1 & r < 1 ;
rng \HM { the } \HM { carrier } \HM { of ( a , X ) | A = [#] ( X | A ) ;
reconsider x = x , y = y as Element of K ;
consider k such that z = f . k and n <= k and k <= len f ;
consider x being element such that x in X \ { p } ;
len ( canFS ( s ) ) = card s & len ( canFS ( s ) ) = card ( s ) ;
reconsider x2 = x1 , y2 = x2 as Element of L2 ;
Q in FinMeetCl ( ( the topology of X ) \/ { x } ) ;
dom ( f | ( dom u ) ) c= dom ( u | ( dom v /\ dom u ) ) ;
attr n divides m & m divides n & n divides m ;
reconsider x = x as Point of I[01] , a be Real ;
a in \mathop { \rm Exec } ( T2 , T2 ) ;
not y in the still of f & not y in the still of f & not y in the bound of f ;
Hom ( ( a \times b ) \times c ) <> {} ;
consider k1 such that p " < k1 and k1 in dom f and f . k1 < f . k1 ;
consider c , d such that dom f = c \ d and f . c = d ;
[ x , y ] in dom g & [ x , y ] in dom k ;
set S1 = sorts ( x , y , z ) , S2 = Following ( s , 2 ) ;
l = m2 & l = m2 & l = m2 & l = m1 & m = m2 & l = m2 & m = m2 & m = m2 & l = m2 & m = m2 & m = m2 & l = m2 & m = m2 & m = m2 &
x0 in dom ( u | A ) /\ ( ( v | A ) " { 0 } ) ;
reconsider p = x as Point of TOP-REAL 2 , a be Real ;
[: I , I :] = [: [: I , I :] , [: I :] :] .= [: I , I :] ;
f . p3 <= f . p2 & f . p2 <= f . p3 ;
( F . x ) `1 <= ( F . x ) `1 & ( F . x ) `2 <= ( F . x ) `2 ;
( x `2 ) ^2 = ( ( W `2 ) ^2 ) ^2 .= ( ( W `2 ) ^2 ) ^2 ;
for n being Element of NAT holds P [ n ] ;
J , K be non empty Subset of I ;
assume 1 <= i & i <= len <* a " *> ;
0 |-> a = <*> the carrier of K & 0 -tuples_on the carrier of K = { {} } ;
X . i in 2 -tuples_on B . i \ B . i ;
<* 0 *> in dom ( e --> [ 1 , 0 ] ) ;
then P [ a ] & P [ succ a ] ;
reconsider sK = seq . ( n + 1 ) as terminal of D ;
( - i ) + 1 <= len - j ;
[#] S c= [#] T & the topology of T c= the topology of T ;
for V being strict RealUnitarySpace holds V in W1 & V in W2 implies V is Subspace of V
assume k in dom mid ( f , i , j ) ;
let P be non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
let A , B be Matrix of K , n , K ;
- a * b = a * b - b * c ;
for A being Subset of A9 , B being Subset of A st A // B holds A // B or B c= A
id ( o2 ) in <* o2 , o1 , o2 *> ;
then ||. x .|| = 0 & x = 0. X & x = 0. X ;
let N1 , N2 be strict Subgroup of G , N1 , N2 be strict Subgroup of G , N2 , N2 be strict Subgroup of G , N1 , N2 be Subgroup of G , N1 , N2 be Subgroup of G , N1 , N2 be Subgroup of N1 , N2
j >= len ( upper_volume ( g , D1 ) | j ) ;
b = Q . ( len Q - 1 ) ;
f2 * f1 /* s is divergent_to+infty & f2 * f1 is divergent_to+infty ;
reconsider h = f * g as Function of I[01] , G ;
assume that a <> 0 and Polynom ( a , b , c ) >= 0 ;
[ t , t ] in the InternalRel of A & [ t , t ] in the InternalRel of A ;
( v |-- E ) | n is Element of ( T . n ) -tuples_on the carrier of T ;
{} = the support of L1 + ( the support of L2 ) .= { v } ;
Directed I is_closed_on Initialized s , P & Directed I is_halting_on Initialized s , P ;
Initialized p = Initialize ( ( p +* q ) +* q ) .= ( p +* q ) ;
reconsider N2 = N1 as strict net of R2 , R2 ;
reconsider Y = Y as Element of <* ( Ids L ) . X , \subseteq ( <* X *> ) . Y ;
"/\" ( { p } , L ) <> p ;
consider j being Nat such that i2 = i1 + j and j in dom f and f /. ( j + 1 ) = f /. ( j + 1 ) ;
not [ s , 0 ] in the InternalRel of S2 & [ s , 0 ] in the InternalRel of S2 ;
m in ( B '&' C ) /\ D ;
n <= len ( P ^ Q ) + len ( Q ^ R ) ;
( x1 `1 ) ^2 = ( x2 `1 ) ^2 .= ( x2 `2 ) ^2 ;
InputVertices S = { x1 , x2 , x3 , x4 , x5 , cin , x4 , x5 , cin , x5 , cin , cin , x4 , cin , cin , x5 , cin , cin , cin , cin , cin , cin , cin , cin , cin , a4 , cin , cin
let x , y be Element of FI1 ( n ) ;
p = |[ p `1 , p `2 ]| .= |[ p `1 , p `2 ]| ;
g * 1_ G = h " * g * h .= h " * g " .= h " * g " * h " .= h " * g " * h " ;
let p , q be Element of PFuncs ( V , C ) ;
x0 in dom ( x1 - x2 ) /\ dom ( x1 - x2 ) ;
( R qua Function ) " = R " * ( R " ) .= R " * ( R " ) ;
n in Seg len ( f /^ ( n -' 1 ) ) ;
for s being Real st s in R holds s <= s2 & t <= s2 holds t <= s2
rng s c= dom ( f2 * f1 ) ;
synonym \mathop { \rm Fin X } for X is Subset of Fin X ;
1_ K * 1_ K = 1_ K * ( 1_ K ) .= 1_ K * ( 1_ K ) .= 1_ K * ( 1_ K ) ;
set S = Segm ( A , P1 , Q1 ) , Q = Segm ( A , B , Q ) ;
ex w st e = sqrt ( w , f ) & w in F & w in G ;
curry k ( P\rbrack , k ) # x is convergent ;
cluster open for Subset of T | A , F be Subset-Family of T ;
len f1 = 1 .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) ;
sqrt ( i * p ) < sqrt ( 2 * p ) ;
let x , y be Element of \mathop { \rm non-empty } ( U0 ) ;
b1 , c1 // b9 , c & c , c9 // c , x ;
consider p being element such that c1 . j = { p } ;
assume that f " { 0 } = {} and f " { 0 } = {} ;
assume IC Comput ( F , s , k ) = n ;
Reloc ( J , card I ) not f in dom ( J .--> a ) ;
Macro ( card I + 1 ) not f is not empty ;
set s3 = LifeSpan ( p3 , s3 ) , P3 = P +* P3 , s3 = P +* P3 , s3 = P3 ;
IC SCMPDS in dom Initialize ( ( intloc 0 ) .--> 1 ) ;
dom t = the carrier of SCM R & dom t = the carrier of SCM R ;
( E-max L~ f ) .. f = 1 & ( E-max L~ f ) .. f = 1 ;
let a , b be Element of V ( V , C ) ;
Cl Int ( union F ) c= Cl Int union union union union union union F ;
the carrier of X1 misses ( A1 \/ A2 ) ;
assume not LIN a , f . a , g . b ;
consider i be Element of M such that i = d1 . i and i in dom f ;
then Y c= { x } or Y = { x } ;
M , v / ( ( y , v ) / ( x , y ) ) |= H1 / ( ( ( y , v ) / ( x , y ) ) / ( x , y ) ) ;
consider m being element such that m in Intersect ( F ) and m in Intersect ( F ) ;
reconsider A1 = support ( u1 ) as Subset of X ;
card ( A \/ B ) = k-1 + ( 2 * 1 ) ;
assume that a1 <> a3 and a2 <> a4 and a3 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4 and a4 <> a4
cluster s \! \mathop { V } -> string for string of S ;
Carrier ( L2 , n ) = L2 . ( n + 1 ) ;
let P be compact non empty Subset of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
assume that r-7 in LSeg ( p1 , p2 ) and r < 1 ;
let A be non empty compact Subset of TOP-REAL n , B be Subset of TOP-REAL n ;
assume [ k , m ] in Indices ( D * ( i , j ) ) ;
0 <= ( sqrt 2 ) * ( p / 2 ) ;
( F . N ) . x = +infty ;
attr X c= Y means : Def1 : Z c= V & X c= Y \ V ;
( y `2 ) * ( z `2 ) * ( z `2 ) <> 0. I ;
1 + card ( X1 \/ X2 ) <= card ( u \/ v ) + card ( v \/ w ) ;
set g = z \circlearrowleft ( L~ z ) , h = z .. z , i = len z ;
then k = 1 & p . k = <* x , y *> ;
cluster -> ( the Element of C ) \mathop { X } -> total ;
reconsider B = A as non empty Subset of TOP-REAL n ;
let a , b , c be Function of Y , BOOLEAN ;
L1 . i = ( i .--> g ) . i .= g . i .= g . i ;
Plane ( x1 , x2 , x3 ) c= P & P [ x1 , x2 , x3 ] ;
n <= indx ( D2 , D1 , j1 ) - 1 ;
( ( g2 ) . O ) `1 = - 1 & ( g2 . I ) `1 = - 1 ;
j + p .. f - len f <= len f - len f ;
set W = W-bound C , E = N-bound C , F = E-bound C , G = E-bound C , f = W-bound C ;
S1 . ( a , e ) = a + e .= a + e .= a + e ;
1 in Seg width ( M * ( ( p * ( len p ) ) ) ) ;
dom ( i * Im ( f , g ) ) = dom ( Im ( f , g ) ) ;
attr attr attr x `1 = W . ( a , p ) *' ( a , p ) ;
set Q = |= |= |= ( g , f , h ) , R = g ;
cluster -> many sorted for ManySortedSet of U1 , the Sorts of U1 ;
attr F = { A } means : Def1 : F is discrete ;
reconsider z9 = One as Element of product \overline G ;
rng f c= rng f1 \/ rng f2 & rng f c= rng f1 \/ rng f2 ;
consider x such that x in f .: A and x in f .: A ;
f = <*> ( the carrier of F_Complex ) & f is FinSequence of COMPLEX ;
E , j |= All ( j , H ) & E , j |= H ;
reconsider n1 = n as Morphism of o1 , o2 ;
assume that P is idempotent and R .: P = R .: P and R .: P = R .: P ;
card ( B2 \/ { x } ) = k + 1 - 1 ;
card ( ( x \ B1 ) /\ ( B \ ( B \ ( B \ B1 ) ) ) ) = 0 ;
g + R in { s : g-r < s & s < g } ;
set q9 = ( q , <* s *> ) := ( s , t ) , q1 = ( q , t ) := ( s , t ) , q2 = ( q , t ) := ( s , t ) , q2 = ( q , t ) := (
for x being element st x in X holds x in rng f1 & x in X
h1 /. ( i + 1 ) = h1 . ( i + 1 ) ;
set cw = max ( B , min ( B , max ( A , B ) ) ) ;
t in Seg width ( I ^ J ) & t in dom ( I ^ J ) ;
reconsider X = dom f /\ C as Element of Fin NAT ;
IncAddr ( i , k ) = halt SCM+FSA .= ( i + k ) ;
N-bound L~ f <= ( q `2 ) / 2 & ( q `2 ) / 2 <= ( q `2 ) / 2 ;
attr R is condensed means : Def1 : R is condensed & R is condensed & R is condensed ;
attr 0 <= a & 1 <= b & a <= 1 & b <= 1 implies a * b <= 1 * b ;
u in ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ;
u in ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ;
len C + 2 - 1 >= 9 + - 1 ;
x , z , y is_collinear & x , z , y is_collinear & x , y , z is_collinear ;
a |^ ( n1 + 1 ) = a |^ ( n1 + 1 ) * a |^ ( n1 + 1 ) ;
<* \underbrace ( 0 , \dots , 0 ) *> in Line ( x , a ) ;
set y9 = <* y , c *> ;
F2 /. 1 in rng Line ( D , 1 ) & F2 /. 1 in rng Line ( D , 1 ) ;
p . m joins r /. m , r /. ( m + 1 ) ;
( p `2 ) ^2 = ( f /. ( i1 + 1 ) ) `2 .= ( f /. ( i1 + 1 ) ) `2 ;
W-min ( X \/ Y ) = W-min ( X \/ Y ) .= W-min ( X \/ Y ) ;
0 + ( p `2 / |. p .| - sn ) <= 2 * r + ( p `2 / |. p .| - sn ) ;
x in dom g & not x in g " { 0 } ;
f1 /* ( s ^\ k ) is divergent_to+infty & f2 /* ( s ^\ k ) is divergent_to+infty ;
reconsider u2 = u as VECTOR of Pmin ( X , Y ) ;
p \! \mathop { \rm \hbox { - } count ( \prod ( Sgm X ) ) = 0 ;
len <* x *> < i + 1 & i + 1 <= len c ;
assume that I is non empty and { x } /\ { y } = { x } ;
set i2 = card I + 4 + ( card I + 4 ) , i2 = ( card I + 4 ) .--> ( ( card I + 4 ) .--> ( ( card I + 4 ) ) ) ;
x in { x , y } & h . x = {} & h . y = T ;
consider y being Element of F such that y in B and y <= x `1 ;
len S = len ( the charact of A ) & len ( the charact of A ) = len ( the charact of A ) ;
reconsider m = M , n = N as Element of X ;
A . ( j + 1 ) = B . ( j + 1 ) \/ A . ( j + 1 ) ;
set N8 = : : ( G is open & G is open & G is open ) ;
rng F c= the carrier of gr { a } & rng F c= the carrier of gr { a } ;
K is Matrix of len F , K & F ( n , n ) is special ;
f . k , f . ( mod n ) ] in rng f ;
h " P /\ [#] ( ( T | P ) | P ) = f " P ;
g in dom f2 \ ( f2 " { 0 } ) ;
gnet X /\ dom f1 = g1 " ( X /\ dom f1 ) .= g1 " ( X /\ dom f1 ) ;
consider n being element such that n in NAT and Z = G . n ;
set d1 = \bf \bf min ( ( x1 , y1 , y2 ) , d2 ) , d2 = dist ( x2 , y2 , z2 ) ;
b `2 + sqrt ( 1 + ( 2 * a ) ^2 ) < sqrt ( 1 + ( 2 * a ) ^2 ) ;
reconsider f1 = f as VECTOR of X , Y ;
attr i <> 0 implies i ^2 mod ( i + 1 ) mod ( i + 1 ) = 1 ;
j2 in Seg len ( ( g2 . ( i2 + 1 ) ) * ( f . ( i2 + 1 ) ) ) ;
dom i = dom ( i - 1 ) .= dom i .= Seg ( len i - 1 ) .= Seg ( len i - 1 ) .= Seg ( len i - 1 ) .= Seg ( len i - 1 ) .= Seg ( len i - 1 ) .= Seg ( len i - 1 ) .= Seg ( len i
cluster sec | ]. PI / 2 , PI / 2 .[ -> one-to-one ;
Ball ( u , e ) = Ball ( f . p , e ) ;
reconsider x1 = x0 as Function of S , T ( ) ;
reconsider R1 = x , R2 = y as Relation of L , L ;
consider a , b being Subset of A such that x = [ a , b ] ;
( <* 1 *> ^ p ) ^ ( <* n *> ^ q ) in R1 ;
S1 +* S2 = S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2 +* S2
( ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * (
cluster -> continuous for Function of C , REAL ;
set C1 = 1GateCircStr ( <* z , x *> , f ) , C2 = 1GateCircStr ( <* x , y *> , f ) , C2 = 1GateCircStr ( <* y , z *> , f ) ;
E . ( e2 , T ) = ( E . ( e2 , T ) ) -T . ( e , T ) ;
( ( arctan ) (#) ( ( arctan ) `| Z ) ) . x = ( ( arctan ) `| Z ) . x ;
upper_bound A = sqrt ( 3 * PI / 2 ) & lower_bound A = 0 ;
