thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
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thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in P ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in C ;
x in X ;
Y `2 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `1 = b `1 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `1 <= b `1 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be set ;
let G be _Graph , a , b be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be element ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in X ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , a be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Nat ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster downarrow x -> being being being being being set ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= \vert s .| ;
G . y <> 0 ;
let X be RealNormSpace , f be PartFunc of X , REAL ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
p0 `1 c= PI ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is_differentiable_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Y1 & b1 >= Y2 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Set Set = number ;
xx is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing ;
IT is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , f be Function of S , X ;
assume P [ n ] ;
assume union S is independent & A is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , a , b be element ;
b `1 c= b9 ;
assume not x in INT + 1 ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B9 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> left ideal ;
q1 in A1 & q1 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { \boldmath $ n $ } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g . x0 in dom f ;
g is_continuous_in x0 & g . x0 in dom f ;
assume O is symmetric ;
let x , y be element ;
let i0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s & P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , f be Function of X , Y ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , a be element ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , a be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> pre-1 ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q -Seg ( a ) ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be void mamaid ;
let N be non empty \cal \cal \cal \cal One ;
let R be RelStr with finite for finite 2 ;
let n , k be Nat ;
let P , Q be Cl RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as FinSeq-Location ;
assume I is not ] ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` ;
cluster S --> T -> object yielding ;
assume t1 <= t2 & t1 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & b1 = b2 ;
dom g1 = A ( ) ;
i < len M + 1 ;
assume not +infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec & y in dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & len g1 >= 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 + f2 ) ;
1 in dom D2 & 1 <= len D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
i = i & i = j ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be mod of \rm \mathbin { - } \rm f } ;
cluster m * n -> invertible ;
let k9 be Nat ;
i -' 1 > m ;
R is transitive implies R is transitive
set F = <* u , w *> ;
p0 `1 c= P3 `1 & p1 `1 <= p2 `1 ;
I is_halting_on t , Q ;
assume [ S , x ] is real ;
i <= len f2 & j <= len f2 ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
B9 c= X0 & B c= X ;
n2 <= ( 2 * n ) / ( 2 * n ) ;
A /\ ( cP ` ) c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , a be Element of V ;
assume n in dom g2 ;
let a be Element of R ;
t `1 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , a be Element of S ;
i . y in rng i ;
REAL c= dom f & for x st x in dom f holds f . x = 0 ;
f . x in rng f ;
mt <= ( r / 2 ) * ( r / 2 ) ;
s2 in r-5 & s1 . a in rng s1 ;
let z , z be complex number ;
n <= N . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 ) ;
let K be add-associative right_zeroed right_complementable associative associative well-unital non empty doubleLoopStr , f be Function of K , L ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & x in dom f ;
n1 < n1 + 1 & n2 < n1 + 1 ;
n1 < n1 + 1 & n2 < n1 + 1 ;
cluster \bf T ( X ) -> non empty ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S . k ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D as set ;
[ a , c ] in E1 ;
assume n in dom h2 ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( Carrier ( L ) ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued finite sequence ;
let k be Element of NAT ;
Integral ( M , f ) < +infty ;
let n be Element of NAT ;
assume z in C -] ( 0 , A ) ;
let i be set ;
n - 1 = n - 1 ;
len ( n-27 ) = n ;
cell ( Z , c ) c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E |^ \omega ;
let B1 be Basis of x , B ;
L2 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b `1 , c `2 ;
LIN q , c , c ;
x in rng f-129 ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable associative non empty doubleLoopStr , f be Function of K , W ;
assume f `1 = f & h `2 = h ;
R1 - R2 is total & R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & [#] ( TOP-REAL 2 ) | K1 is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f | E ) ;
cluster k -\vert -> nesss] ;
not u in { cg } ;
the carrier of f c= B ;
reconsider z = x as Vector of V ;
cluster the RelStr of L -> \rangle ;
r (#) H is ] ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict non-empty MSAlgebra over S , x be set ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in ( { y } ) \ { x } ;
let x , y be Element of X ;
let A , I be \cal E of X ;
[ y , z ] in [: O , A :] ;
that that that card Macro i = 1 and card Macro i = 1 ;
rng Sgm A = A ;
q |- p \! <* All ( y , q ) *> ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z |^ Y ;
( D . 0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , NAT :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f3 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multMagma for multMagma ;
x in support ( ( support t ) | ( support t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 ;
M1 <= sup M1 & M1 <= sup M1 ;
assume x in W ( X ) ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= l1 .= IC s .= IC s ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G , uH = Vertices H , uH = Vertices H , uH = Vertices H , uH = Vertices H , uH
seq " is non-zero implies seq " is non-zero & lim seq = lim seq
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 ( A ) ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X1 , X2 are_separated implies X1 , X2 is_\Vert
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty doubleLoopStr , a be Element of G ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & B is upper ;
let L be non empty reflexive RelStr , f be Function of L , L ;
R is_reflexive X implies R is_transitive X
E , g |= ( H ) => ( H ) ;
dom G /. y = a ;
sqrt ( 1 + 4 ) >= - r ;
G . p0 in rng G ;
let x be Element of FF , a be Element of F ;
D [ P-6 , 0 ] ;
z in dom id ( B ) & z in B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of G ;
rng f\mathbb r c= [: { 0 } , { 1 } :] ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 ;
f . z1 in dom h & h . z1 in dom h ;
P . k1 in rng P ;
M = ( A +* {} ) +* {} .= A ;
let p be FinSequence of REAL , a be Element of REAL ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
h . [. a , b .] = b ;
assume that the distance of V , Q and v in Q ;
let a be Element of ^ ( V ) ;
let s be Element of PP ( ) ;
let Pf be non empty RelStr ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM R & I = ( the InstructionsF of R ) . I ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= ( -> -> implies for j holds F . j = ( j ) * ( j ) ) ;
assume x in downarrow [ s , t ] ;
( x - t ) / 2 in ]. t - t , r + t .[ ;
x in JumpParts ( JumpParts T ) & x in { 0 } ;
let h be Morphism of c , a ;
Y c= ( 1_ K ) .: Y ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x1 , x2 in Y ;
dom <* y *> = Seg 1 .= dom y ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n be Nat holds 0 <= x . n
|[ a , b ]| = [. a , b .] ;
cluster closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & p1 , q2 , q1 is_collinear ;
dom M1 = Seg n & width M1 = n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & g2 . n in dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , a be Element of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( that R * S ) * ( S * R ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume that support L misses rng G and not x in rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT .= dom e ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI ;
o `1 , a9 // o `1 , y `2 ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " . p1 = 0 & f " . p2 = 1 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ A c= conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P \lbrack l \rbrack ;
cluster the TopStruct of J . i -> non empty for TopStruct ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\! > ( W1 + W2 ) ;
assume x in the carrier of R & y in the carrier of R ;
dom ( ( n + 1 ) -tuples_on the carrier of K ) = Seg n ;
s4 misses ( s2 * n ) ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that that that that that that card I c= J and card I = K ;
Im ( lim seq ) = 0 ;
( sin * sin ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z ;
t3 . n = t3 . n .= s . n ;
dom ( cos * cos ) c= dom F ;
W1 . x = W2 . x .= W2 . x ;
y in W -Seg ( x ) \/ W -Seg ( x ) ;
( k + 1 ) <= len ( vv ) + 1 ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I . ( I . p1 ) ;
IT = U /. 1 .= U . 1 ;
f . r1 in rng f & f . r2 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F . 2 ) ;
mode upper_bound of empty multMagma is well unital non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o ;