F ( dom f , - F ) is transformable to F ( dom f , - F ) ;
reconsider p9 = q9 as Point of TOP-REAL 2 , e = e as Point of TOP-REAL 2 ;
g . W in [#] Y & [#] Y c= [#] Y & g . W in [#] Y ;
let C be compact non vertical non empty Subset of TOP-REAL 2 ;
LSeg ( f ^ g , j ) = LSeg ( f , j ) ;
rng s c= dom f /\ ]. - 1 , 1 .[ & rng ( f | ]. - 1 , 1 .[ ) c= dom f /\ dom ( f | ]. - 1 , 1 .[ ) ;
assume x in { ( idseq 2 ) * ( ( idseq 2 ) * ( ( idseq 2 ) * ( ( idseq 2 ) * ( ( Seg 2 ) * ( ( Seg 2 ) * ( ( Seg 2 ) * ( ( Seg 2 ) * ( ( Seg 2 ) * ( ( Seg
reconsider n2 = n , m2 = m - n as Element of NAT ;
for y being ExtReal st y in rng seq holds g <= y holds g <= y
for k st P [ k ] holds P [ k + 1 ]
m = m1 + m2 .= m1 + m2 .= m2 + m2 .= m1 + m2 .= m1 + m2 ;
assume for n holds H1 . n = G . n -H . n ;
set BB = f .: ( the carrier of X1 ) , BB = f .: ( the carrier of X2 ) , BB = f .: ( the carrier of X2 ) ;
ex d being Element of L st d in D & x << d & d << d ;
assume R " ( a , b ) c= R " ( b , a ) ;
t in ]. r , s .[ or t = ]. s , t .[ & t in [. p , q .[ ;
z + v2 in W & x = u + ( z + v2 ) ;
x2 |-- y2 iff P [ x2 , y2 ] & P [ x2 , y2 ] ;
attr x1 <> x2 means : Def1 : |. x1 - x2 .| > 0 & |. x1 - x2 .| > 0 ;
assume p2 - p1 , p3 - p2 - p1 - p2 - p1 , p2 - p1 - p2 - p1 , p3 - p1 - p2 - p2 , p2 - p1 - p2 - p1 , p2 - p1 - p2 - p1 , p2 - p1 - p2 - p2 - p1 , p2 - p2 - p1 - p2 , p2 - p1 - p2 , p2 - p1
set q = Ant ( f ^ <* 'not' A *> ) , r = ( f ^ <* 'not' A *> ) ;
let f be PartFunc of REAL-NS 1 , REAL-NS 1 , REAL-NS 1 , i be Nat ;
( n mod ( 2 * k ) ) ! = n mod ( 2 * k ) ;
dom ( T * the_arity_of o ) = dom <* T . o *> .= dom <* T . o *> ;
consider x being element such that x in w and x in c ;
assume ( F * G ) . ( v . x3 ) = v . x3 ;
assume that the Sorts of D1 c= the Sorts of D2 and f . ( the Sorts of D2 ) = f . ( the Sorts of D2 ) ;
reconsider A1 = [. a , b .] as Subset of R^1 ;
consider y being element such that y in dom F and F . y = x ;
consider s being element such that s in dom o and a = o . s ;
set p = W-min ( C , n ) , q = W-min ( C , n ) ;
n1 - len f + 1 <= len g - 1 + 1 ;
ConsecutiveSet2 ( q , O ) = [ u , v ] ;
set C-2 = ( .n: G ) . ( k + 1 ) ;
Sum ( L * p ) = 0. R .= 0. V .= 0. V ;
consider i being element such that i in dom p and t = p . i ;
defpred Q [ Nat ] means 0 = Q ( $1 ) & $1 = Q ( $1 ) ;
set s3 = Comput ( P1 , s1 , k ) , P3 = Comput ( P2 , s2 , k ) ;
let l be variable of k , A , B be Element of k , A be Subset of l ;
reconsider U = union ( G . H ) as Subset-Family of [: T , T :] ;
consider r such that r > 0 and Ball ( p , r ) c= Q ` ;
( h | ( n + 2 ) ) /. i = p29 . i ;
reconsider B = the carrier of X1 , C = the carrier of X2 as Subset of X ;
po = <* - 1 , 0 , 1 *> & [: { 0 , 1 } , { 1 } :] = [: { 1 , 1 } , { 1 } :] ;
synonym f is real-valued means : Def1 : rng f c= NAT & rng f c= NAT ;
consider b being element such that b in dom F and a = F . b ;
x9 < card ( X1 \/ X2 ) & x9 in card ( X1 \/ X2 ) & x9 in ( X1 \/ X2 ) & x9 in ( X1 \/ X2 ) & x9 in ( X1 \/ X2 ) & x9 in ( X1 \/ X2 ) ;
attr X c= B1 & X c= succ B1 & X c= succ B1 & X c= succ B1 ;
then w in ( Cl x ) \ ( dist ( x , r ) ) & dist ( x , w ) <= r ;
angle ( x , y , z ) = angle ( x , y , z ) ;
attr 1 <= len s means : Def1 : for s being FinSequence holds s . ( len s ) = s . ( len s ) ;
fP1 c= f . ( k + 1 ) ;
the carrier of { 1_ G } = { 1_ G } & { 1_ G } is finite ;
attr p '&' q in TAUT ( A ) means : Def1 : q '&' p in TAUT ( A ) & q in TAUT ( A ) ;
- ( t `2 / t `1 ) < ( - t `1 ) / t `1 ;
U1 . 1 = U1 /. 1 .= U1 . ( 1 ) .= U1 . ( 1 + 1 ) .= U1 . ( 1 + 1 ) .= U1 . ( 1 + 1 ) .= U1 . ( 1 + 1 ) .= U1 . ( 1 + 1 ) ;
f .: ( the carrier of x ) = the carrier of x & f .: ( the carrier of x ) = the carrier of x ;
Indices ( O * ( i , j ) ) = [: Seg n , Seg n :] ;
for n being Element of NAT holds G ( n ) c= G ( n )
then V in M |^ \square & ex x being Element of M st V = { x } ;
ex f being FinSequence of F st f is Matrix & f is Matrix of n , K & f is Matrix & f is Matrix of n , K & f is Matrix of n , K & f is Matrix ;
[ h . 0 , h . 3 ] in the InternalRel of G & [ h . 0 , h . 3 ] in the InternalRel of G ;
s +* Initialize ( ( intloc 0 ) .--> 1 ) = s3 +* Initialize ( ( intloc 0 ) .--> 1 ) .= s3 ;
|[ w , v ]| `1 <> 0. TOP-REAL 2 & |[ w , v ]| `2 <> 0. TOP-REAL 2 ;
reconsider t = t as Element of INT -tuples_on INT ;
C \/ P c= [#] ( ( G \ A ) \ A ) ;
f " V in ( ( X /\ the carrier of S ) /\ D ) /\ ( ( X /\ the carrier of S ) /\ D ) ;
x in [#] ( ( the carrier of ( ( TOP-REAL 2 ) | A ) ) ) ;
g . x <= h1 . x & h . x <= h1 . x & h . x <= h1 . x ;
InputVertices S = { xy , yz , zx , cin , dp , cin , dp , cin , cin } ;
for n being Nat st P [ n ] holds P [ n + 1 ]
set R = Line ( M , i ) * Line ( M , i ) ;
assume that M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line and M1 is being_line and M2 is being_line ;
reconsider a = f1 . ( i0 -' 1 ) as Element of K ;
len ( ( f1 ^ f2 ) ^ ( f2 ^ g2 ) ) = len ( ( f1 ^ f2 ) ^ ( f2 ^ g2 ) ) ;
len ( the connectives of n ) = n & len ( the connectives of n ) = n & width ( the connectives of n ) = n ;
dom ( max ( f , g ) ) = dom ( f + g ) .= dom ( f + g ) ;
( the Sorts of seq ) . n = ( sup Y1 ) * ( sup Y ) ) ;
dom ( p1 ^ p2 ) = dom ( p1 ^ p2 ) .= dom ( p1 ^ p2 ) .= dom p1 \/ dom p2 .= dom p2 ;
M . [ 1 , y ] = 1 * v1 .= ( 1 - 1 ) * v2 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v2 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v1 .= ( 1 - 1 ) * v2 .= ( 1 - 1 ) * v1 .=
assume that W is non trivial and W .cut ( W ) c= the topology of G2 and W is open ;
C6 /. i1 = G1 * ( i1 , i2 ) .= G1 * ( i1 , i2 ) ;
C8 |- 'not' All ( x , p ) 'or' 'not' p 'or' 'not' ( x , p ) ;
for b st b in rng g holds lower_bound rng fD1 <= b & lower_bound rng fD1 <= b
- sqrt ( ( ( - ( ( q `2 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) ) = 1 ;
( LSeg ( c , m ) \/ LSeg ( l , k ) ) c= R ;
consider p being element such that p in LSeg ( x , p ) and p in LSeg ( f , p ) ;
Indices X = [: Seg n , Seg n :] & [: Seg n , Seg n :] = [: Seg n , Seg n :] ;
cluster s => ( q => p ) -> valid ;
Im ( ( Partial_Sums F ) . m ) , ( Partial_Sums ( F ) ) . n ) is_measurable_on E ;
cluster f . ( x1 , x2 ) -> Element of D ( ) ;
consider g being Function such that g = F . t and Q [ t , g ] ;
p in LSeg ( NW-corner Z , NW-corner Z ) & q in LSeg ( NW-corner Z , E-max Z ) ;
set R8 = R / ( 1 / 2 ) , R8 = ]. 1 / 2 , 1 / 2 .[ ;
IncAddr ( I , k ) = AddTo ( d , k ) .= halt SCM+FSA .= ( d , a ) ;
seq . m <= ( the InternalRel of seq ) . k & ( ( the InternalRel of seq ) * ( ( the InternalRel of seq ) * ( ( the InternalRel of seq ) * ( ( the InternalRel of seq ) * ( ( the InternalRel of seq ) * ( ( the InternalRel of seq ) * ( ( the InternalRel of seq )
a + b = ( a ` ) ` ` ` ` ` ` ` ` ` .= ( a ` ) ` ` ` ` ` ` ` ` ` ` ;
id X /\ Y = id X /\ id Y .= id X /\ Y .= id Y ;
for x being element st x in dom h holds h . x = f . x ;
reconsider H = U1 \/ U2 as non empty Subset of U0 ;
u in ( ( c /\ ( d /\ e ) ) /\ f ) /\ j ;
consider y being element such that y in Y and P [ y , lower_bound B ] ;
consider A being finite stable Subset of R such that card A = card ( R * ) and A is finite ;
p2 in rng ( f |-- p1 ) \ rng <* p1 *> ;
len s1 > 0 & len s2 > 0 & len s1 > 0 & len s2 > 0 & len s1 > 0 & len s2 > 0 & len s2 > 0 & len s2 > 0 & len s2 > 0 & len s2 > 0 ;
( E-max ( P ) ) `2 = N-bound ( P ) & ( E-max ( P ) ) `2 = N-bound ( P ) ;
Ball ( e , r ) c= LeftComp Cage ( C , k + 1 ) ;
f . a1 ` ` = f . a1 ` ` ` .= ( f . a1 ` ) ` ;
( seq ^\ k ) . n in ]. - \infty , x0 .[ & ( for n holds seq . n < r ) implies seq is convergent & lim ( seq ) = - r
gg . ( ( g . ( s , t ) ) | G ) = g . ( ( g . ( s , t ) ) | G ) ;
the InternalRel of S is InternalRel of field ( the InternalRel of S ) & the InternalRel of S is InternalRel of S ;
deffunc F ( Ordinal , Ordinal ) = phi . ( $1 , $2 ) ;
F . ( s1 . a1 ) = F . ( s2 . a1 ) ;
x `2 = A ( a ) .= Den ( o , A ( a ) ) ;
Cl ( f " P1 ) c= f " ( ( f " P1 ) " P1 ) ;
FinMeetCl ( ( the topology of S ) \/ { x } ) c= the topology of T ;
synonym o is constructor means : Def1 : o <> \ast & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> * & o <> *
assume that X = Y and card X = card Y and card X <> card Y and card Y <> card X and card X = card Y ;
the InitS of s <= 1 + 1 & ( the InitS of s ) . ( len s ) = ( the InitS of s ) . ( len s ) ;
LIN a , d , c or b , c // d , c & b , c // d , c ;
e . 1 = 0 & e . 2 = 1 & e . 3 = 0 & e . 3 = 0 ;
E in [: S , T :] & E in [: { N } , { N } :] & E in [: { N } , { N } :] ;
set J = ( l , u ) \mathop { I } ;
set A1 = 1GateCircStr ( a1 , bm ( p , q , c ) , cin ) , A2 = Following ( s , 3 ) ;
set c9 = [ <* c , d *> , and2a ] , A1 = [ <* d , c *> , and2a ] , A2 = [ <* c , d *> , and2a ] , A2 = [ <* d , c *> , and2a ] , A2 = [ <* d , c *> , and2a ] , it = [ <* c , d *> , '&' ] ;
x * z " in x * ( z * N ) " * ( x * N ) ;
for x being element st x in dom f holds f . x = g1 . x & f . x = g2 . x ;
Int cell ( GoB f , 1 , G ) c= RightComp f \/ RightComp f \/ RightComp f ;
U is_an_arc_of E-max C , W-min C & W-min C in L~ Cage ( C , n ) & W-min C in L~ Cage ( C , n ) ;
set f-17 = f .: @ g , @ @ f ;
attr S1 is convergent means : Def1 : for S2 , S2 being convergent sequence of S1 , S2 being convergent sequence of S2 holds S1 - S2 is convergent & S2 is convergent & S2 is convergent & lim S2 = x0 ;
f . ( 0 + 1 ) = ( 0 qua Ordinal ) + a .= a ;
cluster -> Subsymmetric for RelStr ;
consider d being element such that R reduces b , d and R reduces c , d and R reduces c , d ;
not b in dom Start-At ( ( card I + 2 ) , SCMPDS ) ;
( z + a ) + x = z + ( a + y ) .= z + ( a + y ) .= z + ( a + y ) ;
len ( l | ( A , 0 ) ) = len l & len ( l | ( A , 0 ) ) = len l ;
t9 . {} is ( {} \/ rng ( t . {} ) ) -valued FinSequence ;
t = <* F . t *> ^ ( C ^ ( p ^ q ) ) ^ ( C ^ q ) ) ;
set p-2 = W-min ( C , n ) , pmin = W-min ( C , n ) , pmin = W-min ( C , n ) ;
k9 -' ( i + 1 ) = ( k - 1 ) + 1 .= ( i - 1 ) + 1 ;
consider u being Element of L such that u = u ` and u in D and u in D and u in D ;
len ( width ( width G ) |-> a ) = width ( ( width G ) |-> a ) .= width ( ( width G ) |-> a ) .= width ( ( width G ) |-> a ) ;
F1 . x in dom ( G * the_arity_of o ) ;
set H2 = the carrier of H2 , H = the carrier of H , G = the InternalRel of H , H = the InternalRel of H , H = the InternalRel of H , H = the InternalRel of H , G = the InternalRel of H , H = the InternalRel of H , G = the InternalRel of H , H ;
set H1 = the carrier of H1 , H2 = the carrier of H2 , H2 = the carrier of H2 ;
( Comput ( P , s , 6 ) ) . intpos ( m + 1 ) = s . intpos ( m + 1 ) ;
IC Comput ( Q , t , k ) = ( l + 1 ) - 1 .= ( l + 1 ) - 1 ;
dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1
cluster <* l *> ^ phi -> ( 1 + 2 ) string of S ;
set b5 = [ <* A1 , cin , c *> , and2 ] , b5 = [ <* cin , dp , c *> , and2 ] , b5 = [ <* A1 , cin , c *> , and2 ] ;
Line ( Segm ( M , P , Q ) , x ) = L * Sgm Q ;
n in dom ( ( the Sorts of A ) * the Arity of S ) & ( the Arity of S ) * the Arity of S = ( the Arity of S ) * the Arity of S ;
cluster f1 + f2 -> continuous for PartFunc of REAL , REAL ;
consider y be Point of X such that a = y and ||. y - x .|| <= r ;
set xy = t . DataLoc ( ( intpos ( 8 + 2 ) + 3 ) , 2 ) , xy = t . intpos ( 8 + 3 ) , xy = t . intpos ( 8 + 3 ) , xy = t . intpos ( 8 + 3 ) , it = t . intpos ( 8 + 3 ) , it = t .
set p-3 = stop I , pg = P +* stop I , P3 = P +* stop I , s3 = P +* stop I , P3 = P +* stop I , P3 = P +* stop I , P4 = P +* stop I , P3 = P +* P3 , P4 = P +* P3 , P4 = P ;
consider a being Point of D2 such that a in W1 and b = g . a and a in A and b in B ;
{ A , B , C , D } = { A , B , C } \/ { D , E , F , J , M } ;
let A , B , C , D , E , F , J , M be set ;
|. p2 .| ^2 - ( 1 - ( p2 `2 / p2 `1 ) ^2 ) >= 0 ;
l -' 1 + 1 = n-1 * ( 1 / ( l + 1 ) ) + 1 ;
x = v + ( a * w + b * y ) + ( c * y ) ;
the TopStruct of L = ( the Scott of L ) | ( the topology of L ) .= ( the Scott of L ) | ( the topology of L ) ;
consider y being element such that y in dom H1 and x = H1 . y and y in dom H2 and x = ( H1 . y ) `2 ;
fn \ { n } = ( \langle n , m *> ) \ { n , m } .= { n , m } ;
for Y being Subset of X st Y is summable holds Y is summable iff Y is summable & Y is summable
2 * n in { N : 2 * Sum ( p | N ) = N & N > 0 } ;
for s being FinSequence holds len ( \HM { the } \HM { Boolean } \HM { of } R ) = len s & len ( s * ( the connectives of R ) ) = len s
for x st x in Z holds ( exp_R * f ) is_differentiable_in x & ( exp_R * f ) . x > 0
rng ( h2 * f1 ) c= the carrier of ( TOP-REAL 2 ) | K1 & rng ( h2 * f1 ) c= the carrier of ( TOP-REAL 2 ) | K1 ;
j + 1- len f <= len f + ( len f - 1 ) - len f + 1 ;
reconsider R1 = R * I as PartFunc of REAL , REAL-NS n ;
C8 . x = s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1 ) .= s1 . ( a - 1
power F_Complex ( z , n ) * ( z , n ) = 1 .= x |^ n .= x |^ n .= x |^ n ;
t is_C ( s , C ) = f . ( the connectives of S ) . t ;
support ( f + g ) c= support f \/ support g & support ( f + g ) c= support f \/ support g ;
ex N st N = j1 & 2 * ( Sum ( ( r | N ) ) | N ) > N & 2 * ( ( r | N ) | N ) > N ;
for y , p st P [ p ] holds P [ y ] & P [ p ] holds P [ y ]
{ [ x1 , x2 ] } is Subset of X1 & { [ x1 , x2 ] } is Subset of X2 & { [ x1 , x2 ] } is Subset of X1 ;
h = ( j |-> h ) . ( id B , h ) .= H . ( j , h ) .= H . ( j , h ) .= H . ( j , h ) ;
ex x1 being Element of G st x1 = x & x1 * N c= A & x1 * N c= A & x1 * N c= A ;
set X = ( |. q .| ) * ( ( |. p .| ) * ( |. q .| ) ) , Y = ( |. q .| ) * ( ( |. q .| ) * ( |. q .| ) ) ;
b . n in { g1 : x0 < g1 & g1 < x0 } ;
f /* s1 is convergent & f /. x0 = lim ( f /* s1 ) implies f /* s1 is convergent & lim ( f /* s1 ) = lim ( f /* s1 )
the lattice of Y = the lattice of Y & the lattice of Y = the lattice of Y & the carrier of X = the carrier of Y ;
'not' ( a . x ) '&' b . x 'or' b . x = TRUE ;
S2 = len ( ( q ^ <* r1 *> ) ^ ( q ^ <* r1 *> ) ) + len ( ( q ^ <* r1 *> ) ^ ( q ^ <* r1 *> ) ) ;
sqrt ( 1 / a * ( ( sec * f1 ) - ( ( sec * f1 ) ) / ( 1 / ( 1 + ( id Z ) * ( ( sec * f1 ) ) ^2 ) ) ) is_differentiable_on Z ;
set K = upper \ ( lim ( A , H ) ) , H = ( lim ( A , H ) ) (#) ( ( lim ( A , H ) ) (#) ( ( lim ( A , H ) ) (#) ( ( lim ( A , H ) ) (#) ( ( lim ( A , H ) ) (#) ( ( lim ( A , H ) ) (#) ( (
assume e in { \frac w + ( v + ( w + ( w + ( w + ( v + ( v + ( w + ( w + ( v + ( w + ( v + ( w + ( w + ( v + ( w + ( w + ( w + ( v + ( w + ( w + ( w + ( w + ( w + (
reconsider d1 = dom a `1 , d2 = dom F , d2 = dom F , d2 = F . ( c , d ) , d2 = F . ( c , d ) as finite set ;
LSeg ( f , j ) = LSeg ( f , j ) \/ LSeg ( f , j ) .= LSeg ( f , j ) ;
assume X in { T ( N2 , K1 ) : h . ( N2 , K1 ) = N2 & h . ( N2 , K1 ) = N2 . ( N2 , K1 ) ;
assume that Hom ( d , c ) <> {} and <* f , g *> * f = <* f , g *> * f ;
dom S29 = dom S /\ Seg n .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) .= dom ( L | n ) .= Seg n .= dom ( L | n ) .= Seg n .= dom ( L | n ) .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg n .= Seg
x in H |^ a implies ex g st x = g |^ a & g in H |^ a & g in H |^ a & h in H
a * ( a , 1 ) = a `1 - ( a * n ) .= a `1 - ( a * n ) .= a `1 - ( a * n ) .= a `1 - ( a * n ) .= a ;
D2 . j in { r : lower_bound A <= r & r <= ( D1 . j ) - r } ;
ex p being Point of TOP-REAL 2 st p = x & P [ p ] & p `2 <= 0 & p `2 <= 0 ;
for c holds f . c <= g . c implies f ^ g ^ g ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ h ^ g ^ h ^ g ^ h ^ h ^ g ^ h ^ g ^ h ^ h ^ g ^ h ^ h
dom ( f1 (#) f2 ) /\ X c= dom ( f1 (#) f2 ) & dom ( f1 (#) f2 ) /\ X c= dom ( f2 (#) f2 ) ;
1 = sqrt ( p * p ) .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * p .= p * ( p * q ) .= p * ( p * q ) .= p
len g = len f + len <* x *> .= len f + len <* y *> .= len f + len <* x *> .= len f + len <* y *> .= len f + len g ;
dom ( F | [: N1 , S :] ) = [: N1 , S :] & dom ( F | [: N1 , S :] ) = [: N1 , S :] & [: N1 , S :] = [: N1 , S :] ;
dom ( f . t ) = dom ( f . t ) .= dom ( ( f . t ) * g ) ;
assume a in ( "\/" ( ( T |^ \alpha ) |^ D ) ) .: D & b in the carrier of S ;
assume that g is one-to-one and rng g /\ rng g c= dom g and g is one-to-one and rng g c= dom g and g is one-to-one ;
( ( x \ y ) \ z ) \ ( ( x \ z ) \ z ) = 0. X ;
consider f such that f * f = id b and f * f = id b and f * f = id b ;
( ( ( cos * sin ) `| [. 0 , PI / 2 .] ) . x ) is increasing ;
Index ( p , co ) <= len LS - Index ( p , co ) + 1 ;
t1 , t2 be Element of [: T , NAT :] , s be Element of [: S , T :] ;
\sqcap ( ( ( Frege ( H ) ) . h ) . h ) <= "/\" ( ( Frege ( ( ( ( curry ( ( K ) . h ) ) . h ) ) , L ) . h ) ;
then P [ f . i0 , f . ( i0 + 1 ) ] & F ( f . i0 , f . ( i0 + 1 ) ) < j ;
Q [ ( D . x ) `1 , F . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . ( D . x
consider x being element such that x in dom ( F . s ) and y = F . s and x in dom ( F . s ) ;
l . i < r . i & [ l . i , r . i ] in Indices G ;
the Sorts of A2 = ( the Sorts of S2 ) --> ( the Sorts of S2 ) .= ( the Sorts of S2 ) +* ( the Sorts of S2 ) .= ( the Sorts of S2 ) +* ( the Sorts of S2 ) ;
consider s being Function such that s is one-to-one and dom s = NAT and rng s = { s } and rng s = { s } ;
dist ( b1 , b2 ) <= dist ( b1 , b2 ) + dist ( b2 , a ) ;
( for n holds ( for C being Subset of TOP-REAL 2 holds C /. ( len C ) ) = W . ( n + 1 ) ) & C is closed & C is closed implies C is closed
q <= ( UMP ( C , n ) ) `2 & ( E-max ( C , n ) ) `2 <= s ;
LSeg ( f | i2 , i ) /\ LSeg ( f | i2 , j ) = {} ;
given a being ExtReal such that a <= Iu and A = ]. a , b .[ and A = ]. a , b .[ and B = ]. a , b .[ ;
consider a , b being complex number such that z = a & y = b & z = a + b and z = a + b ;
set X = { b / n where b is Element of NAT : b in X & c in X } ;
( ( x * y ) \ z ) \ ( x * z ) ) \ ( x * z ) = 0. X ;
set xy = [ <* xy , yz , z *> , f2 ] , yz = [ <* xy , yz , z *> , f3 ] , yz = [ <* xy , yz , z *> , f2 ] , yz = [ <* yz , z *> , f3 ] , [ <* yz , yz , z *> , [ <* yz , yz , z *> , f2 ] ;
Carrier ( l ) /. len ( ( p ^ q ) | ( len ( p ^ q ) ) ) = ( p ^ q ) /. len ( ( p ^ q ) | ( len ( p ^ q ) ) ) .= ( p ^ q ) /. len ( ( p ^ q ) | ( len ( p ^ q ) ) ) ;
sqrt ( ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) = 1 ^2 ;
sqrt ( ( ( p `2 ) / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) < 1 ^2 / ( 1 + sn ) ^2 ;
( ( ( S \/ Y ) \ Y ) \/ ( S \/ Y ) ) `2 = N-bound ( X \/ Y ) ;
( s1 - s2 ) . k = s1 . k - s2 . k .= ( s1 - s2 ) . k .= ( s1 - s2 ) . k ;
rng ( ( h + c ) ^\ n ) c= dom SVF1 ( 1 , f , x0 ) ;
the carrier of X = the carrier of X & the carrier of X = the carrier of X & the carrier of X = the carrier of X ;
ex p3 st p3 = p3 & |. p3 - p4 .| = r & |. p3 - p4 .| = r ;
set h = IExec ( X , A , G ) , i = ( card X + 1 ) * ( i + 1 ) ;
R |^ ( 0 * n ) = \mathop { Il ( X , Y ) } .= R |^ ( 0 * n ) .= R |^ ( 0 * n ) .= R |^ ( 0 * n ) ;
( Partial_Sums ( ( ( F . 0 ) ) ) . n ) . x is nonnegative & ( ( F . 0 ) . x ) . x is nonnegative ;
f2 = C7 . ( ( E8 ) . ( len ( V , len ( ( E , len ( V , len ( V ) ) ) ) ) ) ) ;
S1 . b = s1 . b .= S2 . b .= S2 . b .= S2 . b .= S2 . c .= S2 . c .= S2 . c .= S2 . c ;
p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) & p2 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) ;
dom ( f . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n & dom ( I . t ) = Seg n ;
assume o = ( the connectives of S ) . 11 & the connectives of S = ( the connectives of S ) . 11 ;
set phi = ( l1 , l2 ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u ) , phi = ( l , u
synonym p is_\bf w.r.t. T means : Def1 : HT ( p , T ) = 1 / p & HT ( p , T ) = 1 / p ;
( Y1 `2 ) ^2 = - 1 & ( Y1 `2 ) ^2 = ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * (
defpred X [ Nat , set ] means P [ $1 , $2 , $1 ] & P [ $2 , $2 , $2 ] ;
consider k be Nat such that for n be Nat st k <= n holds s . n < x0 + g ;
Det ( I |^ ( m -' n ) ) = 1_ K & Det I |^ ( m -' n ) = 1_ K ;
sqrt ( b - sqrt ( b ^2 - c ^2 ) ) * 2 * ( 2 * a ) ) < 0 ;
Cd . d = Cd . ( d1 , d2 ) .= ( Cd ) . ( d2 , d2 ) .= ( Cd ) . ( d2 , d2 ) ;
attr X1 is dense means : Def1 : X1 is dense & X2 is dense & X1 /\ X2 is dense & X2 is dense & X1 /\ X2 is dense implies X1 /\ X2 is dense SubSpace of X ;
deffunc F ( Element of E , Element of I ) = $1 * ( $2 , $1 ) ;
t ^ <* n *> in { t ^ <* i *> where i is Nat : Q [ i , T . ( t ^ <* n *> ) ] } ;
( x \ y ) \ x = ( x \ x ) \ y .= y \ x .= 0. X .= 0. X ;
for X being non empty set for X being Subset-Family of X holds X is Basis of <* X , Y *>
synonym A , B means : Def1 : A misses B & A misses B & A misses B & B misses C & A misses C & B misses C ;
len ( - M ) = len p & width ( - M ) = width M & width ( - M ) = width M & width ( - M ) = width M & width ( - M ) = width M & width ( - M ) = width M ;
J = { x where x is Element of K : 0 < x & x < 1 } ;
( Sgm m ) . d - ( Sgm m ) . e - ( Sgm m ) . e <> 0 ;
lower_bound divset ( D2 , k + 1 ) = D2 . ( k + 1 - 1 ) .= D2 . ( k + 1 - 1 ) ;
g . r1 = - 2 * r1 + 1 & dom h = [. 0 , 1 .] & rng h = [. 0 , 1 .] & rng h c= [. 0 , 1 .] & rng h c= [. 0 , 1 .] & rng h c= [. 0 , 1 .] & rng h c= [. 0 , 1 .] & rng h c= dom f & g is continuous ;
|. a .| * ||. f .|| = 0 * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| .= ||. f .|| * ||. f .|| ;
f . x = ( h . x ) `1 & g . x = ( h . x ) `1 & g . x = ( h . x ) `2 ;
ex w st w in dom B1 & <* 1 *> ^ s = <* 1 *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ ( <* 1 *> ^ w ) ^ w ^ w ^ w ^ w ^ w ^ ( <* 1 *> ^ w ) ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
[ 1 , {} , <* d1 , d2 , d2 *> ] in ( { [ 0 , {} , {} ] } \/ ( { [ 0 , {} , {} ] } \/ { [ 1 , {} , {} ] } ) ;
IC Exec ( i , s1 ) + n = IC Exec ( i , s2 ) .= succ IC Comput ( P2 , s2 , n ) .= succ IC Comput ( P2 , s2 , n ) ;
IC Comput ( P , s , 1 ) = IC Comput ( P , s , 1 ) .= 5 .= 5 .= 5 .= 5 .= 5 .= 5 .= 5 .= 5 .= 0 ;
( IExec ( W , Q , t ) ) . intpos i = t . intpos i .= t . intpos i ;
LSeg ( f , i ) misses LSeg ( f , i ) \/ LSeg ( f , j ) ;
assume for x , y being Element of L st x in C & y in C holds x <= y & y in C & x in C & y in C & x in C ;