the carrier of m in B implies B c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F-12 ) = len I - 1 .= len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 |^ \alpha implies the carrier of T = the carrier of T
F . b1 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
redefine attr a <> {} means : Def2 : ( a * a ) = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & not o , b1 on C2 ;
LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let F2 be non empty TopSpace ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x as Subset of m -tuples_on the carrier of K ;
let A , B , C be Element of R ;
cluster strict non empty strict for \cal empty for \mathopen { - } 1 } ;
rng c `1 misses rng e & rng e `1 misses rng e `1 ;
z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) /\ dom ( cot * cot ) ;
the component of Q c= UBD A & ( for x st x in A holds x in Q ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) ;
redefine pred f = u * u ;
for n holds P1 [ prop n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 ;
gcd ( n1 , n2 ) = 1 & gcd ( n2 , n1 ) = 1 ;
set or = ( a * ( INT * r ) ) / 2 ;
|. seq . n .| < |. r1 .| ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / 1 : n in NAT } , h = { n / 1 } ;
k = a or k = b or k = c ;
a9 , b9 are_not collinear & a9 , b9 are_not collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W3 . 1 = s3 . 1 & W . ( 1 + 1 ) = s3 . ( 1 + 1 ) ;
cluster -> trivial for Walk of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^^ C implies A , B are_are that B , C are_that A , B are_that B , C are_that A , C are_that B , C are_that A , B are_that B , C are_that
x in { [ 2 * n + 3 , k ] } ;
1 >= ( ( q `1 ) ^2 + ( q `2 ) ^2 ) ;
f1 is_] ] ] and f2 is_One _of f1 implies f1 = f2
( f . q ) `2 <= ( q `2 ) ^2 ;
h is_in the carrier' of Cage ( C , n ) ;
( b `2 ) ^2 / ( p `2 ) ^2 <= ( p `2 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( f , g ) ;
p2 in ( N . p1 ) & p1 in ( N . p2 ) ;
len ( ( H ) . n ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
redefine attr 1 in C means : Def2 : A c= C ;
assume r1 <> 0 or r2 <> 0 ;
rng q1 c= rng C1 & for n holds P [ n , q1 . n ] ;
A1 , L , A2 , A3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in -> element & not b in { p } ;
then S is negative implies P-2 [ S ] ;
Cl Int [#] T = [#] T & Int [#] T = [#] T ;
f12 | A2 = f2 | A2 & len f12 = len A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
v , v `2 be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 ;
1_ 1 c= ( t * p2 ) * ( \HM { the } \HM { function } ) ;
0 * a = 0. R .= a * 0 .= a * 0 ;
A |^ 2 , 2 |^ 2 = A ^^ A ;
set v\rbrace = v4 /. ( n + 1 ) ;
r = 0. ( \langle \cal E , \Vert * \Vert *> , \Vert *> ) ;
( f . p4 ) `1 >= 0 ;
len W = len ( W ) + len ( W ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 . ( W7 . b ) not contradiction & not ( ex b1 st b1 = b1 . b ) & not ( not b1 = b2 ) ;
reconsider Y1 = X1 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for element ;
( downarrow a /\ downarrow t ) is Ideal of T ;
let X be non empty set , N be non empty set ;
rng f = TS <* S , X *> ;
let p be Element of B , x be the \it Element of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gR in the right of g & FR in the right of g ;
let A1 , A2 be Point of S , x be Point of X ;
x in h " P /\ [#] T1 .= [#] T1 /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X as non empty Subset of T^ <* *> ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 implies ( g2 /. n ) `2 <= ( g2 /. m ) `2
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( : ( - 1 ) * p ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x in ]. a , b .[ ) implies ( f `| ]. a , b .[ ) . x = ( f `| ]. a , b .[ )
rng M5 c= rng D2 & rng M5 c= rng D2 ;
for p be Real st p in Z holds p >= a
( cn ) * ( ( f | X ) . x ) = proj1 . x ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \! \mathop { \rm \hbox { - } Path } M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod ( P , T ) ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , a be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider ii = i - 1 as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the \rbrace of B ) . n & x in ( the Sorts of B ) . n ;
len 0 in Seg ( len f2 ) ;
p0 c= the topology of T & p1 is open & p1 in the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T2 ;
G * ( B * A ) = id o1 .= B * A ;
assume that p , u are_collinear and u , q are_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , S = $1 .. S , T = $1 .. S , N = $1 .. S , T = $1 .. S , N = $1 .. S , T = $1 .. S , N
LIN a1 , a3 , b1 & LIN a1 , a3 , c1 ;
f " ( f .: x ) = { x } ;
dom ( w2 | dom ( r | dom ( r | dom ( r | dom w ) ) ) ) = dom ( r | dom w ) ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q7 . x in rng ( q7 | A ) ;
Carrier ( L-43 ) misses Carrier ( LLet ) ` ;
consider c being element such that [ a , c ] in G ;
assume N|[ o1 , o2 ]| = o8 & I = I ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C ) " { {} } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
redefine attr 0 <= x & x ^2 <= x ;
p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster D -aaa] ( S , T ) -> non empty ;
let x be Element of S ~ ;
cluster \smallfrown \hbox { - } cluster F . ( a , b ) -> one-to-one ;
|. i .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = I ;
} * ( n + 1 ) ! > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // a3 , b3 & a3 , a4 // a3 , a4 ;
then dom A <> {} & dom A <> {} & rng A c= {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x joins X , Y & y in G2 ;
set v2 = v4 /. ( i + 1 ) , v4 = v4 /. ( i + 1 ) ;
x = r . n .= ( r . n ) / ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Upper_Arc ( P ) ;
dom d2 = A2 & dom d2 = A2 & dom d2 = A2 ;
0 < ( p `1 ) ^2 + 1 ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
- - r < Integral ( M , g | B ) ;
cluster O \HM { \tt and F is \HM { operation of X } ;
let U1 , U2 be non-empty MSAlgebra over S , a be Element of U1 ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , a be Real ;
reconsider pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and a is lower & b is bounded ;
Int Cl Int Cl A c= Cl Int Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) <= ( p2 `1 ) ^2 ;
Cl Q ` = [#] ( T | A ) .= [#] ( T | A ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f , n ) , I8 = ' ( f , n ) , I8 = ' ( f , n ) , I8 = ' ( f , n ) , I8 = ' ( f , n ) , I8
len p -' n = len p - n + 1 .= len p - n + n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nL = nY. - L . ( n + 1 ) as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
let qm , qm be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c1 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , a , b be Real ;
y = ( ( f * S ) . x ) . x ;
consider x being element such that x in ) ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( ( n + 1 ) - 1 ) div 2 ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x / 2 ) * ( x / 2 ) as Element of ( len x ) -tuples_on REAL ;
let U1 , U2 be Subspace of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 <= len p1 + 1 ;
let T1 , T2 be Scott Scott Scott Scott Scott of L ;
then x <= y & ( ex x st x in dom y & y = z ) ;
set M = n -tuples_on ( m , n ) ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of o ) | dom ( the_arity_of o ) ) c= dom H ;
z1 " = ( z1 " ) * ( z1 " ) .= z1 " * ( z1 " ) .= z1 " ;
x0 - r / 2 in L /\ dom f ;
then w is strict implies rng w /\ ( L \/ S ) <> {} ;
set x-10 = ( x ^ <* Z *> ) | Z , xX2 = ( x ^ <* Z *> ) | Z ;
len w1 in Seg ( len w1 + len w2 ) & len w1 = len w1 + len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. a . n .| ) ;
( p `1 ) ^2 / ( p `1 ) ^2 <= ( G * ( 1 , 1 ) `1 ) ^2 ;
rng g1 c= L~ ( g | X ) \/ L~ ( g | X ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is \HM { {} } ;
reconsider xx = xx , xx = xx , xx = y as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , d ;
reconsider x1 = x as Element of REAL m -tuples_on REAL ;
assume i in dom ( a * p ) ;
m . : g . [: f , g :] = p . [: f , g :] ;
a / ( s . m ) - ( a / ( s . n ) ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & B1 \/ C2 = B1 \/ B2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
that Following 0. R = a and b-0 0 = b ;
F8 is closed & P8 is closed & P8 is closed & P8 is closed ;
set T = ) for X being non empty TopSpace , x0 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 , x1 , x2 , x3 , x4 ,
Int Cl ( Int R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( FX . x ) = { F . x } ;
G-23 " { c } c= B \/ S ;
f[#] is Relation of [: X , X :] , X & X is reflexive ;
set RQ = the rng of P , RQ = the ^ of Q , RQ = the ^ of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , a be Element of REAL ;
reconsider pp = u as Element of ( TOP-REAL n ) | ( [#] ( TOP-REAL n ) ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N * N ) ;
len ( P-37 ) <= len ( P-35 ) + len ( P-35 ) ;
x " in the carrier of A1 & x " in the carrier of A1 & x " in the carrier of A1 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple & Re ( F . m ) is simple
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL , REAL-NS i , REAL ;
rng f = the carrier of \bf SCM ( A ) .= the carrier of A ;
assume s1 = sqrt ( 2 * ( p `1 ) ^2 + ( p `2 ) ^2 ) ;
attr a > 1 & b > 0 implies a / b > 1 ;
let A , B , C be Subset of Lin I ;
reconsider X0 = X , Y = Y , Z = Z as Subset of REAL ;
let f be PartFunc of REAL , REAL , a be Real ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation ;
Q [ e1 \/ { v-14 } , f ] & P [ e1 , v-5 ] ;
g \circlearrowleft ( W-min L~ z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vrelational ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & z <= x + y ;
sqrt ( 7 * p1 ^2 + e ^2 ) > 0 ;
assume X is BCK-algebra & 0 < X & 0 < 0 ;
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( tan * tan ) . x in dom sec & ( tan * tan ) . x > 0 ;
i2 = ( f /. len f ) * ( f /. ( len f ) ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 \ X1 = ( X1 \ X2 ) \ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & [. a , b .] c= [. a , b .] ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r ;