[' C , f /. x '] = f . ( upper_bound C ) - ( lower_bound C ) ) ;
for F , G being FinSequence st rng F misses rng G & rng F misses rng G holds F ^ G ^ G ^ G ^ F ^ G ^ F ^ G ^ G ^ F ^ G ^ F ^ G ^ G ^ F ^ G ^ F ^ G ^ ( F ^ G ) ^ ( F ^ G ) ^ ( F ^ G ) ^ ( F ^ G ) ^ ( F ^ G ) ^ ( F
||. R /. ( L . h ) - R /. ( L . h ) .|| < e * ( K * ( L . h ) - K * ( L . h ) ) ;
assume a in { q where q is Element of M : dist ( z , q ) <= r } ;
set p3 = [ 2 , 1 , 0 ] .--> [ 2 , 1 , 0 ] ;
consider x , y being Subset of X such that [ x , y ] in F and x in d and y in F and x in d and y in F and x in F and y in F and x in F and y in F ;
for y , x being Element of REAL st y in Y & x in X & y in Y holds x <= y & y <= x & x <= y & x <= y & y <= x
func |. p ^ q .| -> variable equals min ( p , q ) . ( p ^ q ) ;
consider t being Element of S such that x , y '||' z , t and x , z '||' t , t and x , t '||' z , t ;
dom x1 = Seg len ( x1 ^ x2 ) & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x1 = len x2 & len x2 = len x2 & len x1 = len x2 ;
consider y2 being Real such that x2 = y2 and 0 <= y2 and y2 <= 1 and y2 <= 1 and y2 <= 1 / 2 ;
||. f | X - f | X .|| = ||. f /. X - f /. ( X - 1 ) .|| .= ||. f /. X - f /. ( X - 1 ) .|| ;
( the InternalRel of A ) ~ ( x `1 ) = {} .= {} \/ {} .= {} .= {} \/ {} .= {} .= {} ;
assume that i in dom p and for j being Nat st j in dom q holds P [ j , i ] and for j being Nat st j in dom q & j in dom q holds P [ j , i ] ;
reconsider h = f | [: X , Y :] as Function of [: X , Y :] , Y ;
u1 in the carrier of W1 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of W2 & u2 in the carrier of W2 & u1 in the carrier of V implies u1 in the carrier of V
defpred P [ Element of L ] means M <= f . $1 & $1 <= f . $1 & f . $1 <= f . $1 & f . $1 <= f . $1 ;
T . ( u , v ) = s * x + ( - s ) * y .= b * x + ( - s ) * y .= b ;
- ( - ( R1 - R2 ) ) = - x + ( - ( x - y ) ) .= - x + ( - ( x - y ) ) .= - x + ( - ( x - y ) ) .= - x ;
given a being Point of G such that for x being Point of G , a being Point of G , x being Point of G st x in A & a in A holds x in B ;
fA2 = [ dom ( @ f2 ) , cod ( @ g2 ) ] .= [ [ [ f , f2 ] , [ f , f2 ] ] , [ [ f , f2 ] ] ;
for k , n being Nat st k <> 0 & k < n & n < m holds ( n , m ) mod ( n , m ) = 1
for x being element holds x in A |^ d iff x in ( A |^ d ) ` & x in A `
consider u , v being Element of R , a being Element of A such that l /. i = u * v ;
1- sqrt ( ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) > 0 ;
Carrier ( L1 ) . k = L1 . ( F . k ) & F . ( k + 1 ) in dom L1 & F . ( k + 1 ) in dom L2 & F . ( k + 1 ) in dom L2 ;
set i2 = AddTo ( a , i , - n ) , i2 = AddTo ( a , i , - n ) ;
attr B is directed-sups-preserving means : Def1 : -a3 ( B , S ) = ( B , S ) . ( S , T ) ;
a1 " D = { a "/\" d where d is Element of N : d in D & d in D } ;
| ( ( 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 | (
( - f ) . sup A = ( - f ) . sup A .= ( - f ) . sup A .= ( - f ) . sup A ;
( G * ( len G , k ) `1 ) `1 = ( G * ( len G , k ) `1 ) `1 .= ( G * ( 1 , k ) `1 ) `1 .= ( G * ( 1 , k ) `1 ) `1 ;
( Proj ( i , n ) ) . ( ( proj ( i , n ) ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( ( proj ( i , n ) * ( (
f1 + f2 * reproj ( i , x ) is_differentiable_in x0 & f2 * reproj ( i , x ) is_differentiable_in x0 ;
attr the function cos is differentiable means : Def1 : for x st x in Z holds ( ( tan (#) cos ) `| Z ) . x = ( tan . x ) / ( cos . x ) ;
ex t being SortSymbol of S st t = s & h1 . t = h2 . ( x , t ) & t . ( x , t ) = h2 . ( x , t ) ;
defpred C [ Nat ] means ( P [ $1 ] ) & ( ( $1 + 1 ) in A ) & ( ( $1 + 1 ) in A ) & ( ( $1 + 1 ) in A ) & ( ( ( $1 + 1 ) in A ) & ( ( ( $1 + 1 ) in A ) ) & ( ( ( ( ( A ) \/ B ) \/ B ) ) is consistent ) ) ) ;
consider y being element such that y in dom ( p | i ) and q9 . i = ( p | i ) . y and y . i = ( p | i ) . y ;
reconsider L = product ( { x1 } +* ( index ( B ) , l ) ) as Basis of A ;
for c being Element of C , d being Element of D ex e being Element of D st T . ( id c ) = id d & e . ( id d ) = id d
right_cell ( f , n , p ) = ( f | n ) ^ <* p *> .= f | n ^ <* p *> .= f | n ;
( f (#) g ) . x = f . ( g . x ) & ( f (#) g ) . x = f . ( g . x ) ;
p in { |[ 1 , 0 ]| where 1 , j is Nat : G * ( i , j ) `1 < G * ( i + 1 , j ) `2 } ;
f `2 - p = ( - c ) *' ( - ( f , p ) ) .= - ( ( f , p ) *' ) .= - ( ( f , p ) *' ) .= - ( f *' ) *' ( - ( f *' ) ) .= - ( f *' ) *' ( - ( f *' ) ) .= - ( f *' ) ;
consider r being Real such that r in rng ( f | divset ( D , j ) ) and r < m + 1 / 2 ;
f1 . ( |[ ( ( r - ( r - ( 1 - r ) ) / 2 ) , ( ( r - ( 1 - r ) / 2 ) ) / 2 ]| ) in f1 .: ( ( ( 1 - r ) / 2 ) ) ;
eval ( a | n , L ) . x = Ordinal ( a | n , L ) . x .= a . x * ( a | n ) .= a . x * ( a | n ) .= a . x * ( a | n ) .= a . x * ( a . x ) .= a . x * ( a . x ) .= a . x * ( a . x ) ;
z = DigA ( t , x ) .= DigA ( t , x ) .= DigA ( t , x ) .= DigA ( t , x ) .= ( t , x ) . x .= ( t . x ) * ( t . x ) .= ( t . x ) * ( t . x ) .= z . x * ( t . x ) .= z . x * ( t . x ) ;
set H = { ( Intersect S ) . x where S is Subset-Family of X : S in G & x in S } , H = G \ { x } ;
consider S19 being Element of D such that S = [: S , j :] and S `2 = [: S , j :] and [ S , j ] in [: S , { i } :] and [ S , j ] in Indices S and [ S , j ] in Indices S and [ S , j ] in Indices S and [ S , j ] in S and [ S , j ] in S and [ S , j ] in S and S = S and
assume that x1 in dom f and x2 in dom f and f . x1 = f . x2 and f . x2 = f . x2 ;
- 1 <= sqrt ( ( q `1 / |. q .| - sn ) / ( 1 + sn ) ) ^2 ) ;
Carrier ( L ) is Linear_Combination of A & Carrier ( L ) is Linear_Combination of A & Carrier ( L ) = A & Carrier ( L ) c= A & Carrier ( L ) c= A ;
let k1 , k2 , k1 , k2 be Nat , a , b , c be Int-Location , k1 , k2 be Int-Location , k2 be Int-Location ;
consider j being element such that j in dom a and j in dom g and x = g " ( { k } ) and x = a " . j ;
H1 . ( x1 , x2 ) c= H1 . ( x2 , x3 ) or H1 . ( x2 , x3 ) c= H1 . ( x2 , x3 ) & H1 . ( x1 , x2 ) c= H1 . ( x2 , x3 ) ;
consider a being Real such that p = e * p1 + ( a * p2 ) and 0 <= a and a <= 1 and a <= 1 and b <= 1 and a <= 1 ;
assume that a <= c and c <= b and [ a , b ] in dom f and [ b , c ] in dom g and [ a , b ] in dom g and g . a = g . b ;
cell ( Gauge ( C , m ) , m -' 1 , G ) -' 1 , G ) is non empty ;
A5 in { ( S . i ) `1 where i is Element of NAT : i in { ( S . i ) `1 } ;
( T * b1 ) . y = L * ( b * b1 ) .= ( F * b1 ) . y .= ( F * b2 ) . y .= ( F * b2 ) . y ;
g . ( s , I ) . x = s . y & g . ( s , I ) . y = |. s . x .| ;
( log ( 2 , k ) ) / ( 2 to_power ( k + 1 ) ) >= ( log ( 2 , ( 2 to_power ( k + 1 ) ) / ( 2 to_power ( k + 1 ) ) ) / ( 2 to_power ( k + 1 ) ) ) ;
then that p => q in S and not x in the still of S and not p in the \frac of S & not p in the \frac { x } , S & not p in S & p in S & q in S & not p in S & p in S & q in S & p => q in S & p => q in S & p => q in S ;
dom ( the InitS of rr ) misses dom ( the InitS of ( rr ) ) & dom ( the InitS of ( ( the InitS of ( ( the InitS of ( ( the InitS of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( the Sorts of ( ( (
synonym f is extended real-valued means for for x , y being set st x in rng f & y in rng f & x in rng f holds x = y ;
assume for a being Element of D holds f . { a } = a & for X being Subset of D holds f .: ( X \/ { a } ) = f .: ( X \/ { a } ) ;
i = len p1 .= len p2 .= len p1 + len <* x *> .= len p1 + len <* x *> .= len p2 + 1 .= len p1 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 + 1 .= len p2 ;
( l - 1 ) * ( ( 1 - l ) * ( 1 - l ) ) = ( g - l ) * ( ( 1 - l ) * ( 1 - l ) ) + ( ( 1 - l ) * ( 1 - l ) ) .= ( ( 1 - l ) * ( 1 - l ) ) * ( 1 - l ) ;
CurInstr ( P2 , Comput ( P2 , s2 , l ) ) = halt SCM+FSA .= halt SCM+FSA .= CurInstr ( P2 , Comput ( P2 , s2 , l ) ) .= halt SCM+FSA ;
assume for n be Nat holds ||. ( seq . n ) - ( lim seq ) .|| <= ( |. seq .| ) . n & ( |. seq . n - ( lim seq ) ) .| < e ;
sin . ( cos . ( cos . ( 2 * PI ) ) ) = sin . ( cos . ( 2 * PI ) ) .= 0 ;
set q = |[ g1 `1 , g2 `2 ]| , r = |[ g1 `1 , g2 `2 ]| , s = |[ g1 `1 , g2 `2 ]| , t = |[ g2 `1 , t `2 ]| ;
consider G be sequence of S such that for n be Element of NAT holds G . n in func WSet ( F . n ) ;
consider G such that F = G and ex G1 , G2 st G1 in SX & G2 in SX & G1 in ( the carrier of X ) & G2 = ( the carrier of X ) \/ ( the carrier of Y ) ;
the root of ( [ x , s ] -tree ( [ x , s ] , [ x , s ] ) ) in ( the Sorts of Free ( C , X ) ) . s ;
Z c= dom ( ( exp_R / ( 3 / ( 4 * ( f1 + f2 ) ) ) ) ) ;
for k be Element of NAT holds ( r . k ) = ( upper_volume ( f , S ) ) . k
assume that - 1 < n and - 1 < n and n < 1 and n <= len ( - 1 ) and ( - 1 ) * ( n + 1 ) < ( - 1 ) * ( n + 1 ) ;
assume that f is continuous and a < b and c < d and f . a = c and f . b = d and f . c = d and f . d = c ;
consider r being Element of NAT such that s2 = Comput ( P1 , s1 , i ) and r <= q and r <= q ;
LE f /. ( i + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) , f /. ( j + 1 ) ) ;
assume that x in the carrier of K and y in the carrier of K and inf { x , y } in the carrier of K and x in the carrier of K and y in the carrier of K and x in the carrier of K and y in the carrier of K and x in the carrier of K ;
assume f +* ( i1 , \xi ) in ( proj ( F , ( F . ( i1 , i2 ) ) ) ) " ( ( proj ( F , ( F . ( i1 , i2 ) ) ) " ) " { f . ( ( F . ( i1 , i2 ) ) " ) ;
rng ( ( Flow M ) ~ ) c= the carrier' of M & rng ( ( the InternalRel of M ) * the InternalRel of M ) c= the carrier' of M & rng ( ( the InternalRel of M ) * the InternalRel of M ) c= the carrier' of M ;
assume z in { ( the carrier of G ) \times { t } where t is Element of T : t in { t } } ;
consider l be Nat such that for m be Nat st l <= m holds ||. s1 . m - x0 .|| < g / 2 ;
consider t be VECTOR of product G such that [: t , t :] = ||. ( D . t ) - ( D . t ) .|| and ||. t .|| <= 1 ;
assume that the topology of v = 2 and v ^ <* 0 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p and v ^ <* 1 *> in dom p ;
consider a being Element of the points of X1 , X such that a on A and not a on A and a on A ;
( - x ) |^ ( k + 1 ) * ( ( - x ) |^ ( k + 1 ) ) " = 1 / ( ( - x ) |^ ( k + 1 ) ) " ;
for D being set st for i st i in dom p holds p . i in D holds p . i in D . i
defpred R [ element ] means ex x , y st [ x , y ] = $1 & [ y , x ] in $1 & [ x , y ] in $1 & [ y , x ] in $1 ;
L~ f2 = union { LSeg ( p1 , p2 ) , LSeg ( p2 , p3 ) } .= { LSeg ( p1 , p2 ) , LSeg ( p2 , p3 ) } ;
i -' len ( h1 - 2 ) + 1 - 1 < i - ( len h1 - 2 ) + 1 - 1 + 1 - 1 ;
for n be Element of NAT st n in dom F holds F . n = |. ( F . n ) . x .|
for r , s1 , s2 being Real holds r in [. s1 , s2 .] & s1 in [. r , t .] & s1 <= s2 & s2 <= t implies [. s1 , t .] c= [. r , t .]