|. ( f /* s ) . k - G3 .| < r ;
len Line ( A , i ) = width A & width A = width A ;
S' ^2 = ( S . g ) ^2 .= ( S . g ) ^2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom Initialized ( p +* I ) ;
i1 , i2 , i3 , i3 , iff not ( ex b1 , b2 , b3 , b2 , b3 , b3 , b2 ) & ( not b1 , b2 , b3 , b2 is_collinear ) & not b1 , b2 , b3 is_collinear
( 1 + ( r - 1 ) * ( 1 - r ) ) / ( 1 - r ) = r ;
for x st x in Z holds f2 is_differentiable_in x & f . x > - 1 ;
reconsider q2 = ( q / x ) / ( 2 * x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & j + 1 <= j ;
assume f in the carrier of [ X \to [#] Y , [#] Y ] ;
F . a = H / ( ( x , y ) \leftarrow ( y , z ) ) ;
TRUE = TRUE & ( C , u ) *> . ( 1 , u ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 <= len G ;
( ( p2 `1 ) - x1 ) - x1 > - g . x1 ;
|. r1 - ] = |. a1 .| * |. \neq .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b ;
DkW = DkW .( n ) + 1 ;
i1 = ma + n & i2 = [: K , K :] & i1 = - 1 ;
f . a [= f . ( f . O1 "\/" f . a ) ;
redefine pred f = v & g = u + v ;
I . n = Integral ( M , F | E ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R * ( i , j ) ) ;
set h = the continuous Function of X , R ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is negative holds P [ H ] ;
set b8 = S5 ^\ ( i + 1 ) , P8 = ( i + 1 ) + 1 ;
Hom ( a , b ) c= Hom ( a `1 , b ) ;
sqrt ( 1 + ( n + 1 ) ) ^2 < ( 1 + s ) ^2 ;
( l . 1 ) `1 = [ dom l , cod l ] ;
y +* ( i , y /. i ) in dom g ;
let p be Element of CQC-WFF ( Al ) , x be Element of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ ( X /\ X1 ) ;
p2 in rng ( f /^ p1 ) & p1 in rng ( f /^ p1 ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in K1 /\ ( ( TOP-REAL 2 ) | K1 ) \/ ( ( TOP-REAL 2 ) | K1 ) ;
- 1 <= ( ( f2 . O ) `2 ) ^2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 -' k2 + 1 .= k1 -' k2 + 1 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & x0 < g2 . n ;
sgn ( p `1 , K ) = - 1_ K .= - 1_ K ;
consider u be Nat such that b = ( p |^ y ) * u ;
ex A being the the len of f is \rbrack or ex A st a = Sum A ;
Cl ( union H ) = union ( ( Cl H ) ` ) .= Cl ( ( Cl H ) ` ) ` ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 + len t1 ;
v-29 = v + w |-- ( v + A ) ;
v <> DataLoc ( t3 . GBP , 3 ) & v . DataLoc ( v . GBP , 3 ) = s . DataLoc ( v . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= s . s ;
( \dot y ) . s = s . ( y . s . s ) ;
{ s : s < t } in INT & t = {} implies t = {} ;
s ` \ s = s ` \ ( 0. X ) \ ( 0. X ) .= 0. X ;
defpred P [ Nat ] means B + $1 in A & B + $1 in B ;
339 * 339 = 3322222223 * ( 3339 + 1 ) ;
U ( ) = ( T ( ) ) . ( ( id A ( ) ) . ( ( id A ( ) . ( a , b ) ) ) ) ) ;
reconsider y = y as Element of COMPLEX ( len y ) ;
consider i2 being Integer such that y0 = p * i2 and not contradiction ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) ;
set f = ( S , U ) \mathop { {} , {} } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a , b be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , a be Element of REAL n ;
reconsider l = 0. ( V ) , m = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a ' 'or' ( b ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( xx - g ) . 2 - g .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & a , c // b9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 - k1 & k2 + 1 = k2 -' k1 + 1 ;
sqrt ( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
sqrt ( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in cell ( RR , 1 , 1 ) & E-max L~ R in L~ R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b `1 or p `1 , a `2 = b ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is being being set ;
F | ( N2 | S ) = CircleMap * ( F | S ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( cos * cos ) = [. - 1 , 1 .] .= [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable & Im ( seq ) is summable ;
||. ( vseq . n ) - ( t-5 . n ) .|| < e ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 -element string of S2 , x be Element of S2 ;
reconsider x-29 = seq . ( n + 1 ) as sequence of ( TOP-REAL n ) | REAL ;
assume that that C meets L~ go and L~ co /\ L~ co = { go /. 1 } and L~ go /\ L~ co = { pion1 /. 1 } ;
- ( ( - 1 ) * x ) < F . n - ( x - 1 ) * x ;
set d1 = be element of being Real , z1 , z2 be Point of REAL ;
2 to_power ( 1 - 1 ) = 2 to_power ( 1 - 1 ) - 1 ;
dom ( v2 * ( len d2 * ( len d ) ) ) = Seg len ( d2 * ( len d ) ) ;
set x1 = - ( k2 + 1 ) + |. k2 + 1 .| ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( Carrier ( Carrier ( L2 ) ) + ( Carrier ( L2 ) ) ) ) c= I ;
'not' All ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. of A ;
Z c= dom ( ( sin * ( sin * f1 ) ) ^2 ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - sn ) .| < r ;
ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ C ) & ConsecutiveSet2 ( A , succ B ) c= ConsecutiveSet2 ( A , succ C ) ;
E = dom ( L . m ) & Carrier ( L . m ) c= E . m ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ( s . IC s ) = P . IC s .= s . IC s .= s . IC s ;
redefine attr x > 0 means : Def2 : ( 1 - x ) * x = x ^2 - x ^2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , R .] ;
b , c are_connected & - C , - C are_connected implies - C , C are_homotopic
assume f = id the carrier of O & f is Function of O , O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) ;
reconsider g = f " as Function of U2 , U1 ;
A1 in the carrier of ( ( G . k ) | X ) & A2 : not ex x being Point of X st x in the carrier of ( G . k ) | X & not x in the carrier of ( G . k ) | X ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = ) +* ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib n ) >= 4 * log ( n , 5 ) ;
v3 /. ( k + 1 ) = v3 . ( k + 1 ) ;
0 mod i = ( - i * ( 0 qua Nat ) ) / ( 0 qua Nat ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = n ;
Line ( SIT , j ) = SIT . j .= 0. K ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , x1 ] ;
|. f .| - Re ( |. f .| (#) ( ( b - a ) (#) h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ <* b1 *> ^ b1 ^ b1 ^ b2 ;
Mr is closed implies IExec ( I , P , s ) . b = s . b
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= s . DataLoc ( s . a , 4 ) ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , q ;
f^ . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= ( y1 + z1 ) + ( z1 + z1 ) ;
f_ ] . a = ffrom ( a , b ) . v & v in InputVertices S ;
( p `1 ) ^2 / ( p `2 ) ^2 <= ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E-max L~ Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 + ( p `2 ) ^2 ;
consider p such that p = pp and s1 < p & p < s2 & p < p & p < s2 ;
|. ( f /* ( s * F ) ) . l - G . ( s * F ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= REAL ;
n = k * ( 2 * t ) + ( n mod 2 ) ;
dom B = 2 -tuples_on the carrier of V & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 ;
for L being complete LATTICE holds <* <* \mathclose ( A ) , a *> *> , [ b , a ] *> is isomorphic ;
[ gi , gj ] in [: I , I :] \ I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for x st x in dom ( f1 + f2 ) holds f1 . x = - 1 / ( a + x ) ^2 ;
reconsider y = ( a ` ) / ( F . ( a ` ) ) , F . ( a ` ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) ) . c <= h . c ;
set G2 = the subgraph of G , s3 = the consider of G , v = the Vertex of G , w = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m .| / |. p . m .| < d / ( p . m ) ;
for x being element st x in ~ ( u ) holds x in ~ ( t )
P = the carrier of ( TOP-REAL n ) | K1 .= the carrier of ( TOP-REAL n ) | K1 .= the carrier of ( TOP-REAL n ) | K1 ;
assume p10 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) |^ ( m + 1 ) = 0. X ;
let g be Element of Hom ( cod f , dom g ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be PartFunc of X , Y , x be Point of X ;
set h = Hom ( a , g ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg n & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the right of H ;
x in dom ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( cos ^ ) ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , q1 , P & LE p1 , p2 , P & LE q1 , q2 , P ;
attr B is closed means : Def2 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means : Def2 : the_rank_of ( M * a ) = the_rank_of ( a * M ) ;
consider j such that j in dom ' of of dom ' and I = len ' + j ;
consider x1 such that z in x1 and x1 in ( P . n ) and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ] ;
set Cs1 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2 , i + 1 ) , Cs2 = Comput ( P2 , s2
set cv = 3 / ( { a , b , c } \/ { b , c } ) ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( arccot * arccot ) `| Z ) ;
r3 <= s3 + ( r2 - |. v2 .| ) / ( 2 * ( 1 - r2 ) ) ;
dom ( f (#) f3 ) = dom f /\ dom f3 .= dom f /\ dom f3 .= dom ( f (#) f3 ) /\ dom ( f (#) f3 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg ( l (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g1 = gp as Point of ( TOP-REAL n ) | K1 , g2 = ( TOP-REAL n ) | K1 , g1 = ( TOP-REAL n ) | K1 , g2 = ( TOP-REAL n ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom let *> <* *> implies ( ( Frege A ) * ( ( Frege A ) . o ) ) . y = ( ( Frege A ) . o ) . y ;
for I being non degenerated commutative doubleLoopStr holds the carrier of I is commutative commutative commutative doubleLoopStr ;
set s2 = s +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* ( ( intloc 0 ) .--> 1 ) , s2 = P +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* ( ( intloc 0 ) .--> 1 ) , s2 = P
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim s1 in [. a , b .] ;
v . ( l-13 . i ) = ( v *' lpp ) . i .= ( v *' lmin ) . i ;
consider n be element such that n in NAT and x = seq . n and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x & F1 . x <> 0 ;
card Funcs ( X , 0 , x1 , x2 , x3 , x4 ) = { E } ;
j + ( 2 * ( k + 1 ) ) > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n2 , n3 , n3 , n4 , n4 , n4 ) ;
( mg1 ) . ( HT ( g2 , T ) ) = 0. L ;
then H1 , H2 are_are \frac of H1 , H2 & Cl ( H1 , H2 ) , Cl ( H2 , H1 ) are_isomorphic ;
( N-min L~ f ) .. f > 1 & ( E-max L~ f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) & x2 in L~ g ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x0 be Element of REAL ;
DigA ( t-23 , z9 ) . ( k + 2 ) is Element of k -tuples_on k ;
I \mathop { \rm 2222223 } = k2 & I \mathop { \rm \hbox { - } Int 2 } = k2 ;
uu = { [ a , u9 ] } & { [ a , u9 ] } = { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and x = v + u2 ;
for y st y in rng F ex n st y = a |^ n & a <= n ;
dom ( ( g * ( ( ( id V ) \dot \to C ) ) | K ) ) = K ;
ex x being element st x in ( [#] U0 ) \/ A . s & x in ( the Sorts of U0 ) . s ;
ex x being element st x in ( ( ( ( ( ( ( O ) \/ A ) ) \/ B ) \/ B ) \/ A ) . s ) ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} implies ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p10 , p2 ) c= { p10 } /\ { p10 } ;
sqrt ( b + ( b-2 ) / 2 ) in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G8 such that z = y and P [ z ] ;
( the sequence of ( ( the sequence of ( the carrier of X ) ) | the carrier of Y ) ) . ( the carrier of X ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + ( len w + 1 ) ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 & q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E-4 = g | ( ( E | D ) ` ) .= g | ( E | D ) ` .= g | ( E | D ) ` ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , H = x4 , 7 = x2 , 8 = x3 , 8 = x4 , 8 = x4 , 8 = 7 , 8 = 8 as Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex x0 being Element of NAT st f |^ x0 is K & f . x0 is 1 ;
Seg len ( ( the multF of b2 ) * ( ( the multF of b2 ) * ( ( the Arity of b2 ) * ( the Arity of b2 ) ) ) ) ) = dom ( the Arity of b2 ) ;
( Complement ( A ) ) . m c= ( Complement ( A ) ) . n ;
f1 . p = p9 & g1 . ( p9 . p ) = d . ( p9 . p ) & g1 . ( p9 . p ) = d . ( p9 . p ) ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) , ^ ( F | Y ) | Y ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) to_power ( n + 1 ) ) <= ( ( r2 ) to_power ( n + 1 ) ) ;
Sum ( F-12 ) = Sum f & dom ( F-12 ) = dom g & for x be Element of X st x in dom ( F . n ) holds ( F . n ) . x = f . x ;
assume for x , y being set st x in Y holds x /\ y in Y ;
assume W1 is Subspace of W2 & W2 is Subspace of W3 implies W1 + W2 is Subspace of W2 & W1 + W2 is Subspace of W1
||. ( t-15 . x ) .|| = lim ||. ( x . x ) .|| .= ||. ( x . x ) .|| .= ||. ( x . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( 1 + ( p `1 ) ^2 ) ;
g | Sphere ( p , r ) = id ( the carrier of TOP-REAL 2 ) | K1 .= ( the carrier of TOP-REAL 2 ) | K1 .= the carrier of ( TOP-REAL 2 ) | K1 ;
set Nmin = ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= width B .= width B .= width B ;
attr a <> 0 means : Def2 : ( A ^^ B ) Y. = ( A Y. ) ^^ ( B -- C ) ;
then f is_\cal One for u , 1 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 , w1 , w2 , w2 } ;
p2 /. IC s = p2 . IC s .= ( IC Comput ( p2 , s , k ) ) .= ( IC Comput ( p2 , s , k ) ) + ( card I + 1 ) ;
ind ( T-10 | b ) = ind b - ind B .= ind B - ind B - ind B .= ind B - ind B - 1 / B + 1 / B - 1 / B - 1 / B + 1 / B - 1 / B <= - 1 / B - 1 / B ;
[ a , A ] in the InternalRel of G_ ( k , X ) & [ a , A ] in the InternalRel of element ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o2 ) ;
( ( ( a , CompF ( PA , G ) ) | ( z , G ) ) ) . z = TRUE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , N = ( S , D ) | 1 , M = ( S , D ) | 2 , N = ( S , D ) | 2 , N = ( S , D ) | 2 , M = ( S , D ) | 2 , N = ( S , D ) |
len s1 - ( len s2 - 1 ) > 0 + 1 - 1 ;
\delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f21 ] in the carrier' of A ( ) ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( TOP-REAL 2 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( V ) /\ the carrier of V .= { 0. V } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one & P [ P2 ] ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s . m - x0 < s . m ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p3 *> ^ <* p1 *> .= h ;
c /. [ b , c ] = c /. ( [ a , c ] ) .= c /. ( [ a , c ] ) .= c /. ( [ a , c ] ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p1 , t1 = p2 , t2 = p1 , t2 = p2 , t2 = p1 , t1 = p2 , t2 = p1 , t2 = p2 , t2 = p1 , t2 = p2 , t1 = p1 , t2 = p2 , t2 = p1 , t2 = p2 ;
sqrt ( 1 + ( 2 * ( 1 + ( 2 * ( 1 / 2 ) ) / 2 ) ) ^2 ) ) in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( ( p1 `1 ) `2 + D ) `2 .= C * ( ( p1 `1 ) `2 + D ) `2 ;
R . b `1 = 2 * PI - b .= 2 * PI - b .= b ;
consider ] such that B = ( 1 - ( 2 * A ) + ( - ( 2 * A ) ) ) * ( 0 - ( 2 * A ) ) ;
dom g = dom ( ( the Sorts of A ) * ( the_arity_of o ) ) .= dom ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
[ P . l1 , P . l2 ] in => ( ( T . l1 ) , T . l2 ) ;
set s2 = Initialize s , P2 = P +* I , s3 = P +* I , P3 = P +* I , s4 = Comput ( P3 , s3 , 1 ) , P4 = P3 ;
reconsider M = mid ( z , i2 , i1 ) as non empty Element of ( TOP-REAL 2 ) * ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 <= ( |[ 0 , 1 ]| ) `1 ;
assume x in the left of g or x in the left of g & x in the right of f ;
consider M being strict Subgroup of A9 such that a = M and T is Subspace of M ;
for x st x in Z holds ( ( ( ( ( ( ( 1 / 2 ) * f ) ) * f ) ) `| Z ) . x ) <> 0 ;
len W1 + len W2 + m = 1 + len W2 + len W1 + m .= len W1 + len W2 + len W1 + m ;
reconsider h1 = ( v-12 . n - t-16 . n ) - t-16 . n as Lipschitzian Function of X , Y ;
( - ( len p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is conjunctive and F in the |= of ( s2 ) and F in the |= of ( s2 ) and F in the |= of ( s2 ) ;
( ( gcd ( x , y ) , 3 ) ) * ( x , y ) = gcd ( x , y , 3 ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = from [: p , q :] \/ [: p , q :] ;
x in { X where X is Ideal of L : X is Ideal of L & X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 /\ ( the carrier of V ) & the carrier of W1 /\ W2 c= the carrier of V ;
( 1 / a + b ) * id ( a + b ) = ( 1 / a + b ) * id ( a + b ) ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - ( 2 |^ ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 * f1 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } and b2 . r = { c2 . r } ;
ex P st a1 on P & a2 on P & a1 on P & a2 on P & a3 on P & a1 on P & a2 on P ;
reconsider gf = g `1 * f `1 , hg = h `1 * g `2 as strict Subgroup of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and P [ v1 ] and P [ v1 ] ;
n in { i where i is Nat : i < n + 1 & n < len f + 1 } ;
( F /. ( i , j ) ) `2 >= ( ( F /. m ) `2 ) * ( ( F /. m ) `2 ) ;
assume K1 = { p : ( p `1 / |. p .| - sn ) / ( 1 + sn ) >= sn & p `2 / |. p .| - sn ) >= sn } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I2 = goto 3 , I2 = goto 4 , I2 = goto 0 , I2 = goto 0 , I2 = goto 4 , I2 = goto 0 , I2 = goto 0 , I2 = goto 4 , I2 = goto 0 , s4 = goto 4 , s4 = goto 0 , s4 = goto 1 , s4 = goto 0 , s4 = goto
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & P [ x9 ] ;
reconsider e1 = e1 , e2 = f , fe = f , fe = g , fe = f , fe = g , e2 = h , e1 = h , e2 = f , e2 = h , e2 = h , e = h , e2 = h , e = h , f = h , e = h , f = h , e = h , f = h , e = h , f = h ,
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl O ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ ( A + $1 ) & A = ( A + $1 ) --> ( A + $1 ) ;
the left of - g = the left of g & the right of - g = the right of g ;
reconsider pp = x , pp = y , \overline = z , \overline = x , \overline = y , \overline = z , \overline = z , \overline = z , \overline = z , card = z , card = z , card = z , card = z , card = z , card ( z , card ( z , card ( z , card ( z , , card ( z , card ( z , , p , z
consider g2 such that g2 = y and x <= g2 & g2 <= x0 and g2 <= x0 and x0 <= g2 & g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] ;
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 + ( len x1 ) ) .= len ( x2 + ( len x2 ) ) .= len ( x2 + ( len x1 ) ) ;
for x being element st x in X holds x in the set of the set of \HM { 0 , 1 } & x in the set of \HM { 0 , 1 }
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} .= {} ;
func ) consider X ( ) -> set means : Def2 : for x being set holds x in X ( ) iff x in X ( ) & x in X ( ) ;
len ( ( ( C /. len C ) ) | ( len C -' 1 ) ) <= len ( C /. 1 ) ;
attr K has a , b means : Def2 : a <> 0. K & v . ( a |^ i ) = i * v . ( a |^ i ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o = [ o , the carrier of S ] ;
for x st x in X ex y st x in X & y in X & y is a holds f . x = f . y
IC Comput ( P-6 , s2 , k ) in dom ( ( n + k ) -tuples_on the carrier of K ) ;
attr q < s & r < s implies ]. r , s .[ c= ]. p , q .[ & ]. r , s .] c= ]. p , q .[ ;
consider c being Element of Class ( F . c , 3 ) such that Y = ( F . c , 3 ) . ( c , 2 ) ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def2 : for x being Element of S2 , y being Element of S2 , a being Element of S2 , b being Element of S2 , s being Element of S2 , s being Element of S2 , a being Element of S1 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( ( ( ( 1 / 2 ) * ( arccot ) ) * ( arccot ) ) ) * ( arccot ) ) ) `| Z ) ) ;
r-7 in Int cell ( GoB f , i , width GoB f -' 1 ) \ L~ f & r-7 in L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) * ( i + 1 ) ) ^2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + ( len f -' 1 ) - len f + 1 - len f + 1 ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 ) & h . n > 0 ;
set s0 = ( ( a , I , p ) , s ) . i , p = s . i , q = p . i , s = p . i , s = p . i , s = p . i , s = p . i , p = s . i , r = p . i , s = p . i , s = p . i , s = p . i
( p . k ) . 0 = 1 or ( p . k ) . 0 = - 1 & ( p . k ) . 1 = - 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x = [ x9 , y9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( len p + k ) .= ( q | ( len p + k ) ) . ( len p + k ) ;
g + h = gg + hh & A1 + h = g + h + h + X ;