assume that v in { G where G is Subset of T2 : G in B & G c= B1 & G c= B2 & B c= B1 & B c= B2 } ;
let g be INT-Expression of A , INT , X be set , b be Element of Z , c be Element of Z , d be Element of Z , b be Element of Z , b be Element of Z , a , b be Element of Z , c be Element of Z , d be Element of Z , b be Element of Z , e be Element of Z ;
min ( g . [ x , z ] , k ) . [ y , z ] = ( min ( g . [ y , z ] , k ) ) . [ y , z ] ;
consider q1 be Real_Sequence such that for n holds P [ n , q1 . n ] ;
consider f being Function such that dom f = NAT and for n being Element of NAT holds f . n = F ( n ) and f . n = F ( n ) ;
reconsider B-6 = B /\ B , Z = O /\ Z , Z = O as Subset of B ;
consider j being Element of NAT such that x = the {} of K and 1 <= j and j <= len f and 1 <= j & j <= len f and 1 <= j & j <= len f and 1 <= i & i <= len f and f /. j = f /. i ;
consider x such that z = x and card ( x . O ) in card ( x . O ) and x in ( x . O ) & x in ( x . O ) & x in ( x . O ) ;
( C * _ { n , n2 } ) . 0 = C ( ( ( <* k , n2 *> ) . 0 ) ) . 0 .= C ( ( ( ( n , n2 ) . 0 ) ) . 0 ) ;
dom ( X --> rng f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X & dom ( X --> f ) = X ;
N-bound L~ Cage ( C , n ) <= ( N-bound L~ Cage ( C , n ) ) / 2 & N-bound L~ Cage ( C , n ) <= ( N-bound L~ Cage ( C , n ) ) / 2 ;
synonym x , y means : Def1 : { x , y } = y & { x , y } c= { x , y } ;
consider X being element such that X in dom ( f | ( n + 1 ) ) and ( f | ( n + 1 ) ) . X = Y ;
assume that Im k is continuous and for x , y being Element of L st x = x & y = y holds x << y iff x << y & y << x & x << y ;
sqrt ( 1 / 2 * ( ( - 1 ) * ( ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) ) ) ) is_differentiable_on REAL ;
defpred P [ Element of omega ] means ( the partial of A1 ) . $1 = A1 . $1 & ( the partial of A2 ) . $1 = A2 . $1 & ( the partial of A1 ) . $1 = ( the partial of A2 ) . $1 ;
IC Comput ( P , s , 2 ) = succ IC Comput ( P , s , 2 ) .= succ IC Comput ( P , s , 2 ) .= 6 .= 6 ;
f . x = f . ( g1 * f ) .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x .= ( g1 * f ) . x ;
( M * ( F . n ) ) . n = M . ( ( F . n ) . n ) .= M . ( ( ( ( P * ( ( P * ( F * ( ( P * ( F * ( F * ( F * ( ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F * ( F *
the support of L1 + L2 c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the support of L1 \/ ( the support of L2 ) c= ( the carrier of L2 ) \/ ( the carrier of L2 ) ;
attr a , b , c is_collinear means : Def1 : for x , y being Element of X st x , y \rbrace in R & x , y are_orthogonal w.r.t. R & x , y are_orthogonal w.r.t. R & x , y are_elements of R & x , y are_w.r.t. R ;
( the PartFunc of product s ) . n <= ( the PartFunc of product s ) . n & ( the PartFunc of product s ) . n <= ( the PartFunc of product ( G , n ) ) . ( ( the Sorts of A ) . n ) ;
attr 1 <= r & r <= 1 & r <= 1 implies ( ( 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 )
s2 in { p ^ <* n *> where n , p is Nat : p ^ <* n *> in T1 & p ^ <* n *> in T1 } ;
|[ x1 , x2 , x3 ]| . 2 - |[ x1 , x2 ]| . 2 = x2 . 1 - x3 . 2 .= x3 . ( 1 - x1 ) .= x2 . ( 1 - x1 ) ;
attr F . m is nonnegative means : Def2 : F . m is nonnegative & F . m is nonnegative ;
len ( ( G . z ) * ( ( G . y ) * ( ( G . y ) * ( ( G . y ) * ( ( G . z ) * ( ( G . y ) * ( ( G . y ) * ( ( G . z ) * ( ( G . y ) * ( ( G . y ) * ( ( G . y ) * ( ( G . y ) * ( ( G . z ) * ( ( G . y ) * ( (
consider u , v being VECTOR of V such that x = u + v and u in W1 and v in W2 and v in W2 and u in W2 and v in W2 and v in W2 and v in W1 and u in W2 and v in W2 ;
given F be FinSequence of NAT such that F = x and dom F = n and rng F = { 0 , 1 } and Sum F = 0 and Sum ( F ) = 1 and Sum ( F ) = 0 ;
0 = L1 * cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos cos
consider n be Nat such that for m be Nat st n <= m holds |. ( f # x ) . m - ( f # x ) . n .| < e ;
cluster non empty there exists \hbox { $ ( 1 / 2 ) ) st c is Boolean & ( { ( ( ( { c } ) ) } ) , ( { ( { c } ) } ) is Boolean & ( { ( ( { c } ) ) } ) is Boolean & ( { ( ( { c } ) ) } ) <> empty ) ;
"/\" ( B , L ) = "/\" ( ( B , L ) "/\" ( B , L ) ) .= "/\" ( ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "/\" C ) ) ) ) ) ) ) .= "/\" ( ( B "/\" ( B "/\" ( B "/\" ( B "/\" ( B "\/" ( B "\/" ( B "\/" ( B "\/" C ) ) ) ) ) ) .= "/\" ( ( ( B
sqrt ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) <= sqrt ( ( r ^2 + ( r ^2 ) ) ^2 ) + sqrt ( ( r ^2 + ( r ^2 ) ^2 ) ) ;
for x being element st x in A /\ dom ( f `| X ) holds ( f `| X ) . x >= r2
2 * ( r1 - c ) * ( 2 * ( r1 - c ) ) = 0. TOP-REAL 2 * ( r1 - c ) * ( r1 - c ) ) ;
reconsider p = P /. 1 , q = P " * ( ( - 1 ) * ( ( - 1 ) * ( 1 / 2 ) ) ) as FinSequence of K ;
consider x1 , x2 being element such that x1 in uparrow s and x2 in uparrow t and x = [ x1 , x2 ] and x = [ x1 , x2 ] ;
for n be Nat st 1 <= n & n <= len q1 holds q1 . n = ( upper_volume ( g , M ) ) . n
consider y , z being element such that y in the carrier of A and z in the carrier of A and y in the carrier of A and i = [ y , z ] ;
given H1 , H2 being strict Subgroup of G such that x = H1 & y = H2 and H1 = H2 and H2 = H1 & H2 = H2 and H1 = H2 and H2 = H1 and H2 = H2 ;
for S , T being non empty RelStr , d being Function of T , S st d is complete & d is directed-sups-preserving holds d is monotone & d is monotone & d is monotone & d is monotone
[ a + i , b ] in ( the carrier of V ) & [ b , c ] in ( the carrier of V ) & [ b , c ] in the carrier of V & [ a , b ] in the carrier of V & [ b , c ] in the carrier of V & [ b , c ] in the InternalRel of V & [ b , c ] in the InternalRel of V & [ b , c ] in the InternalRel of V & [ b , d ] in the InternalRel of V & [ b , d ] in the InternalRel of V & [ b ,
reconsider mF = max ( ( len ( F . n ) * ( p . n ) ) , x ) as Element of NAT ;
I <= width GoB ( GoB ( GoB f , i1 ) , width GoB f ) & I = ( GoB f ) * ( len GoB f , 1 ) `2 ;
f2 /* q = ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 /* ( f1 /* s ) ) ^\ k .= ( f2 /* ( f1 /* s ) ) ^\ k ;
attr A1 \/ A2 is linearly-independent means : Def1 : for A , B being Subset of V st A misses B & B misses A & A c= B holds A /\ B = B & A /\ B = B /\ A & B /\ B = B /\ A ;
func A -NAT -> set equals union { A ( s ) where s is Element of ( the carrier of R ) * : s in A ( ) } ;
dom ( Line ( v , i + 1 ) ) (#) ( ( Line ( v , i ) ) (#) ( ( ( ( v ^ ( p , i ) ) (#) ( ( v ^ ( q , i ) ) (#) ( ( v ^ ( p , i ) ) (#) ( ( v ^ ( q , i ) ) (#) ( ( v ^ ( p , i ) ) ) ) ) ) = dom ( ( v ^ ( q , i ) ) ) ;
cluster [ x , y ] , [ x , y ] ] -> LSeg ( x , y ) & [ x , y ] in [: { x , y } , { y } :] ;
E , All ( x2 , All ( x2 , x1 , x2 ) ) |= All ( x2 , x2 , x3 ) => All ( x2 , x3 , x4 ) ;
F .: ( id X , g ) . x = F . ( id X , g ) . x .= F . ( id X , g ) . x .= F . ( g . x , g . x ) .= F . ( g . x , g . x ) .= F . ( g . x , g . x ) ;
R . ( h . m ) = F . ( x0 + h . m ) - ( g . m ) ) ;
cell ( G , ( X -' 1 , t ) , ( Y -' 1 ) ) \ ( ( X -' 1 ) \ ( X -' 1 ) ) meets ( ( ( X -' 1 ) \ ( X -' 1 ) ) \ ( ( X -' 1 ) \ ( X -' 1 ) ) ) ;
IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) = IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= ( IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= ( IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) ) .= IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= ( IC Comput ( P2 , s2 , LifeSpan ( P2 , s2 , LifeSpan ( P2 , s2 ) ) .= IC Comput ( P2 ,
sqrt ( 1 - ( ( ( - ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) ) ^2 ) > 0 ;
consider x0 being element such that x0 in dom a and x0 in g and x0 in dom g and g . x0 = a . ( g . x0 ) and g . x0 = a . ( g . x0 ) ;
dom ( r1 (#) ( f , A ) ) = dom ( ( f , A ) * ( g , A ) ) .= dom ( ( f , A ) * ( g , A ) ) .= A ;
d1 . [ y , z ] = ( ( y - 1 ) * z ) `1 .= ( ( y - 1 ) * z ) `1 .= ( ( y - 1 ) * z ) `1 ;
attr for i being Nat holds C . i = A . i /\ B . i & C . i c= B . i /\ B . i ;
assume that x0 in dom f and f is continuous and for x st x in dom f holds ||. f /. x - f /. x0 .|| <= r ;
p in Cl A implies for K being Basis of p st K in K & for Q being Basis of T st Q in K & Q c= K holds A meets Q
for x be Element of REAL n st x in Line ( x1 , x2 ) holds |. ( x1 - x2 ) - ( x2 - x3 ) .| <= |. ( x1 - x2 ) - ( x2 - x3 ) .|
func Sum <* a *> -> Ordinal means : Def1 : a in it & for b being Ordinal st a in it holds it . b = a & it . b = b ;
[ a1 , a2 ] in ( the InternalRel of A ) \/ ( the InternalRel of A ) & [ a2 , a3 ] in ( the InternalRel of A ) \/ ( the InternalRel of A ) ;
ex a , b being element st a in the carrier of S1 & b in the carrier of S2 & x = [ a , b ] & [ a , b ] in the InternalRel of S1 & [ b , a ] in the InternalRel of S1 & [ b , a ] in the InternalRel of S2 & [ b , a ] in the InternalRel of S1 & [ b , a ] in the InternalRel of S1 ;
||. ( v . n ) - ( v . m ) .|| * ||. x - y .|| < e * ||. x - y .|| ;
then for Z being set st Z in { Y where Y is Element of I : F ( Z ) c= Z } holds z in Z & z in Z & z in Z & z in Z } ;
sup ( { s , t } ) = [ sup { s , t } , sup { s , t } ] .= sup { s , t } ;
consider i , j being Element of NAT such that i < j and [ y , f . i ] in [: I , J :] and [ f . i , f . j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [: I , J :] and [ i , j ] in [:
for D being non empty set , p , q being FinSequence st p c= q & p ^ q in D & p ^ q in D holds p ^ q in D
consider e1 being Element of the carrier of X such that c , e1 // e2 , e2 and a , e // e1 , e2 and a , e // e1 , e2 and a , e // e1 , e2 and c , e // e2 , e2 and a , e // e1 , e2 and a , e // e1 , e2 ;
set U = I \! \mathop { x } , C = I \! \mathop { y } , F = I \! \mathop { x } , T = I -\mathop { y } ;
|. q1 .| ^2 = ( ( ( - 1 ) / ( ( ( 1 - ( q `2 / |. q .| - sn ) ) / ( 1 - sn ) ) ^2 ) / ( 1 - sn ) ) ^2 ) .= ( ( ( - 1 ) / ( ( 1 - sn ) / ( 1 - sn ) ) ^2 ) / ( 1 - sn ) ^2 ) ^2 .= ( ( ( ( ( 1 - sn ) / ( 1 - sn ) ) ^2 ) / ( 1 - sn ) ) ^2 ) ^2 / ( 1 - sn ) ) ^2 ) ^2 ) ^2 .= ( ( ( ( ( 1 - sn ) ) ^2 )
for T being non empty TopSpace , x , y being Element of [: the topology of T , { x } :] st x "\/" y = x & x "\/" y = y holds x "\/" y = x "\/" ( y "\/" ( x "\/" y ) )
dom ( ( the charact of U1 ) * the Arity of U2 ) = dom ( the Arity of U1 ) & dom ( the Arity of ( U1 ) * the Arity of U2 ) = dom ( the Arity of ( U1 ) * the Arity of U2 ) & dom ( the Arity of ( U1 ) * the Arity of U2 ) = dom ( the Arity of ( U1 ) * the Arity of U2 ) ;
dom ( h | X ) = dom h /\ X .= dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h | X ) .= X /\ dom ( h
for N1 , N2 being Element of G , K st dom ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N & rng ( h . N1 ) = N ;
( mod ( u , m ) ) . i = ( mod ( v , m ) ) . i + ( mod ( v , m ) ) . i .= ( mod ( v , m ) ) . i ;
- ( q `1 / |. q .| - cn ) < - 1 & - ( q `2 / |. q .| - cn ) < - 1 & - ( q `2 / |. q .| - cn ) < 1 & - ( q `2 / |. q .| - cn ) < 1 & - ( q `2 / |. q .| - cn ) < 1 ;
attr r1 = f1 & r2 = f2 . ( len f1 ) , r2 = f2 . ( len f1 ) , s2 = f1 . ( len f2 ) , s1 = f2 . ( len f1 ) , s2 = f2 . ( len f2 ) , s2 = f2 . ( len f1 ) , s2 = f2 . ( len f2 ) ;
v-4 . m is bounded Function of X , the carrier of Y & for x be Element of X , Y st x in the carrier of ( ( ( the carrier of ( X . m ) ) \/ { x } ) holds x9 . x = ( ( the Sorts of ( X . m ) ) . x ) . x ;
attr a <> b & c <> c & a , b , c is_collinear & not ex a , b st angle ( b , c , d ) = 0 & angle ( a , b , c ) = 0 & angle ( a , b , c ) = 0 & angle ( a , c , d ) = 0 ;
consider i , j being Nat such that p1 = [ i , j ] and i = [ i , j ] and j = [ i , j ] and i = j and j = k and i = k and j = k and j = k ;
|. p .| ^2 + ( 2 * ( p `2 / |. p .| - sn ) ) ^2 = |. p .| ^2 + ( 2 * ( p `2 / |. p .| - sn ) ) ^2 ;
consider p1 , p2 being FinSequence of [: X , Y :] such that y = p1 ^ p2 and p1 ^ p2 = p2 ^ p3 and p2 ^ p3 = p3 ^ p4 and p1 ^ p2 ^ p3 ^ p4 ^ p4 ^ p4 ^ p4 ( X , Y ) ^ p2 ^ p3 ^ p4 ^ p4 ^ p4 ^ p4 ^ p4 ^ Consider p1 ^ p2 , p2 ^ p3 being FinSequence such that y = p1 ^ p2 ^ p2 ^ p3 ^ p4 ^ p4 ( X , Y ) ;
gcd ( A , r1 , r2 ) = sqrt ( ( 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 /
( proj2 .: A ) `2 = lower_bound ( proj2 .: ( ( proj2 .: A ) /\ ( proj2 .: ( ( proj2 .: A ) /\ ( ( proj2 .: ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( proj2 .: A ) /\ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) )
s , ( k |= H1 '&' H2 ) iff s |= H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 implies s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |= H2 & s |= H1 '&' H2 & s |= H1 '&' H2 & s |=
len ( s + 1 ) = card ( support ( b1 + b2 ) ) .= ( support ( s ) ) + ( support ( b1 + b2 ) ) .= ( support ( s ) ) \/ ( support ( b1 + b2 ) ) .= ( support ( s ) ) \/ ( support ( b1 + b2 ) ) .= ( support ( s ) ) \/ { b1 + b2 } .= { b1 + b2 ) .= ( support ( s ) ) \/ ( support ( b1 ) .= ( support ( s ) .= ( support ( s ) .= ( support ( s ) ) \/ ( support ( s ) ) .= ( support ( s )
consider z being Element of L1 such that z >= x and z >= y and for z being Element of L1 st z >= y & z >= x holds z >= y ;
LSeg ( UMP D , |[ ( E-bound D ) / 2 , ( N-bound D ) / 2 ) / 2 ) /\ D = { ( UMP D ) / 2 , ( N-bound D ) / 2 } /\ D .= { ( sup D ) / 2 } ;
lim ( ( ( f `| N ) /* b ) /* c ) = lim ( ( f `| N ) /* c ) .= lim ( ( f `| N ) /* c ) .= lim ( ( f `| N ) /* c ) ;
P [ i , pr1 ( f , i ) , pr1 ( f , i ) ] means pr1 ( f , i ) = pr1 ( f , i ) ;
for r be Real st 0 < r ex m be Nat st for k be Nat st m <= k holds ||. ( seq . k ) - ( lim ( seq . k ) ) .|| < r
for X being set , P being a_partition of X , x being set st x in P & P [ x ] holds P [ x ]
Z c= dom ( ( ( ( id Z ) (#) ( ( exp_R * f ) ) `| Z ) ) & f | Z is continuous ;
ex j being Nat st j in dom ( l ^ <* x *> ) & j in dom ( l ^ <* x *> ) & z = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j & z = ( l ^ <* x *> ) . j ;
for u , v being VECTOR of V st 0 < r & u in N & v in N holds r * u + ( r * v ) in N
A , Int A \/ Int A , Int A / B / B / B / B , B / A / B / A .] \/ B / B , C / A / B / B / A / B , D / B / A / B , D / B / A / B , D / B , E / A / B , E / B , E / A / B , E / B , D / A / A , E / B , A / B , E / A / A / A / A / B , E / B , E / B / A , E / B , F / B , F / A
- Sum ( <* v , u *> , w ) = - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= - ( v + u ) .= ( v + u ) + ( v + u ) .= ( v + u ) ;
( Exec ( a := b , s ) ) . IC SCM R = ( Exec ( i := b , s ) ) . IC SCM R .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC s .= succ IC t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= succ t .= t . IC t .= t . IC t .= t . IC t .= t .
consider h being Function such that f . a = h and dom h = I and for x being element st x in I holds h . x = ( the support of J ) . x ;
for S1 , S2 being non empty reflexive RelStr , D being non empty Subset of S1 , f being Function of S1 , S2 st f = ( the carrier of S1 ) \/ ( the carrier of S2 ) & f is directed holds f is directed & f is directed & f is directed
card X = 2 implies ex x st x in X & ex x st x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X & x in X
E-max L~ Cage ( C , n ) in rng ( Cage ( C , n ) ) & E-max L~ Cage ( C , n ) in rng Cage ( C , n ) ;
for T , T being decorated Tree , p , q being Element of dom T st p -tree q = q holds ( T -tree p ) . ( p ^ q ) = T . ( p ^ q )
[ i2 + 1 , j2 ] in Indices G & [ i2 + 1 , j2 ] in Indices G & f /. k = G * ( i2 , j2 ) & f /. k = G * ( i2 , j2 ) & f /. k = G * ( i2 , j2 ) ;
cluster -> prime means : Def1 : k divides ( n , n ) & k divides ( n , m ) & ( for m being Nat st m divides n holds k divides m ) & ( n divides m ) & ( n divides m ) & ( n divides m ) implies n divides m ) & ( n divides m ) & n divides m ) ) ;
dom F " = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F = the carrier of X1 & rng F = the carrier of X2 & rng F = the carrier of X2 & rng F =
consider C being finite Subset of V such that C c= A and card C = n and card C = n and card C = n and card C = n + 1 and k in dom A and A c= A and A c= B and B c= C and A c= B and B c= C and A c= C and B c= C ;
V is prime implies for X , Y being Subset of [: the topology of T , { 0 } :] st X /\ Y c= V & X c= Y & Y is open & X is open & Y is open & Y is open & X is open & Y is open & Y is open & X is open & Y is open & X is open & Y is open & Y is open & X is open & Y is open & Y is open & X is open & Y is open & Z c= V & Y is open & Z c= V & Z c= V & Z c= V & Z c= V & Z c= V & Z c= V &
set X = { F ( v1 ) where v1 , v2 is Element of B : P [ v1 , v2 ] } , Y = { F ( v1 , v2 ) where v1 , v2 is Element of B : P [ v1 , v2 ] } , Z = { F ( v1 , v2 ) where v1 , v2 is Element of A : P [ v1 , v2 ] } ;
angle ( p1 , p3 , p2 ) = 0 .= angle ( p2 , p3 , p2 ) .= angle ( p3 , p2 , p3 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) .= angle ( p2 , p3 , p2 ) ;
- sqrt ( 1 - ( ( q `2 / |. q .| - sn ) / ( 1 - sn ) ) ^2 ) = - ( ( q `2 / |. q .| - sn ) ) ^2 / ( 1 - sn ) ^2 .= - ( ( q `2 / |. q .| - sn ) ) ^2 / ( 1 - sn ) ^2 ) .= - ( ( q `2 / |. q .| - sn ) ^2 ) .= - ( ( q `2 / |. q .| - sn ) ^2 / ( 1 - sn ) ^2 ) .= ( ( q `2 / |. q .| - sn ) ^2 ) / ( 1 - sn ) ^2 ) / (
ex f being Function of I[01] , TOP-REAL 2 st f is continuous one-to-one & rng f = P & f . 0 = p1 & f . 1 = p2 & f . 1 = p3 & f . 0 = p4 & f . 1 = p4 & f . 1 = p4 & f . 1 = p3 & f . 1 = p4 & f . 1 = p4 ;
attr f is PartFunc means : Def1 : SVF1 ( 2 , f , u ) is continuous & SVF1 ( 2 , f , u ) is continuous ;
ex r , s st x = |[ r , s ]| & ( G * ( len G , 1 ) `1 <= r & r < G * ( 1 , 1 ) `2 ) & s < G * ( 1 , 1 ) `2 & s < G * ( 1 , 1 ) `2 } c= { G * ( 1 , 1 ) `2 }
assume that f is special and 1 <= t and t <= len G and t <= width G and G * ( t , width G ) `2 >= G * ( t , width G ) `2 and t <= G * ( t , width G ) `2 and t <= G * ( t , width G ) `2 and t <= G * ( t , width G ) `2 ;
attr i in dom G means : Def1 : r * ( reproj ( i , x ) ) = r * ( reproj ( i , x ) ) ;
consider c1 , c2 being bag of o1 such that ( ex c being bag of o1 st c /. k = <* c1 , c2 *> ) & c /. k = <* c1 , c2 *> & c /. k = c1 /. ( c + 1 ) & c /. ( c + 1 ) = c2 /. ( c + 1 ) & c /. ( c + 1 ) = c1 /. ( c + 1 ) ;
y0 in { |[ r1 , s1 ]| : r1 < s1 & s1 < s1 & s1 < s2 } & s1 in { |[ r1 , s1 ]| : s1 < s2 } ;
Cl X ^ Y = the carrier of X .= ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ ( ( C ^ Y ) . ( ( C ^ Y ) . ( ( C ^ ( ( C ^ ( C ^ ( ( C ^ Y
attr M1 = len M2 & width M1 = width M1 & width M1 = width M2 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & width M1 = width M1 & M1 = M1 & M1 = M2 implies M1 = M2 ;
consider g2 be Real such that 0 < g2 and { y where y is Point of S : ||. y - x0 .|| < g2 } c= N2 and ||. y - x0 .|| < g2 & ||. y - x0 .|| < g2 & ||. y - x0 .|| < g2 } c= N ;
assume x < sqrt ( - b * c ) + sqrt ( 2 * a ) or x > - sqrt ( 2 * a ) ;
( G1 '&' G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( <* 3 *> ^ ( G1 ^ G2 ) ) . i & ( G1 ^ G2 ) . i = ( ( G1 ^ G2 ) . i ) . i & ( G1 ^ G2 ) . i = ( G1 ^ G2 ) . i ;
for i , j st [ i , j ] in Indices M1 & [ i , j ] in Indices M1 holds ( M1 * M2 ) * ( i , j ) < M1 * ( i , j )
for f being FinSequence of NAT , i being Element of NAT st i in dom f & i in dom f & f /. i = f /. ( i + 1 ) holds f /. ( i + 1 ) = f /. ( i + 1 ) & f /. ( i + 1 ) = f /. ( i + 1 )
assume that F = { [ a , b ] where a , b is set , c is set : a in B & b in B & c in A & c in A } and a in B and b in A and c in A and c in B and a in A and b in B and c in A and c in B and a , b // c , d ;
b2 * ( q2 - q1 ) + ( - ( q2 - q2 ) ) * ( q2 - q1 ) + ( - ( q2 - q2 ) ) * ( q2 - q1 ) + ( - ( q2 - q2 ) * ( q2 - q1 ) ) = 0. TOP-REAL n + ( - ( q2 - q2 ) * ( q2 - q1 ) ) ;
Cl ( F ) = { D where D is Subset of T : ex B being Subset of T st B = Cl ( B ) & B is open & B is closed & B is closed & A c= B } ;
attr IT is summable means : Def1 : for s being Real_Sequence holds s is summable & s is summable & s is summable & s is summable & s is summable & for n be Nat holds s . n = ( s . n ) * ( s . n ) ;
dom ( ( cn " ) | D ) = ( the carrier of ( TOP-REAL 2 ) ) /\ D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D .= D ;
[ X \to Z ] is full full SubRelStr of ( X |^ ( [#] Y ) ) |^ the carrier of Z & [ X , Z ] is full SubRelStr of ( X |^ ( [#] Y ) ) |^ the carrier of Z ;
( G * ( 1 , j ) ) `2 = ( G * ( 1 , j ) ) `2 & ( G * ( 1 , j ) ) `2 <= ( G * ( 1 , j ) `2 ) `2 ;
pred m1 c= m2 means for p , q being set st p in P & q in P & p in P & q in P holds p <= q & q <= p & p <= q & p <= q & p <= q & q in P & p in P & q in P & p in Q & q in Q & p in Q & q in Q & p in Q & q in Q & p in Q implies p = q ;
consider a being Element of B ( ) such that x = F ( a ) and a in { G ( ) where b is Element of B ( ) : P [ b ] } and P [ a , b ] ;
We say that the multiplicative empty empty loop over R means : Def1 : the carrier of it = { the carrier of R , the carrier of R } & the carrier of it = { the carrier of R } ;
ConsecutiveSet2 ( a , b , 1 ) + Morphism ( c , d ) = b + c .= b + c .= b + d .= b + d .= c + d .= b + d .= d + d .= d + c + d .= d + d + c .= d + d + c + d .= d + d + d .= d + c + d .= d + d ;
cluster strict for Function of INT , INT means for Function of INT , INT , i , j being Element of INT holds it . ( i , j ) = i + j & i in dom ( i , j ) & j in dom ( i , j ) implies ( i , j ) = j + ( j , i )
1- ( ( 2 * ( 1 - r ) ) / 2 ) * ( ( 2 * ( 1 - r ) ) / 2 ) = ( ( 2 * ( 1 - r ) ) / 2 * ( ( 1 - r ) / 2 ) ;
eval ( ( a | n ) *' , x ) = eval ( a | n , x ) * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * eval ( p , x ) .= a * ( p , x ) ;
assume that the TopStruct of S = the TopStruct of T and for D being non empty Subset of S st D is non empty & D is non empty holds D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open & D is open ;
assume that 1 <= k and k <= len w + 1 and k <= len ( ( ( q , w ) `2 ) / 2 ) and ( ( q , w ) / 2 ) * ( ( q , w ) / 2 ) = ( ( ( q , w ) / 2 ) * ( ( q , w ) / 2 ) ) * ( ( q , w ) / 2 ) ) * ( ( ( q , w ) / 2 ) ) * ( ( ( ( q , w ) / 2 ) ) ;
2 * ( n + 1 ) + ( 2 * ( n + 1 ) ) >= ( 2 * ( n + 1 ) + ( 2 * ( n + 1 ) ) + ( 2 * ( n + 1 ) ) + ( 2 * ( n + 1 ) ) + ( 2 * ( n + 1 ) ) + ( 2 * ( n + 1 ) ) ) ;
M , v / ( ( x. 3 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) = ( x. 4 , m ) ;
assume that f is_differentiable_on l and for x0 st x0 in l holds 0 < g . x0 and for x1 st x1 in dom f & x1 in dom f & x0 < x1 & x1 in dom f & f /. x1 in dom f & f /. x0 in dom f ;