L1 is distributive & L2 is distributive implies L1 * L2 is distributive & L1 * L2 is distributive & L1 * L2 is distributive & L1 * L2 is distributive & L1 * L2 is distributive & L1 * L2 is distributive
redefine pred x in rng f & y in rng ( f | x ) & f | y = f | y ;
assume that 1 < p and ( 1 + p ) * q + q = 1 and 0 <= p & p <= 1 ;
F* ( f , the ^ of t ) = rpoly ( 1 , the carrier of F_Complex ) *' t .= ( 1 , the carrier of F_Complex ) *' t ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} implies A = {}
( ( NW-corner X ) `1 ) ^2 + ( ( E-max X ) `1 ) ^2 + ( ( E-max X ) `2 ) ^2 ) <= ( ( E-max X ) `1 ) ^2 + ( ( E-max X ) `2 ) ^2 ;
for c being Element of the \rbrack of A , a being Element of the bound of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . GBP , 2 ) .= s . DataLoc ( s . DataLoc ( s . GBP , 2 ) , 2 ) ;
for a , b being Real holds [ a , b ] in ( y >= 0 implies a >= 0 ) & b >= 0 implies a >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) ` ;
mode BCK-algebra of i , j , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n ;
set x2 = |( Re ( y ) , Im ( y ) , Im ( y ) , Im ( y ) , Im ( y ) , Im ( y ) , Im ( y ) , Im ( y ) , Im ( y ) ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= ( delta ( S2 ) . n ) & |. delta ( S2 ) . n .| < e / 2 ;
( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) <= ( - ( q `1 / |. q .| - sn ) ) / ( 1 + sn ) ;
set A = ( 2 * b ) / ( 2 * a ) ;
for x , y being set st x in R-1 holds x , y are_\hbox { - }
deffunc F ( Nat ) = b . ( $1 + 1 ) * ( M . $1 ) * ( G . $1 ) ;
for s being element holds s in ^2 ( f \/ g ) iff s in ^2 ( f \/ g ) \/ ^2 ( f \/ g )
for S being non empty non void non void holds S is connected holds S is connected ;
max ( ( degree ( K , n , r ) ) / ( 1 + ( d + 1 ) ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) & Lin ( B ) = Lin ( B ) ;
set n-15 = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN , M = ( n + 1 ) -tuples_on BOOLEAN ;
f " V in Hom ( X , p ) & f " ( f " V ) in D & f " ( f " V ) . p in D ;
rng ( ( a ) *> \mathbin ( 1 , b ) +* ( 1 , c ) ) c= { a , c , b } ;
consider y being | of G1 such that y `1 = y and dom y `2 = y -Wmod G2 ;
dom ( 1 / ( ( 1 / ( x + h ) ) * f ) ) /\ ]. x0 , x0 + h .[ c= ]. x0 , x0 + h .[ ;
( -> Matrix of i , j , n , r ) , - r , s be Element of TOP-REAL n ;
v ^ ( ( n |-> 0 ) ^ ( ( n |-> 0 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1 ) ^ ( ( n --> 1
ex a , k1 , k2 st i = a := k1 & j = b := k2 & i = c := k2 ;
t . NAT = ( ( NAT .--> ( i1 , i2 ) ) . NAT ) . NAT .= ( ( NAT .--> i1 ) . NAT ) . NAT .= ( NAT --> i1 ) . NAT .= ( NAT --> i1 ) . NAT .= ( NAT --> i1 ) . NAT ;
assume that F is bbbfamily and rng p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) and for i be Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN b , c , a & not LIN c , a , b
( L1 \HM { L2 } ) \& O c= ( L1 \HM { L1 } ) \HM { {} } & ( L1 \HM { L2 } ) \HM { {} } c= ( L1 \HM { L2 } ) \HM { {} }
consider F be ManySortedFunction of E , REAL such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( -w ) = b * ( -w ) and 0 < b & b < 1 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum |. $1 .| & |. Sum ( $1 ) .| <= Sum |. $1 .| ;
u = cos / ( x , y ) . v * x + cos / ( x , y ) . v .= cos . ( x , y ) * x + cos . ( x , y ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| : |. p .| = {} & {} = ( the Sorts of A ) . ( id the carrier of A ) ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y & X is finite and X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & g <= h & l1 <= h & h . l1 <= h . l1 } ;
vol ( ( G . n ) vol ) <= vol ( ( G . n ) vol ) , vol ( G . n ) ) ;
f . y = x .= x * 1. L .= x * 1. L .= x * 1. L .= x * 1. L .= x ;
NIC ( halt SCMPDS , i1 ) = { i1 , succ ( i1 , succ ( i1 , succ ( i1 , k ) ) ) } .= { i1 , succ ( i1 , k ) } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } .= { p1 , p2 } ;
Product ( ( ( the support of I1 ) +* ( i , { 1 } ) ) +* ( i , { 1 } ) ) in Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S1 ) ;
W-bound ( Q1 ) <= ( q1 `1 ) / ( ( q1 `1 ) / ( q1 `1 ) ) ^2 & ( q1 `1 ) / ( q1 `1 ) <= ( q1 `1 ) / ( q1 `1 ) ^2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + 1 ) & f /. ( ( i1 + len g -' 1 ) + 1 ) = f /. ( i1 + 1 ) ;
M , v / ( x. 3 , x. 4 ) / ( x. 4 , x. 0 ) / ( x. 4 , x. 0 ) |= H ;
len ( ( P ^ ) | ( len P + 1 ) ) in dom ( ( P ^ ) | ( len P + 1 ) ) ;
A |^ ( \mathbb n , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
TOP-REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y = p1 . n1 and x = p1 . n1 and y = p1 . n1 ;
consider X being set such that X in Q and for Z being set st Z in X & Z <> {} holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCMPDS & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) = halt SCMPDS ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id id id id id ( the carrier of V ) ) . v .| & ||. v .|| = |. ( id ( the carrier of V ) ) . v .|
for phi holds not phi in X implies not phi in X & not phi in X & not phi in X & not phi in X
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom |. f .| ) ) ) ) | dom ( f | dom ( |. f .| ) ) ) ) c= dom ( f | dom |. f .| ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & b = c ;
the_arity_of ( ( \HM { the } \HM { Y } , c ) ) = <* \mathop { \rm dom } ( b , c ) , \mathop { \rm cod } ( b , c ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous & f | X is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 & a4 = b3 & a4 = b3 & a4 = b3 & a4 = b3 ;
D2 . indx ( D2 , D1 , n1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ]. r , s .[ ) = ]. |. r .| , r .] /. 1 .= <* r *> . 1 .= r ;
consider n be Nat such that for m be Nat st n <= m holds Cseq . m = Cseq . m ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & a <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= x0 + ( K * |. h .| ) ;
attr F is commutative means : Def2 : for b being Element of X holds F -\hbox { b } = f . b ;
p = ( - 1 ) * ( ( p0 `1 ) + ( p2 `2 ) ) .= 1 * ( ( p2 `1 ) + ( p2 `2 ) ) .= ( p2 `1 ) * ( p2 `1 ) + ( p2 `2 ) * ( p2 `1 ) ) .= ( p2 `1 ) * ( p2 `1 ) ;
consider z1 such that b , x3 , x3 is_collinear and o , x1 , z1 is_collinear & o <> x1 & o <> x2 ;
consider i such that Arg ( ( Rotate ( s , 2 ) ) . q ) = s + Arg ( q ) . i and s . i = ( 2 * PI ) . i ;
consider g such that g is one-to-one & dom g = card ( f . x ) & rng g c= dom g & g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and for i , j st i <> j & j <> i & i < j & j < j & j < n & i < j & j < n & i < j & j <= n ;
attr F is associative means : Def2 : F .: ( F .: ( f , g ) , h ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z & m < i & i < m ;
consider k2 be Nat such that k2 in dom ( P-2 . ( k2 + 1 ) ) and ( for k be Nat st k in dom P-2 . ( k2 + 1 ) ) holds ( ( k + 1 ) + 1 ) . ( k + 1 ) ) . ( k2 + 1 ) = 0 ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . ( n + 1 )
F1 . [ id a , a ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D2 } ;
consider z being element such that z in dom ( ( dom F ) | ( dom F ) ) and ( ( dom F ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f holds x = f . y & y = f . x
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , the carrier of K , \langle J , J *> ) ) . ( j + 1 ) ;
- 1 / ( ( 1 / 2 ) (#) D ) = ( ( mm ) (#) D ) | n .= ( mm ) (#) D .= ( ( m * n ) (#) D ) | n .= ( Det M ) | n .= ( Det M ) | n ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x ;
len ( f1 . j ) = len f2 /. ( j + 1 ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( 'not' a , A , G ) , B , G ) |= All ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k ) \/ RightComp Cage ( C , k + 1 ) ;
x \ ( a |^ m ) = x \ ( ( a |^ k ) * a ) .= ( x \ a ) \ a .= ( x \ a ) \ a ;
k -] -insucc ( I ) = ( ( commute ( I ) ) . k ) . ( k + 1 ) ) .= ( ( commute ( I ) ) . k ) . ( k + 1 ) .= ( ( commute ( I ) ) . k ) . ( k + 1 ) ) .= ( ( commute ( I ) ) . k ) . ( k + 1 ) .= ( ( commute ( I ) ) . k ) . ( k + 1 ) ) ;
for s being State of A2 holds Following ( s , n + 2 * n ) . ( n + 2 * n ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 . x <> 0
support ( ( support ( n ) ) \/ support ( m ) ) c= support ( ( support ( n ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ( ) , the carrier of B ( ) :] ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( b . a ) = g . a & phi . ( b . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i = len ( F ^ <* p *> ) + 1 ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x6 , x6 , x5 , x6 , x5 , x6 } = { x1 } \/ { x2 , x3 , x4 , x5 , x5 } \/ { x1 , x2 , x3 , x4 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U2 & the Sorts of U2 c= the Sorts of U2 ;
( - ( 2 * a * ( b * a ) + b * c ) ) / ( 2 * a * b ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r ;
Z = dom ( ( exp_R * ( arccot * ( arccot * f1 ) ) ) ) ;
lim ( f , S ) is convergent & lim ( f , S ) = lim ( f , S ) & lim ( f , S ) = lim ( f , S ) ;
( X . ( a9 . ( f . ( x . ( x . ( x . x ) ) ) ) ) => ( ( x . ( x . ( x . ( x . ( x . x ) ) ) ) ) ) => ( X . ( x . ( x . ( x . ( x . x ) ) ) ) ) ) ) in consider such that X [ x , y ] ;
len ( M2 * M1 ) = n & width ( M2 * M1 ) = n implies M1 * M1 = M2 * M1
attr X1 union X2 is open means : Def2 : X1 , X2 where X1 , X2 is open SubSpace of X : X1 , X2 \HM { p } is open & X1 , X2 \HM { p } is open & X1 , X2 \mathclose { p } is open ;
for L being lower-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1be Function of [: M . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . x ) ) ) ) ) ) ) ) ) ) ) , M . ( b . ( b . ( b . ( b . ( b . ( b . ( b .
consider w being FinSequence of I such that the InitS of M , <* s *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= 1_ G .= 1_ G .= 1_ G ;
assume for i be Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 /\ ( the carrier' of C2 ) .= the carrier' of C1 /\ ( the carrier' of C2 ) .= the carrier' of C1 /\ ( the carrier' of C2 ) .= the carrier' of C1 /\ ( the carrier' of C2 ) ;