for G1 being _Graph , W being Walk of G1 , e being Vertex of G2 , e being Vertex of G , e being Vertex of G st e in W & e in W holds e in ( the \frac of G ) . e & e in ( the \frac e ) . e holds e in ( the \frac e ) . e
not c1 is not empty iff ( not ( ex y1 , y2 st y1 is not empty & not ( ex y1 st y1 is not empty & y1 is not empty & y1 is not empty & not y1 is not empty & y1 is not empty & not y1 is not empty & not y1 is not empty & not y1 is not empty & not y1 is not empty & not y1 is not empty ) & not y1 is not empty & not not y1 is not empty & not q2 is not empty & not ( not ( ex q2 is not empty & not ( ex q2 is not empty & not ( ex q2 st q2 is not empty & not ( not ( not ( ex q2 is not empty & not ( ( ex x1 is not empty & not ( ex x1 is not empty & not ( x1 is
Indices GoB f = [: Seg ( len GoB f , width GoB f ) , Seg ( width GoB f ) :] & f /. len f = ( GoB f ) * ( len GoB f , width GoB f ) & f /. len f = ( GoB f ) * ( len f , width GoB f ) & f /. len f = ( GoB f ) * ( len f , width GoB f ) ;
for G1 , G2 being strict Subgroup of O , G , H being Subgroup of O st G1 is Subgroup & G2 is Subgroup of O & H is Subgroup of O & H is Subgroup of O & H is Subgroup & H is Subgroup of G holds H is Subgroup of G & H is Subgroup of H & H is Subgroup of G & H is Subgroup of G & H is Subgroup of G & H is Subgroup & H is Subgroup & H is Subgroup & G is Subgroup & H is Subgroup & H is Subgroup & G is Subgroup & G is Subgroup of G & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup & G is Subgroup
UsedIntLoc ( ( intloc 0 ) := f ) = { ( intloc 0 ) .--> ( ( intloc 0 ) .--> ( ( intloc 0 ) .--> ( ( intloc 0 ) .--> 1 ) ) ) , ( ( intloc 0 ) .--> 1 ) ;
for f1 , f2 being FinSequence of F st f1 ^ f2 is p -element & f1 ^ f2 is ( len f1 ) -element & ( for i being Element of NAT st i in dom f1 holds P [ i , f1 . i ] ) & Q [ i , f2 . i ] holds Q [ i , f2 . i ]
sqrt ( ( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 + ( p `2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) ^2 ) = sqrt ( ( ( p `2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ^2 ) + ( p `2 / sqrt ( 1 + ( p `2 / p `2 ) ^2 ) ) ;
for x1 , x2 , x3 being Element of REAL n holds |. ( x1 - x2 ) - x3 .| = |. ( x1 - x2 ) - x3 .| & |. ( x1 - x2 ) - x3 .| = |. ( x1 - x2 ) - x3 .|
for x holds - x in dom ( ( - 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 )
for T being non empty TopSpace , P being Subset-Family of T st P c= the topology of T & P is Basis of T ex x being Point of T st P [ x , P . x ]
( a 'or' b ) . x = 'not' ( ( a 'or' b ) . x ) 'or' 'not' ( ( a 'or' b ) . x ) .= 'not' ( ( a 'or' b ) . x ) 'or' 'not' ( ( a 'or' b ) . x ) .= 'not' ( ( a 'or' b ) . x ) 'or' 'not' ( ( a 'or' b ) . x ) .= 'not' ( ( a 'or' b ) . x ) 'or' 'not' ( ( a 'or' b ) . x ) .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE .= TRUE 'or' TRUE ) 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE 'or' TRUE .= TRUE
for e being set st e in [: X1 , X2 :] ex X1 being Subset of X st e = X1 & ( ex Y1 being Subset of X st Y1 = Y1 & Y1 is open & ( ex Y1 being Subset of X st Y1 is open & Y1 is open & Y1 is open & Y1 is open ) & ( ex Y1 being Subset of X st Y1 is open & Y1 is open ) ) holds e is open )
for i being set st i in the carrier of S for f being Function of [: S , T :] , ( the carrier of S ) , ( the carrier of T ) st f = H & f is Function of [: S , T :] , ( the carrier of T ) st f = F ( i ) holds f is Function of [: S , T :] , ( the carrier of T ) , ( the carrier of S ) st f is Function of S , T & f is Function of S , T & f is Function of S , T & f is Function of S , T & f is Function of S , T & f is Function of S , T & f is Function of S , T & f is Function of S , T & f
for v , w st for x st x <> y holds w . x = v . y holds J . ( v . x ) = v . ( ( v . x ) . ( v . y ) )
card D = card ( D1 . ( i + 1 ) ) - card ( D1 . ( i + 1 ) ) .= 2 * ( D1 . ( i + 1 ) ) - card ( D1 . ( i + 1 ) ) .= 2 * ( D1 . ( i + 1 ) ) - card ( D1 . ( i + 1 ) ) .= 2 * ( D1 . ( i + 1 ) ) - card ( D1 . ( i + 1 ) ) .= 2 * ( D1 . ( i + 1 ) - 1 .= 2 * ( D1 . ( i + 1 ) .= 2 * ( D1 . ( i + 1 ) .= 2 * ( D1 . ( i + 1 ) .= 2 * ( D1 . ( i + 1 ) .= 2 *
IC Exec ( i , s ) = ( s +* ( i .--> ( s . ( 0 , SCM+FSA ) ) ) ) . 0 .= ( ( s +* ( i .--> ( s . ( 0 , SCM+FSA ) ) ) . 0 .= ( s +* ( i .--> ( s . ( 0 , SCM+FSA ) ) ) . 0 .= ( s . ( 0 , SCM+FSA ) ) . 0 .= ( s . ( 0 .--> 1 ) ) . 0 .= ( ( s . ( 0 , 1 ) ) . 0 .= ( ( s . 0 ) ) . 0 .= ( ( s . 0 ) ) . 0 .= ( ( s . 0 ) . 0 .= ( ( s . 0 ) ) . 0 .= ( ( s . 0 ) ) . 0 .=
len f -' ( i1 -' 1 ) + 1 = len f -' ( i1 -' 1 ) .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' ( i1 -' 1 ) .= len f -' ( i1 -' 1 ) .= len f -' ( i1 -' 1 ) .= len f -' 1 + 1 .= len f -' 1 .= len f -' ( i1 -' 1 ) + 1 .= len f -' 1 + 1 .= len f -' 1 + 1 .= len f -' 1 .= len f -' 1 + 1 .= len f -' 1 .= len f -' 1 .= len f -' 1 .=
for a , b , c being Element of NAT st 1 <= a & 2 <= b & c < d holds a + b < c + d or a + b < c + d & c + d < b + d
for f being FinSequence of TOP-REAL 2 , p being Point of TOP-REAL 2 st p in LSeg ( f , p ) & p in LSeg ( f , p ) & p in LSeg ( f , p ) holds Index ( p , f , p ) = Index ( p , f )
lim ( ( curry ( P , k ) # x ) # x ) = lim ( ( curry ( P # x ) ) # x ) + ( lim ( ( curry ( P # x ) # x ) ) ) ;
z2 = g /. ( n -' 1 ) .= g /. ( i -' 1 + 1 ) .= g /. ( i -' 1 + 1 ) .= g /. ( i -' 1 + 1 ) .= g /. ( i -' 1 + 1 ) .= g /. ( i -' 1 + 1 ) .= g /. ( i -' 1 + 1 + 1 ) .= g /. ( i + 1 + 1 ) ;
[ f . 0 , f . 3 ] in id ( the carrier of G ) \/ ( the InternalRel of G ) or [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 0 , f . 3 ] in the InternalRel of G & [ f . 1 , f . 2 ] in the InternalRel of G ;
for G being Subset-Family of B st G = { [ X , B ] where X is Subset of A : X in G & Y in G & x in F & X in G } holds ( ( Intersect ( G ) ) . X = ( Intersect ( G ) ) . X ) & ( ( Intersect ( G ) ) . X = ( Intersect ( G ) ) . X )
CurInstr ( P1 , Comput ( P1 , s1 , m ) ) = CurInstr ( P1 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) .= CurInstr ( P2 , s2 , m ) .= CurInstr ( P2 , s2 , m ) .= CurInstr ( P2 , s2 ) .= CurInstr ( P2 , s2 , m ) .= CurInstr ( P2 , s2 , m ) .= CurInstr ( P2 , s2 , m ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) .= CurInstr ( P2 , Comput ( P2 , s2 , m ) .=
assume that a on M and b on N and c on N and d on N and d on M and d on N and d on N and a on M and b on N and d on N and d on N and d on M and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N and d on N
assume that T is \hbox { T _ 4 } and ex F be Subset-Family of T st F is closed & for n being Nat st n in dom F ex F be Subset-Family of T st F is finite-ind & card F <= n & F is finite-ind & ( for n being Nat st n <= 1 holds F . n <= n ) & ind F <= n ;
for g1 , g2 st g1 in ]. r1 , r2 .[ & g1 in ]. x0 - 1 , x0 .[ & r1 < g1 & g1 < x0 holds |. ( f - g ) . n - 1 .| <= ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) / ( ( f - g ) ) ^2 ) ) ) ) ) holds |. ( f - g ) ^2 ) ) * ( ( ( ( f - g ) ) ) * ( ( f - g ) ) * ( ( f - g
exp_R /. ( z1 + z2 ) = exp_R . ( z1 + z2 ) + exp_R /. ( z2 + z2 ) .= exp_R . ( z1 + z2 ) + exp_R /. ( z2 + z2 ) .= ( exp_R /. ( z1 + z2 ) + ( exp_R /. ( z2 + z2 ) ) + ( exp_R /. ( z1 + z2 ) ) ;
F . i = F /. i .= F /. i .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= F /. ( n + 1 ) .= <* F /. ( n + 1 ) *> .= <* F /. ( n + 1 ) *> .= <* F /. ( n + 1 ) *> .= <* F /. ( n + 1 ) *> .= <* F /. ( n + 1 ) *> ;
ex y being set , f being Function st y = f . n & dom f = NAT & rng f = { y } & f . 0 = { y } & for n being Nat holds f . ( n + 1 ) = { f . n , f . ( n + 1 ) } ;
func f * F -> FinSequence of V means : Def1 : len it = len F & for i be Nat st i in dom F holds it . i = F ( i ) * F ( i ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x9 , cin , x9 , x9 } = { x1 , x2 , x3 , x4 , x5 , x9 , x9 , x9 , x9 } \/ { x2 , x3 , x4 , x5 , x5 , x9 , x9 , x9 , y9 , x9 , x9 } .= { x1 , x2 , x3 , x4 , x4 , x5 , x9 } \/ { x2 , x4 , x9 , x9 } ;
for n being Nat for x being set st x = h . n holds h . ( n + 1 ) = o ( x , n ) & x in InputVertices S ( x , n ) & x in InputVertices S ( x , n ) & x in InputVertices S ( x , n ) & x in InputVertices S ( x , n ) & y in InputVertices S ( x , n ) & x in InputVertices S ( x , n )
ex S1 being Element of VERUM ( Al , Al ) st ( SubP ( P , l ) ) . ( e , e ) = S1 & ( ( for e being Element of [: Al , Al :] st e in S & e in S & e in S & e in S holds ( e = e ) & ( e = ( e , S ) ) . ( e , e ) ) ) ;
consider P being FinSequence of ( G ) such that pT = Product P and for i being Element of NAT st i in dom P ex t being Element of the carrier of T st P . i = t & t . i = t . i & t . i = t . i ;
for T1 , T2 being strict non empty TopSpace , T1 , T2 being topology of T2 st the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 holds the topology of T1 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T1 & the topology of T2 = the topology of T2 & the topology of T2 = the topology of T2 & the topology of T2 & the topology of T2 = the topology of T1 & the topology of T2 = the topology of T2 & the
assume that f is partial differentiable on GoB f and r (#) ( f , 3 ) is_differentiable_in x0 and r (#) ( f , 3 ) is_differentiable_in x0 and r (#) ( f , 3 ) . x0 = r * ( f , x0 ) + r * ( f , x0 ) * ( f , x0 ) ;
defpred P [ Nat ] means for F , G being FinSequence st len F = $1 & len G = $1 & len G = $1 & len F = $1 & len G = $1 & for i being Element of NAT st i in dom F holds F . i = G . i & G . i = F . i & G . i = G . i & G . i = F . i ;
ex j st 1 <= j & j < width GoB f & ( GoB f ) * ( 1 , j ) `2 <= s & s <= G * ( 1 , j + 1 ) `2 & ( GoB f ) * ( 1 , j + 1 ) `2 <= s & s <= G * ( 1 , j + 1 ) `2 } c= { ( ( GoB f ) * ( 1 , j + 1 ) ) `2 }
defpred U [ set , set ] means ex F being Subset-Family of T st $1 = F & ( for x being Element of T st x in F ex y being Subset-Family of T st y in F & x in F & y in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F & x in F & y in F ) ) & P [ F , G ] ;
for p2 being Point of TOP-REAL 2 st LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 holds LE p2 , p3 , P
f in St ( E , H ) & for y st g . y <> f . y holds ( for x st x in dom ( f . x ) holds f . x = y ) implies f . ( x , y ) = f . ( x , y ) ) & ( for x st x in dom ( f . x ) holds f . x = f . ( x , y ) )
ex p2 being Point of TOP-REAL 2 st x = p2 & p2 `1 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & 1 <= p2 `2 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 1 & p2 `2 <= 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 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1 & 1 <= 1
assume for d1 being Element of NAT st d1 <= ( n + 1 ) & d1 <= ( n + 1 ) holds ( for t being Element of NAT st t in { ( n + 1 ) * ( t ) ) & ( for s being Element of NAT st s in { ( n + 1 ) * ( t . s ) ) holds t . ( ( n + 1 ) * ( t . s ) ) = t . ( ( n + 1 ) * ( t . s ) ) ) & ( t . ( ( n + 1 ) ) = t . ( ( n + 1 ) ) & ( t . ( ( n + 1 ) ) ) & ( t . ( ( n + 1 ) ) = t . ( ( n + 1 ) ) & ( t . ( ( n + 1 ) ) & ( t . ( ( n + 1 ) ) = t . ( ( n + 1 ) ) & ( t .
assume that s <> t and s is Point of Closed-Interval-TSpace ( x , r ) and not ex e being Point of TOP-REAL 2 st e in LSeg ( x , r ) & not e in LSeg ( x , r ) & e in LSeg ( x , r ) & e in LSeg ( x , r ) ;
given r such that 0 < r and for s being Point of C st 0 < s ex x1 , x2 being Point of TOP-REAL 2 st x1 in dom f & x2 in dom f & f /. ( x1 - x2 ) = f /. ( x1 - x2 ) & f /. ( x1 - x2 ) = f /. ( x1 - x2 ) & f /. ( x1 - x2 ) = f /. ( x1 - x2 ) ;
( p | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x
assume that x , h + x in dom sec and cos | [. x , h .] . x = sqrt ( 4 * PI * PI * PI * PI ) . x and cos . x = sqrt ( 4 * PI * PI * PI ) . x ;
assume that i in dom A and len A > 1 and B > 0 and A is Matrix of 0 , REAL and B is Matrix of 0 , REAL and B is Matrix of 0 , REAL and B is Matrix of 0 , REAL and B is Matrix of 0 , REAL and B is Matrix of 0 , REAL ;
for i being non zero Element of NAT st i in Seg n holds ( i divides n ) & ( i divides n & i divides n & i divides n & i divides n & i divides n ) & ( i divides n implies h . i = ( n gcd i ) . ( h . i ) ) & ( h . i = ( n gcd i ) . ( h . i ) ) & h . ( h . i ) = ( n gcd i ) . ( h . i ) & h . ( h . i ) = ( n -' i ) ) & h . ( h . i ) = ( n -' i ) * ( h . i ) ) & h . ( h . i ) ) & ( h . i ) = ( n -' i ) * ( h . i ) * ( h . i ) * ( h . i ) * ( h . i ) = ( n -' i ) * ( h . i )
( ( ( b1 => b2 ) '&' ( c1 '&' c2 ) ) '&' ( ( ( b1 'or' b2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) ) '&' ( ( ( c1 '&' c2 ) '&' ( c2 '&' c2 ) ) '&' ( ( c1 '&' c2 ) '&' ( c1 '&' c2 ) ) '&' ( ( c1 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) ) '&' ( ( c1 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' ( ( c2 '&' c2 ) ) '&' ( ( c1 '&' c2 ) '&' ( ( c1 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c1 '&' c2 ) '&' ( c2 '&' c2 ) ) '&' ( ( ( c2 ) '&' ( c2 '&' c2 ) '&' ( c2 '&' c2 ) '&' ( c2 ) '&' ( c2 ) '&' ( c2 ) ) '&' ( c2
assume that for x holds f . x = ( ( - 1 ) (#) ( cos * sin ) ) . x and for x st x in dom ( ( - 1 ) (#) ( cos * sin ) ) holds ( ( - 1 ) (#) ( cos * sin ) ) . x = - 1 * ( cos . x ) * sin . x ) ;
consider R8 , R8 being Real such that R8 = Integral ( M , ( F . n ) ) , I , I be PartFunc of X , REAL such that R8 = Integral ( M , ( F . n ) ) and Integral ( M , ( F . n ) ) = Integral ( M , ( F . n ) ) + Integral ( M , ( F . n ) ) ) ;
ex k be Element of NAT st k = k & 0 < d & for q be Element of product G st q in X & ||. f /. ( q - 1 ) - f /. ( q - 1 ) .|| < r holds ||. f /. ( q - 1 ) - f /. ( q - 1 ) .|| <= r ;
x in { x1 , x2 , x3 , x4 , x5 , x5 , x9 , y9 , x9 , x9 } & x in { x1 , x2 , x3 , x4 , x5 , x9 , x9 } & x in { x1 , x2 , x3 , x4 , x5 , x9 , x9 , y9 } & x in { x1 , x2 , x3 , x4 , x5 } & x in { x1 , x2 , x3 , x4 , x5 } ;
( G * ( j , i ) ) `2 = ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i ) `2 .= ( G * ( 1 , i
f1 * p = p .= ( the Arity of S1 ) . ( ( the Arity of S1 ) * the Arity of S2 ) .= ( the Arity of S1 ) . ( ( the Arity of S2 ) . o ) .= ( the Arity of S1 ) . o .= ( the Arity of S1 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 ) . o .= ( the Arity of S2 )
func tree ( T , P , T ) -> FinSequence means : Def1 : for q st q in it ex r st r in P & q in P & r < q & p = q & r < q & p = q & r < q & q in P & r < q & p = q & r < q & p = q & r <= q & q <= r ;
F /. ( k + 1 ) = F . ( k + 1 ) .= F . ( p . ( k + 1 ) ) .= F /. ( p . ( k + 1 ) ) .= F /. ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( k + 1 ) ) .= F . ( p . ( p . ( k + 1 ) .= p . ( p . ( p . ( k + 1 ) .= p . ( p . ( p . ( k + 1 ) .= p . ( p . ( p . ( k + 1 ) .= p . ( p . ( k + 1 ) .= p . ( p . ( p . ( k + 1 ) .= p . ( p . ( p . ( p . ( k + 1 ) ) .= p . ( p . ( k +
for A , B , C being Matrix of K st len A = len B & len B = width C & len B = width C & len A = width B & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C = width C & width B = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C = width C & width B = width C & width B = width C & width B = width C = width C & width B = width C = width C = width C & width B = width C = width C = width C = width
seq . ( k + 1 ) = 0. X + ( seq . ( k + 1 ) ) .= ( seq . ( k + 1 ) ) . ( ( ( seq ^\ k ) * ( seq ^\ k ) ) . n ) .= ( ( seq ^\ k ) * ( seq ^\ k ) ) . n .= ( ( seq ^\ k ) * ( seq ^\ k ) ) . n ;
assume that x in ( the carrier of C ) & y in ( the carrier of C ) & z in the carrier of C and x in the carrier of C and y in the carrier of C and z = [ x , y ] ;
defpred P [ Element of NAT ] means for f being FinSequence of NAT st len f = $1 & f . ( $1 + 1 ) = ( VAL g ) . ( ( VAL g ) . ( $1 + 1 ) ) & ( VAL g ) . ( ( VAL g ) . ( $1 + 1 ) ) = ( VAL g ) . ( ( VAL g ) . ( ( VAL g ) . ( $1 + 1 ) ) ) ;
assume that 1 <= k and k + 1 <= len f and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) and f /. k = G * ( i , j ) ;
assume that s < 1 and ( for q st q in X & q in X holds |. q .| < 1 ) & |. q .| < 1 & |. q .| < 1 & |. q .| < 1 & |. q .| < 1 ) implies |. q .| >= 1 & |. q .| >= 1 & |. q .| >= 1 & |. q .| >= 1 & |. q .| >= 1 ;
for M being non empty MetrSpace , x being Point of M , f being Function of M , ( TOP-REAL n ) | A st x = f . x holds ex n being Element of M st x = Ball ( x , f . n ) & f . n = Ball ( x , f . n )
defpred P [ Element of omega ] means $1 in Z & ( for x st x in Z holds f1 . x = ( 1 / ( $1 + 1 ) ) * ( f1 . x ) ) & ( f1 . x = ( 1 / ( $1 + 1 ) ) * ( f1 . x ) ) * ( f2 . x ) ) & ( ( 1 / ( $1 + 1 ) ) * ( f1 . x ) ) * ( f2 . x ) = ( 1 / ( ( ( ( ( 1 / ( ( ( $1 ) ) * ( f1 . x ) ) * ( f2 . x ) ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) * ( f2 . x ) ) - ( ( ( f2 . x ) ) - ( ( ( f2 . x )
defpred P1 [ Nat , Point of C ] means ( $1 < r & ( ex g st g in Y & ( for x st x in Y & x in X & x in Y & g in Y ) & ( not ex f st f . x = f . x ) & ( for x st x in X holds f . x = g . x ) ) & ( for x be Point of X st x in X holds f . x = f . x ) ) implies ( for x be Point of X st x in X holds f . x = ( f . x ) & ( x in X & f . x = ( f . x ) & ( f . x = ( f . x ) & ( for x be Point of X st x in X & f . x = ( f . x ) & ( f . x = ( f . x ) & ( f . x ) & f . x = ( f . x ) & f . x = ( f . x ) & ( f . x = ( f
( f ^ mid ( g , 2 , len g ) ) . i = ( mid ( g , 2 , len g ) ) . i .= g . ( len g + 1 ) .= g . ( len g + 1 ) .= g . ( len g + 1 ) .= g . ( len g + 1 ) .= g . ( len g + 1 ) .= f . ( len g + 1 ) .= f . ( len g + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f . ( len f + 1 ) .= f .
sqrt ( 1 / 2 * ( n + 1 ) ) * ( 2 / ( n + 1 ) ) ) = ( 1 / 2 ) * ( ( n / 2 ) * ( n + 1 ) ) .= ( 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 / 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 *
defpred P [ Nat ] means for G being finite Group , A being strict over G st G is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite & A is finite holds A is finite ;
assume that not ( not ( ex u st 1 <= u & u <= len f & u in LSeg ( f , u ) & not f . u in LSeg ( f , u ) & not f . u in LSeg ( f , u ) ) & not f . u in LSeg ( f , u ) and not f . u in LSeg ( f , u ) ;
defpred P [ Element of NAT ] means ( Partial_Sums ( cos ) ) . $1 = ( Partial_Sums ( cos ) ) . $1 - ( Partial_Sums ( cos ) ) . $1 * ( Partial_Sums ( cos ) ) . $1 * ( Partial_Sums ( cos ) ) . $1 - ( Partial_Sums ( cos ) ) . $1 * ( Partial_Sums ( cos ) ) . $1 ) ;
for x being Element of product F , x being Element of product F st x in dom ( the Sorts of F ) & x in dom ( the Sorts of F ) & x in dom ( the Sorts of F ) & x in dom ( the Sorts of F ) holds x = ( the Sorts of F ) . x
( x " ) |^ ( n + 1 ) = ( x " ) |^ ( n + 1 ) .= ( x " ) |^ ( n + 1 ) .= ( x |^ n ) " .= x |^ ( n + 1 ) .= x |^ ( n + 1 ) .= x |^ ( n + 1 ) .= x |^ ( n + 1 ) .= x |^ ( n + 1 ) .= x |^ ( n + 1 ) ;
DataPart Comput ( P +* I , Initialized s ) = DataPart Comput ( P +* I , Initialized s ) .= DataPart Comput ( P +* I , Initialized s ) .= DataPart Comput ( P +* I , Initialized s ) .= DataPart Comput ( P +* I , Initialized s , LifeSpan ( P +* I , Initialized s ) ) ;
given r such that 0 < r and ]. x0 - r , x0 .[ c= dom ( f1 + f2 ) and for g st g in dom ( f1 + f2 ) /\ dom ( f2 + g2 ) holds f1 + f2 <= ( f1 + f2 ) . g ;
assume that X c= dom f1 /\ dom f2 and ( f1 | X ) . x in dom f1 & ( f1 | X ) . x in dom f1 & ( f1 | X ) . x = f1 . x ;
for L being complete complete LATTICE for X being Subset of L st X = "\/" ( X , L ) ex x being Element of L st x = "\/" ( X , L ) & x is prime & x is prime
Support ( e ) in dom ( m *' p ) & for i being Element of NAT , p being Polynomial of n , L st i in dom ( m *' p ) & p in Support ( m *' p ) holds ex m being Polynomial of n , L st p in Support ( m *' p ) & p is Polynomial of n , L
( f1 - f2 ) /. ( lim ( f1 - f2 ) ) = lim ( f1 - f2 ) .= ( f1 - f2 ) /. ( lim ( f1 - f2 ) ) .= ( f1 - f2 ) /. ( lim ( f1 - f2 ) ) .= ( f1 - f2 ) /. ( lim ( f1 - f2 ) ) .= ( f1 - f2 ) /. ( lim ( f2 - f2 ) ) .= ( f1 - f2 ) /. ( lim ( f1 - f2 ) ;
ex p1 being Element of QC-WFF ( Al ( ) ) st F ( p ( ) ) = g ( p ( ) ) & for g being Function st g ( p ( ) ) = f ( g ( p ( ) ) ) & g ( p ( ) ) = g ( p ( ) ) ;
( mid ( f , i , len f ) ) /. j = ( mid ( f , i , len f ) ) /. j .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) .= f /. ( j + 1 ) ;
( ( p ^ q ) | ( len p + k ) ) . ( len p + k ) = ( ( p ^ q ) | ( len p + k ) ) . ( len p + k ) .= ( ( p ^ q ) | ( len p + k ) ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q ) . ( len p + k ) .= ( p ^ q )
len mid ( ( D2 , D1 , j ) + 1 ) = indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) + 1 .= indx ( D2 , D1 , j ) ;
x * y = x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( y * z ) * z ) .= x * ( ( x * z ) * z ) .= x * ( ( x * z ) * z ) .= x * ( ( x * z ) * z ) .= x * ( ( x * z ) * z .= x * ( x * z ) * z .= x * ( x * z ) * z .= x * ( x * z ) .= x * ( x * z ) .= x * ( ( x * z ) .= x * ( x * z ) .= x * ( x * ( x * z ) .= x * ( x * ( x * z ) .= x * ( x * z ) .= x * (
v . ( <* x , y *> ) * ( <* x0 , y0 *> ) = proj ( 1 , 1 , 1 ) * ( proj ( 1 , 1 ) ) + proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) + proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) ) + proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) + proj ( 1 , 1 ) * ( proj ( 1 , 1 ) ) ;
i * i = <* 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * ( 1 / 2 ) , 0 * i * i * i * i + i * i + i * i + i * i + i * i + i * i + i * i + i * i + i * i + i * i + i * i * i + i * i + i * i ) *> .= <* i * i + i * i * i * i * i * i * i * i * i + i * i * i + i * i + i * i + i * i + i * i + i * i * i + i * i + i * i + i * i * i + i * i + i * i + i * i + i
Sum ( L (#) F ) = Sum ( ( F (#) G ) ^ ( G (#) F ) ) .= Sum ( ( F (#) G ) ^ ( G (#) F ) ) .= Sum ( ( F (#) G ) ^ ( G (#) F ) ) .= Sum ( ( F (#) G ) ^ ( G (#) F ) ) .= Sum ( ( F (#) G ) ^ ( G (#) F ) ) .= Sum ( ( F (#) G ) (#) ( G (#) F ) ) .= Sum ( ( F (#) ( G (#) F ) ) .= Sum ( ( F (#) F ) ) + ( G (#) F ) ) .= Sum ( ( G (#) F ) ) .= Sum ( ( F (#) F ) (#) ( G (#) F ) ) .= Sum ( ( ( G (#) F ) (#) ( G (#) F ) ) .= Sum ( G (#) ( G (#) ( G (#) F ) + ( G (#) F ) ) .= Sum ( G (#) ( G (#) F ) ) + ( G (#) ( G (#) F ) (#) ( G (#) F ) .= Sum ( G (#) ( G (#) F ) .= Sum ( G (#) ( G (#) F ) ) + (
ex r be Real st for e be Real st 0 < e ex Y be finite Subset of X st Y c= e & Y c= Y & for x be Element of X st x in Y & x in Y holds |. ( x - x0 ) .| < r ;
( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) & ( GoB f ) * ( i , j ) = f /. ( k + 1 ) ;
( - cos ) . x = ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos ) . x .= ( - cos . x .= ( - cos . x .= ( - cos . x .= ( - sin . x ) .= ( - sin . x .= ( - sin . x .= ( - sin . x ) .= ( - sin . x .= ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * ( - sin . x ) * (
x- ( a , b ) * sqrt ( 2 * a , c ) + - sqrt ( 2 * a , c ) * sqrt ( 2 * a , c ) > 0 & - ( - ( - a ) * sqrt ( 2 * a , c ) ) < 0 & - ( - ( - a ) * sqrt ( 2 * a , c ) ) < 0 ;
assume that inf ( \mathopen { \uparrow X , L ) ) in X and ex_sup_of X , L and for X st X in X holds "/\" ( X , L ) = "/\" ( X , L ) and "/\" ( X , L ) = "/\" ( X , L ) and "/\" ( X , L ) = "/\" ( X , L ) ;
( ( B ) . j ) = ( i |-> ( j , i ) ) ** ( B . j ) & ( ( B . i ) --> ( j , i ) ) . j = ( j |-> ( i , j ) ) . j ;