reconsider o9 = o `1 , p = o `1 , q = o `1 , r = o `2 , s = o `2 , n = s , m = s , n = s , m = s , n = t , n = t , m = t , n = s , n = t , m = t , n = s , m = t , n = t , n = t , m = t , n = t , n = t , m = t , n = t , n = t
1 * x1 + ( 0 * x2 + x3 ) = x1 + ( 0 * x2 + x3 ) .= x1 + 0 * x2 .= x1 + 0 * x2 .= x1 + 0 * x2 + 0 * x3 .= x1 + 0 * x2 + 0 * x3 + 0 * x3 + 0 * x3 + 0 * x3 + 0 * x3 .= 1 * x1 + 0 * x2 + 0 * x3 + 0 * x3 + 0 * x3 + 0 * x4 + 0 * x4 + 0 * x2 + 0 * x3
( ( E . 1 ) qua Function ) " = ( ( E . 1 ) qua Function ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " .= ( E . 1 ) " ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . l1 + 1 ) - f . ( s1 . l1 + 1 ) .| < ( 1 / |. M . ( l1 + 1 ) .| + 1 ) ;
LSeg ( ( Rev Cage ( C , n ) ) * ( i , j ) , ( Rev Cage ( C , n ) ) * ( i + 1 , j ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of A and len ( the \kern1pt X ) = width A and width ( f ) = width A and width ( f ) = width A ;
len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( TOP-REAL n ) | ( i + 1 ) ) ;
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) . 1 = proj ( 2 , 2 ) . 1 ;
attr a <> 0 & b <> 0 & Arg ( a ) = Arg ( b ) & Arg ( b ) = Arg ( b ) & Arg ( b ) = Arg ( b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the empty empty empty set , a , b , c )
assume that V1 is linearly-independent and V2 is linearly-independent and V1 = { v + u : v in V1 & u in V1 & v in V1 & u in V1 & v in V1 & v in V1 & u in V1 & v in V1 & v in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * ( y1 - z ) + ( 1 - z ) * x2 in N ;
rng ( ( ( ( ( ( ( 1 - 1 ) ) * ( S ) ) * ( S . k ) ) ) * ( S . k ) ) ) ) ) = Seg card ( S . ( card ( S . k ) ) ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n ;
h2 " . n = h2 . n & 0 < ( - 1 / ( 2 * n ) ) / ( 2 * n ) & 0 < ( - 1 / ( 2 * n ) ) / ( 2 * n ) ;
( Partial_Sums ( ||. seq .|| ) ) . m = ||. ( seq .|| ) . m - ( ||. seq .|| ) . m .|| .= ||. ( seq . m ) - ( seq . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) .= sup ( k .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) , A |^ ( k , l ) ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for n being Nat holds ( a * b ) + ( b * n ) = ( a * b ) + ( b * n )
consider A5 being countable Subset of CQC-WFF ( Al ) such that r is Element of CQC-WFF ( Al ) & A5 is ( Al ) -NAT & A5 is ( Al ) -NAT ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st x in M & y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 , y2 } & { [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , x2 } ;
h . O = |[ A * ( ( f . O ) `1 ) + B , C * ( ( f . O ) `2 ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) `1 in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n -> prime for Nat means : Def2 : for p being Nat holds p divides m & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being Lattice , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a "/\" b <= c
consider b being element such that b in dom ( H / ( ( x , y ) \leftarrow ( x , y ) ) ) and z = H / ( ( x , y ) \leftarrow ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 1 , G & e in W . 3 & W . 1 = W . 3 ;
( ( h (#) f ) `| Z ) . ( 2 * n ) . x = ( h (#) f ) . ( 2 * n ) . x + h . ( 2 * n ) . x ;
j + 1 = j - len ( h11 + 2 ) + 1 .= i + 1 - len ( h11 + 2 ) - 1 .= i + 1 - len ( h11 + 2 ) - 1 .= i + 1 - len ( h11 + 2 ) - 1 ;
( S *' S ) . f = S . ( S *' f ) .= S . ( S . ( S . f ) ) .= S . ( S . f ) .= S . ( S . f ) .= S . ( S . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is \ means : Def2 : for p , q st p in R & q <> q & p <> q & p <> q & q <> r & p <> r ;
dom product ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f .= dom f .= dom f .= dom f ;
sup ( proj2 .: ( Upper_Arc C ) /\ the carrier of ( TOP-REAL 2 ) | D ) <= sup ( proj2 .: ( Upper_Arc C ) /\ D ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - S . m .| < r
i * fN - fH = i * fN - ( i * y ) .= i * ( f - y ) - i * ( f - y ) .= i * ( f - y ) ;
consider f being Function such that dom f = 2 -tuples_on X & for Y be set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y & g2 in union C & g2 in union C & g = [ g1 , g2 ] ;
func d gcd n -> Nat means : Def2 : d divides n & d divides ( d |^ n ) & d divides ( d |^ n ) ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . D or t = h . E or t = h . F ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) <= sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) ;
h1 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 ) .= h2 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { [ o , x2 ] } such that a = [ o , x2 ] and o <> [ o , x2 ] ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a ;
||. h1 .|| . n = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| ;
( ( - ( exp_R * exp_R ) ) `| Z ) . x = f . x - exp_R . x .= ( - exp_R * exp_R ) . x .= ( - exp_R * exp_R ) . x .= ( - exp_R * exp_R ) . x ;
redefine attr r = F .: ( p , q ) means : Def2 : len r = len p & for i st i in dom r holds r . i = F . ( p . i , q . i ) ;
sqrt ( ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ) / 2 ) ) + ( r ^2 + ( r ^2 + ( r ^2 + 1 ) ) / 2 ) ) <= sqrt ( r ^2 + ( r ^2 + 1 ) ) ;
for i being Nat , M being Matrix of n , K st i in Seg n & i > n holds Det ( M , i ) = Sum ( \cal L ( M , i ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a " * v = a * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r . ( i -' 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) * ( ( R /* h ) ^\ n ) " ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = f .: ( the carrier of H2 ) ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) * ( the Arity of S ) ) . o .= ( the Arity of S ) . o ;
H1 = n + 1 & ( |. 2 to_power ( n + 1 ) .| + h ) to_power ( n + 1 ) = n + 1 + 1 .= n + 1 ;
( O `1 ) * ( O `1 ) = 0 & ( O `1 ) * ( O `1 ) = 1 & O `2 * ( O `1 ) = 0 & O `1 * ( O `1 ) = 0 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a = e implies b = e ) & ( a = e ) & ( a = b implies a = b ) & ( a = b implies a = b ) ;
dom ( ( f +* g ) | D ) = dom ( ( f +* g ) | D ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( ( f +* g ) | D ) .= D ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a & v in B * a & u in C * a ;
g `1 * P " = g `1 * ( g " * P ) .= g " * ( g " * P ) .= g " * ( g " * P ) .= g " * ( g " * P ) .= g " * ( g " * P ) .= g " * ( g " * P ) ;
consider i , s1 such that f . i = s1 & not ( ex s st s = s1 & not ( s = s1 & s = s1 ) & not ( s = s1 ) & s = s1 ) & not ( s = s1 ) & s = s2 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] in R & [ s2 , t2 ] in R & [ s1 , t2 ] in R ;
then H is negative & H is non negative & H is non negative & H is non negative implies H is non negative ;
attr f1 is total means : Def2 : ( 1 / 2 ) (#) ( f1 /* c ) = ( f1 . c ) (#) ( f2 . c ) (#) ( f2 . c ) (#) ( f2 . c ) (#) ( f2 . c ) (#) ( f2 . c ) ) ;
z1 in W2 " ( { z2 } ) or z1 = z2 " ( { z2 } ) & z1 in W2 " ( { z2 } ) & z2 in ( { z1 } ) \ { z2 } ) ;
p = 1 * p .= a " * p .= a * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a * ( b * q ) ;
for rseq be Real_Sequence for K be Real st for n be Nat holds rseq . n <= K holds upper_bound rng ( seq ^\ K ) <= K * ( seq ^\ K )
TOP-REAL 2 meets L~ go \/ L~ co or x in L~ go \/ L~ co or x in L~ go \/ L~ co or x in L~ go \/ L~ co & x in L~ co \/ L~ co ;
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . ( k + 1 ) - g . ( k + 1 ) .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ) +* ( C .--> D ) +* ( E .--> E ) +* ( F .--> J ) +* ( M .--> N ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) ) ;
|. ( ( delta ( H , n ) ) `| A ) . k - ( ( lower ( H , n ) ) | A ) . k ) .| <= e * ( ( delta ( H , n ) ) . k ) ;
( ( the Sorts of A ) . ( i , the carrier of S ) ) . v = [ ( the Arity of S ) . ( i , the carrier of S ) , ( the Sorts of A ) . ( i , the carrier' of S ) ] ;
{ x1 , x1 , x2 , x3 , x4 , x5 , x5 , x6 , x6 , x6 , x5 , x6 , x6 , x5 , x6 , x6 , x5 , x6 , x6 , x6 , x6 , x6 , x1 , x2 , x3 , x4 } = { x1 , x2 , x3 , x4 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R * cos ) , A ) = 0 and integral ( ( exp_R * cos ) , A ) = 0 ;
p `1 is Permutation of dom f1 & p `1 = ( Sgm Y ) " * ( Sgm Y ) & p `2 = ( Sgm Y ) " * ( Sgm Y ) " * ( Sgm X ) ;
for x , y st x in A holds |. ( 1 / 2 ) * ( f . x - f . y ) .| <= 1 * |. f . x - f . y .|
( |. p2 .| ) ^2 = |. q2 .| * ( ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) ) .= |. q2 .| * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) ;
for f be PartFunc of the carrier of C , REAL st dom f is compact & f | X is continuous holds rng f = dom f & f | X is compact & for x be Element of C st x in X /\ dom f holds f . x is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , PA , G ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 & for k be Nat st k in n1 holds Q [ k , F3 . k , F3 . k ] ;
ex u , u1 st u <> u1 & u , u1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , v2 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , u1 , v1 , u1 , v1 , v1 , u1 , v1 , v1 , v1 , u1 , v1 , v1 , v1 , u1 , v1 , v2 , v1 , u1 , v1 , u1 , v1 , v2 , v1
for G being Group , A , B being non empty Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * ( N ` B )
for s be Real st s in dom F holds F . s = integral ( R ^2 ) (#) ( ( R ^2 ) (#) ( f ^ ) ) , s ) . x )
width AutMt ( f1 , b1 , b2 ) = len ( f2 * ( b1 , b2 ) ) .= len ( f2 * ( b1 , b2 ) ) .= len ( ( f2 * ( b1 , b2 ) ) * ( b2 , b2 ) ) .= len ( ( f2 * ( b1 , b2 ) ) * ( b2 , b2 ) ) ;
f | ]. PI / 2 , PI / 2 .[ = f & dom f = ]. PI / 2 , PI / 2 .[ & for x st x in ]. PI / 2 , PI .[ holds f . x = - 1 / 2 * x + 1 / 2 * x )
assume that X is closed and a in X and a in X & y in a & x in { [ n , x ] } \/ { [ n , x ] } \/ { [ n , x ] } ;
Z = dom ( ( ( ( ( ( ( 1 / 2 ) * ( arctan ) ) * ( arctan ) ) ) * ( arctan ) ) ) `| Z ) /\ dom ( ( ( ( 1 / 2 ) * ( arctan ) ) * ( arctan ) ) `| Z ) ;
func CQC-WFF ( V ) -> Subset of V means : Def1 : for k st 1 <= k & k <= len it holds it . k = V . ( k + 1 ) & it . k = V . ( k + 1 ) ;
for L being non empty TopStruct , N being net of L , M being net of L st c is net of N for c being Element of L st c is cluster cluster cluster cluster -> convergent for Function of N , L holds c is convergent & for x being Element of N st x in N holds x is convergent & x is convergent & N . x is convergent
for s being Element of NAT holds ( ( ( id the carrier of Al ) + ( id the carrier of Al ) ) + ( id the carrier of Al ) ) . s = ( ( id the carrier of Al ) + ( id the carrier of Al ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( E-max L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Nat ) *> .= len p + 1 ;
assume that Z c= dom ( ( - ( ( ln * f ) ) * ( f ^ ) ) ) and for x st x in Z holds f . x = exp_R . x / ( x + a ) ) and f . x > 0 ;
for R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) & f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) ;
for S being -1 Functor of C , B for c being Object of C holds card S . id ( c , c ) = id ( ( the Arrows of C ) . c ) , ( the Arrows of C ) . c )
ex a st a = a2 & a in f6 /\ f5 & b in ( f " { 0 } ) & a in ( f " { 0 } ) /\ ( f " { 0 } ) & b in ( f " { 0 } ) /\ ( f " { 0 } ) ;
a in Free ( H2 / ( x. 4 , x. k ) ) \/ ( ( x. 4 , x. k ) / ( x. k , x. k ) ) ;
for C1 , C2 being non empty set , f being Function of C1 , C2 st f is stable & f is stable holds f = g & f = g implies f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go + E-bound L~ co .= ( W-bound L~ go + E-bound L~ co ) `1 .= E-bound L~ go + E-bound L~ co .= ( W-bound L~ go + E-bound L~ co ) `1 ;
Suppose u = <* x0 , y0 , z0 *> and f is partial & u = y0 & u = y0 & f . ( 3 + 1 ) = y0 ; Then SVF1 ( 3 , f , 1 ) . u = SVF1 ( 3 , f , 1 ) . u + SVF1 ( 3 , f , 1 ) . u
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = ( t . {} ) `1 & t . {} = ( t . {} ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b ;
func Class ( R , a ) -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & a in A & a in A & a in A ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( G ) . $1 ) ) `1 ) ) ) `1 ) `1 ) + ( ( ( G . $1 ) `1 ) `2 ) `1 ) + ( ( G . $1 ) `2 ) `2 ) + ( ( G . $1 ) `2 ) `2 ) = ( ( G . $1 ) `2 ) + ( ( G . $1 ) `2 ) ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 1 and dim ( W1 ) = 0 and dim ( W1 ) = 1 and dim ( W1 ) = 0 ;
mama_empty ( m . t ) = ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 ;
d11 = ( x9 ^ <* d *> ) . ( ( y9 , d ) | ( x | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | y ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .= f . ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y
consider g such that x = g & dom g = dom f0 & for x being element st x in dom fP holds g . x in f . x ;
x + 0. F_Complex = x + ( len x |-> 0. F_Complex ) .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x ;
( ( f /^ ( k -' 1 ) ) + 1 ) in dom ( f /^ ( ( k -' 1 ) ) | ( k -' 1 ) ) /\ ( ( k -' 1 ) | ( k -' 1 ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p1 , p2 , p3 } and P1 = { p1 , p2 , p4 } and P1 = { p1 , p2 , p4 , p4 , p4 , p4 , p4 , p2 , p4 , p4 , p4 , p4 , p4 , p2 , p4 , p1 , p2 , p2 , p4 , p2 , p2 , p4 , p2 , p2 , p4 , p2 , p4 , p4 , p4 , p1 , p2 , p1 ,
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c1 = d , c2 = c , c1 = d , c1 = d , c2 = c , c1 = d , c1 = d , c2 = c , c1 = d , c2 = d , c1 = d , c2 = d , c1 = d , c1 = d , c2 = d , c2 = c , c1 = d , c2 = d , c1 = d , c2 = d , c1 = c , c1 = d , c1 = d , c1 = d , c1 = d , c2 = d , c1 = d , c1 = d , c2 = d , c1 = d , c1 = d , c1 = d , c1 = d , c1 = d , c1 = d , c1 =
reconsider GtbFFFbf = G1 . ( t . b ) * F1 . b as Morphism of G1 . ( a * b ) , G2 . ( b * b ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 ) ) \/ LSeg ( f , i + 1 -' 1 ) ;
Integral ( P . m , P . n ) | dom ( P . n ) <= Integral ( M . ( n + m ) , P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 ) , ( G * ( i , 1 ) `1 ) `1 ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Element of G , b being Element of H st a = b holds a |^ b = b |^ a * ( a |^ b )
consider B being Function of [: Seg ( S + L ) , the carrier of V :] , the carrier of V such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where p2 is Point of TOP-REAL 2 : P [ p2 ] & p2 `1 <= p2 `1 & p2 `2 <= p1 `1 & p2 `1 <= p2 `1 & p2 `2 <= p2 `1 & p2 `1 <= p2 `1 & p2 `1 <= p2 `1 & p2 `1 <= p2 `1 } as Subset of TOP-REAL 2 ;
sqrt ( ( ( NW-corner C ) `1 - W-bound C ) / 2 ) ^2 + ( W-bound C ) ^2 <= ( W-bound C ) ^2 + ( W-bound C ) ^2 + ( W-bound C ) ^2 ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x .| <= P . x & |. Im ( F . n ) .| . x <= P . x
len @ @ r = len ( @ @ @ s ) + len @ @ @ s .= len @ ( @ s ) + len @ @ s .= len @ s + len @ @ r .= len @ s + len @ @ s ;
v / ( ( x. 3 ) . ( m1 , m1 ) ) . ( ( x. 0 ) . ( m1 , m2 ) ) = ( x. 4 ) . ( ( x. 0 ) . ( m1 , m2 ) ) ;
consider r being Element of M such that M , v / ( ( x. 3 ) . ( x. 4 ) ) / ( x. 4 ) / ( x. 4 ) / ( x. 4 ) ) |= r ;
func w1 \ w2 -> Element of Union ( G , RH ) means : Def2 : for i being Element of NAT holds ( the NH of G ) . ( i , j ) = ( the NH of G ) . ( i , j ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= ( Exec ( n2 , s2 ) ) . b2 .= s . b2 .= s . b2 .= s . b1 .= s . b1 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k ) + Partial_Sums ( |. seq .| ) . n )
set F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N } , G = S \! \mathop { N } , G = S \! \mathop { N } , F = S \! \mathop { N } , G = S \! \mathop { N }
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) ) + Partial_Sums ( seq ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x = L . ( x- x ) + R . ( x- x ) ;
func the closed of \HM { a , b , c , d } -> Subset of TOP-REAL 2 equals ( the Element of \HM { a , b , c } ) ` ;
a * b ^2 + ( a * c ) + ( b * c ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) + ( c * a ) +
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) ;
( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) ) ) ) ) ) ) ) ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) ) ) ) ) ) = ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) | ( ( Q ^ <* x *> ) ) ;
Sum ( ( F |^ ( n1 + 1 ) ) * Sum ( C ^\ ( n1 + 1 ) ) ) = ( C |^ ( n1 + 1 ) ) * Sum ( C ^\ ( n1 + 1 ) ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 .= ( GoB f ) * ( 1 , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( Partial_Sums ( s ) ) . $1 + ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 ) + ( Partial_Sums ( s ) ) . $1 * ( Partial_Sums ( s ) ) . $1 ;
( the_arity_of g ) . x = ( the Arity of S ) . ( g . x ) .= ( ( the Arity of S ) . ( g . x ) ) | ( ( the Arity of S ) . ( g . x ) ) .= ( the Arity of S ) . ( g . x ) .= ( the Arity of S ) . ( g . x ) ;
( X ~ Y ) ^ Z is_differentiable_on X * Y & card ( X ~ ) = card ( X * Y ) & card ( X * Y ) = card ( X * Y ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . ( n + 1 ) & b = F . ( n + 1 ) \ G . ( n + 1 ) holds b = F . ( n + 1 ) \ G . ( n + 1 )
E , f / ( ( x. 2 ) . ( x. 2 ) ) / ( x. 2 ) |= ( x. 2 ) . ( x. 2 ) => ( x. 2 ) . ( x. 2 ) ) / ( x. 2 ) ;
ex R2 be 1-sorted st R2 = ( p | ( n2 -' i ) ) . ( i -' 1 ) & ( the carrier of p ) | ( n2 -' i ) = the carrier of ( p | ( n2 -' i ) ) ;
[. a , b + sqrt ( 1 - ( 1 - ( a + b ) ) / ( 1 - ( a + b ) ) ) / ( 1 - ( a + b ) ) ) , a + b + sqrt ( 1 - ( a + b ) ) / ( 1 - ( a + b ) ) / ( 1 - ( a + b ) ) ) .] is Element of REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a , s ) .= Exec ( a , s ) . IC s .= s . 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 s .= succ IC s .= succ IC s ;
card ( h1 | k ) = ( power ( F_Complex , k ) ) * Sum ( ( - 1_ F_Complex ) | k ) .= ( ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) ) * Sum ( - 1_ F_Complex ) ) .= ( ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) ) * Sum ( - 1_ F_Complex , k ) .= ( ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) ) * Sum ( - 1_ F_Complex ) .= ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) .= ( - 1_ F_Complex ) * Sum ( - 1_ F_Complex ) * Sum ( - ( - 1_ F_Complex ) ) * Sum ( - 1_ F_Complex )
( ( f (#) g ) /* c ) = f /. c * ( g /* c ) " .= ( f (#) ( g /* c ) ) " .= ( f (#) ( g /* c ) ) " .= ( f (#) ( g /* c ) ) " .= ( f (#) ( g /* c ) ) " .= ( f (#) ( g /* c ) ) " ;
len C9 - len ( ( ( C | ( len C | ( len C | ( len C ) ) ) ) ) ) ) = len ( ( C | ( len C -' 1 ) ) ) - len ( ( C | ( len C -' 1 ) ) ) ) - len ( ( C | ( len C -' 1 ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= X /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= X /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= X /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= dom ( r (#) f ) /\ X .= X /\ X .= X /\ X .= X /\ X .= X /\ X .=
defpred P [ Nat ] means 2 * Fib ( $1 + 1 ) = Fib ( $1 + 1 ) * Fib ( $1 + 1 ) + Fib ( $1 + 1 ) * Fib ( $1 + 1 ) * Fib ( $1 + 1 ) * Fib ( $1 + 1 ) + Fib ( $1 + 1 ) * Fib ( $1 + 1 ) * Fib ( $1 + 1 ) * Fib ( $1 + 1 ) + Fib ( $1 + 1 ) * Fib ( $1 + 1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and f is onto & for n being Nat st n < k & n < n holds f " { f . n } = { f . n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( S , BOOLEAN ) and ( for A being Element of S holds ( A \/ B ) . A = Prob ( c9 , A ) . ( A \/ B ) ) and ( for A being Element of S holds ( A \/ B ) . A = Prob ( c , A ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of Y ( ) : P [ x ] } , L ( x , y ) ) and P [ y , x ] ;
assume that A c= Z and f = ( - 1 ) (#) ( ( exp_R * exp_R ) `| Z ) and f = ( exp_R * exp_R ) `| Z ) . ( ( exp_R * exp_R ) . x ) and for x st x in Z holds ( ( exp_R * exp_R ) `| Z ) . x = exp_R . x / ( exp_R . x ) ^2 and f . x > 0 ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q2 ) = { j + len Seq q2 where j is Nat : j <= len Seq q2 & len Seq q2 = len q1 + len q2 & len q1 = len q2 + len q2 & len q1 = len q2 + len q2 & len q1 = len q2 ;
consider G1 , G2 , G1 being Element of V such that G1 <= G2 & G2 <= G2 & f = G1 & g = G2 & f = G1 & g = G2 & f = G2 & g = G2 & f = G1 & g = G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c be Element of C st c in C /\ dom it holds it /. c = - f /. c + f /. ( c ) ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v st v <> a holds union L , v |= ( union L ) , v |= ( union L ) , v |= H , v / ( a , b ) ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( k + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt ( p . ( i + 1 ) ) = ( i - n ) * ( i + 1 ) & ( i - n ) * ( i + 1 ) = ( i - n ) * ( i + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * arccot ) `| Z ) and for x st x in Z holds ( ( ( arccot * arccot ) `| Z ) `| Z ) . x = - 1 & ( ( ( arccot * arccot ) `| Z ) `| Z ) . x = - 1 ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( L~ f ) ` ) \ ( ( L~ f ) ` ) \ ( ( L~ f ) ` ) \ ( L~ f ) ` ) c= ( L~ f ) \ ( ( L~ f ) ` ) ;
ex Q1 being open Subset of X st s = Q1 & ex F1 being Subset of Y st F1 is open & F is open & ( for n being Nat st n >= m holds F1 . n = F ( n ) ) & ( for n being Nat st n >= m holds F1 . n = F ( n ) ) ;
gcd ( A , r1 ) . ( r2 , r2 ) = 1 / ( 2 * r1 ) * gcd ( A , r2 ) .= 1 / ( 2 * r1 ) * ( 2 * r1 ) .= 1 / ( 2 * r1 ) * ( 2 * r1 ) .= 1 / ( 2 * r1 ) ;
R8 = ( ( ( ( ( ( , s2 ) ) . ( m2 + 1 ) ) ) . m2 ) . m2 ) . m2 .= ( ( ( ( , s2 ) . m2 ) . m2 ) . m2 ) . m2 ) . m2 .= ( ( ( ( 3 , s2 ) . m2 ) ) . m2 .= ( ( 3 , s2 ) . m2 ) . m2 ;
CurInstr ( P-6 , Comput ( P3 , s3 , m1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) ;
func -> Subset of [: A ( ) , A ( ) :] means : Def2 : a in dom f iff ex i st i in dom f & a = f . i & a = f . i & a = f . i & b = f . i ;
for a , b being Element of F_Complex st |. a .| > |. b .| & a >= 1 holds a * ( - b ) is >= 1 implies a * ( - b ) is >= 0
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_isomorphic and g = f . 1 and for s1 , s2 being State of C1 , a being Element of C2 st s1 = g . a & s2 = f . a holds s1 is stable & a = f . a & a = f . a & b = f . a ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) & 0 <= ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) ;
for F being Subset-Family of T7 st F is open & not {} in F & for A , B being Subset of T7 st A in F & B in F & A misses B & A misses B & B misses B holds card F = card ( A \/ B )
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . k * G . k ;
i |^ ( ( mod ( n + k ) ) - i ) = i |^ ( s + k ) - i .= i |^ ( s + k ) - i |^ ( s + k ) - i ) .= i |^ ( s + ( s + k ) - i ) .= i |^ ( s + ( s + k ) - i ) ;
consider q being oriented Chain of G such that r = q & q <> {} & q . ( len q + 1 ) = v1 & ( F . ( len q + 1 ) ) . ( len q + 1 ) = v2 & ( F . ( len q + 1 ) ) . ( len q + 1 ) = v2 ;
defpred P [ Element of NAT ] means $1 <= len ( ( g , Z ) | ( len G + 1 ) ) & ( ( ( ( g , Z ) | ( len G + 1 ) ) ) | ( len G + 1 ) ) . $1 = ( ( ( ( g , Z ) | ( len G + 1 ) ) ) . ( len G + 1 ) ) . ( len G + 1 ) ) ;
for A , B being square Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width A implies A * B = B * A
consider s being FinSequence of the carrier of R such that Sum s = u & for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a * b = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x , Re y ) , ( Im y ) , ( Im y ) , ( Im y ) , ( Im y ) , ( Im y ) , ( Im y ) , ( Im y ) , ( Im y ) + ( Im y ) + ( Im y ) ) ;
consider g2 being FinSequence of FF such that g2 is continuous & rng g2 c= A & g2 . 1 = x1 & g2 . ( len g2 ) = x2 & g2 . ( len g2 ) = x2 & g2 . ( len g2 ) = x1 & g2 . ( len g2 ) = x2 ;
then n1 >= len p1 & n2 >= len ( crossover ( p1 , p2 , n1 , n2 , n3 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 ) ;
( q `1 ) * a <= ( q `1 ) * a & - ( q `2 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a & - ( q `1 ) * a <= ( q `1 ) * a ;
( ( F . ( p . ( len p ) ) ) | ( len p ) ) = ( ( F . ( len p ) ) | ( len p ) ) ) . ( len p ) .= ( ( F . ( len p ) ) | ( len p ) ) . ( len p ) .= v . ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k1 = ( ( a := intloc 0 ) , 1 ) := ( ( a := intloc 0 ) , 1 ) ) := ( ( a := intloc 0 ) := ( ( a := intloc 0 ) , 1 ) ) ;
consider B9 being Subset of B1 , y1 being Function of B1 , B2 such that B1 is finite and B1 is finite and B1 is finite and B1 is finite & B1 is finite & B1 is finite & B1 is finite & B1 is finite & B1 is finite ;
v2 . b2 = ( curry F2 ( F2 , g ) ) . b2 .= ( ( curry F2 ( F2 , g ) ) . b2 ) . b2 .= ( ( curry F2 ( F2 , g ) ) . b2 ) . b2 .= ( ( curry F2 ) . b2 ) . b2 .= ( ( curry F2 ) . b1 ) . b2 .= ( ( curry F2 ) . b2 ) . b2 ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) ) .= the carrier of SCMPDS .= the carrier of SCMPDS ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d holds |. h .| " * ||. ( R2 * ( h + R1 ) ) .|| < e / ( 2 * ( h + R1 ) ) ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , G * ( len G , 1 ) ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , p , P , p1 , p2 & LE q , p , P , p1 , p2 } ;
( ( - x ) .|. y ) = - ( ( - 1 ) * ( x .|. y ) ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 ) ^2 + ( p `2 ) ^2 ) * sqrt ( 1 + ( p `2 ) ^2 ) ;
sqrt ( ( U * ( ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * R ) ) ) ) ) ) ) ) ) ) ) ) * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * R ) ) ) ) ) ) ) ) ) ) ) = ( ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * ( W * (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def2 : dom it = dom it & for x be Element of REAL m st x in dom it holds it . x = ( - h ) . x + h . x * h . x ;
assume that 1 <= k and k + 1 <= len f and [ i + 1 , j ] in Indices G and f /. k = G * ( i , j ) and f /. k = G * ( i + 1 , j ) ;
assume that not y in Free H and x in Free H and not x in Free ( H ) and not x in Free ( H ) \/ { x } and not x in Free ( H ) \/ { y } ;
defpred P11 [ Element of NAT , Element of NAT ] means P [ p ] & ( for n being Element of NAT st $1 = p |^ n & $2 = ( $1 |^ n ) * ( 2 |^ $1 ) ) * ( 2 |^ $1 ) ) & ( $1 |^ $1 ) * ( 2 |^ $1 ) = ( $1 |^ ( 2 |^ $1 ) * ( 2 |^ $1 ) ) * ( 2 |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X st A c= it holds A is closed & for W being Subset of X st W in it & W is open & A c= W holds W is open & W is open & W is open & W is open & W is open & W is open & W is open & W is open & W is open & W is open implies W is open ;
[#] ( ( ( dist ( P ) ) .: Q ) ) = ( ( dist ( P ) ) .: Q ) .: Q ) .= ( ( dist ( P ) ) .: Q ) .: Q .= ( ( dist ( P ) ) .: Q ) . Q .= ( ( dist ( P ) ) .: Q ) . Q ;
rng ( F | ( [: S , S :] , 2 :] ) = {} or rng ( F | ( [: S , S :] , 2 :] ) ) = { 1 } or rng ( F | ( [: S , S :] , 2 :] ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 = { p1 , p2 } and P1 = { p1 , p2 } and P1 is closed and P2 is closed and p1 in P1 and p2 in P1 and p1 in P2 and p1 in P1 and p2 in P2 and p1 in P1 and p1 in P2 and p1 in P2 and p1 = p2 /\ P1 and p1 = p2 /\ P1 and p1 = p2 /\ P1 and p1 = p2 /\ P1 and p1 = p2 /\ P1 and p1 = p2 /\ P1 /\ P1 and p1 = p2 /\ P1 and p2 = p1 /\ P1 and p1 = p2 /\ P1 /\ P1 and p1 = p1 /\ P1 /\ P1 and p1 = p1 /\ P1 /\ P1 /\ P1 and p2 = p1 /\ P1 and p1 = p1 /\ P1 /\ P1 /\ P1 and p2 = p1 /\ P1 and p1 = p1 /\ P2 and p1 = p1 /\ P2 and p1 = p1 /\ P2 /\ P2 /\ P2 /\ P2 and p1 = p1 /\ P2 /\ P2 /\ P2 /\ P2 /\ P2 /\ P2 and p1 = p1 /\ P2 /\ P2 /\
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 ) , ( p2 `1 ) ^2 + ( p2 `1 ) ^2 ;
( ( \lbrace a , X , Y , Z , Y , Z , Y , Z , Y , Z , Y , Z , Y , Z , Y , Z , W , W , W , W , Z , W , W , Z , W , W , W , W , Z , W , W , W , W , W , W , W , W , W , W , W , W , W , W , W , W , W , W , W , Z , W , W , W , Z , s , W , W , W , Z , s , W , W , Z , s , W , W , W , W , W , W , W , W , W , s , W , W , W , s , W , W , W , W , s , W , s , W , s , W , s , W , s , W , s , W , s , s , s , s , s , s , s , s , s , s , s , s , s , s , s , W , W , s , s , s , W , s ,
for T being non empty TopSpace , A , B being closed Subset of T , A being Subset of T , B being Subset of T st A <> {} & B misses A & A misses B holds A is closed & B is closed & A misses B & B misses A implies A is closed
for i , j being strict Subgroup of G for G1 being strict Subgroup of G st i + 1 in dom F & G1 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G1 is strict Subgroup of G st G1 = F . ( i + 1 ) holds G1 is strict Subgroup of G
for x st x in Z holds ( ( ( arctan - arccot ) * ( arctan - arccot ) ) `| Z ) . x = ( ( ( arctan - arccot ) * ( arctan - arccot ) ) `| Z ) . x / ( ( arctan - arccot ) . x ) ^2
synonym f /* ( a ^\ k ) means : Def2 : for x0 st x0 in dom f & rng ( a ^\ k ) c= ]. x0 , x0 + r .[ & for x st x in Z holds f . x = f . x - f . x0 ) ;
then X1 , X2 misses X1 & X1 , X2 misses X2 & X1 , X2 misses X1 & X2 , X1 misses X2 & X1 , X2 misses X1 & X2 , X1 misses X2 implies X1 , X2 misses X2 & X2 , X1 misses X1 & X1 , X2 misses X2 & X2 , X2 misses X1 & X1 , X2 misses X2 & X2 , X1 misses X2 & X1 , X2 misses X2 & X2 , X2 misses X1 implies X1 , X2 misses X2 ;
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be R be ( 1 / 2 ) (#) ( SVF1 ( 1 , f , u ) ) ) . ( h . ( h . ( h . ( h . x ) ) ) ) st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - f . ( h . x ) ) = L . ( h . x - f . x ) ;
sqrt ( ( p2 `1 ) ^2 + ( p2 `1 ) ^2 ) + ( p2 `1 ) ^2 + ( p2 `2 ) ^2 + ( p2 `2 ) ^2 + ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) >= ( p2 `1 ) ^2 + ( p2 `1 ) ^2 ;
( ( 1 / t ) * ||. ( t1 - t1 ) .|| ) to_power m = ( ( 1 / t ) * ||. ( t1 - t1 ) .|| ) to_power m & ( ( 1 / t ) to_power m ) to_power m = ( 1 / t ) to_power m ;
assume that for x holds f . x = ( ( - 1 / 2 ) * ( sin . x ) ) * ( sin . x ) - ( sin . x ) ) and for x st x in Z holds ( ( sin . x ) ^2 - ( sin . x ) ^2 ) * ( sin . x ) ^2 < 1 ;
consider X1 being Subset of Y , Y1 being open Subset of X such that t = X1 & Y1 = Y1 & X1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card S . ( ( d |^ n ) + 3 ) = card { d |^ ( ( d |^ 3 ) + 1 ) + 3 ) where d is Element of GF ( p ) : not contradiction } .= 3 + 1 ;
sqrt ( ( W-bound D ) ^2 - ( W-bound D ) ^2 + ( W-bound D ) ^2 ) = ( W-bound D ) ^2 - ( W-bound D ) ^2 + ( W-bound D ) ^2 - ( W-bound D ) ^2 ) .= ( W-bound D ) ^2 - ( W-bound D ) ^2 ;