thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S opp is convergent
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of 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 `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
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 dom f ;
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 = } ;
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_R 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 , x be Point of TOP-REAL 2 ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> sup ;
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 ;
n >= ss ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be PartFunc of Y , BOOLEAN ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-bounded ;
rng f = Y ;
( G . n ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `2 = a `2 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= PI & pp c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is_continuous_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 : x is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b1 >= Z1 ;
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 ] ;
.= z (#) f ;
xx is increasing & xx is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
( G . n ) is non-decreasing ;
( G . n ) is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is finite independent & finite S 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 be Element of S ;
b ` c= b9 ` ;
assume not x in RAT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> Ideal of L ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & bn < 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 is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric ;
let x , y be element ;
let j0 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 = P +* I ;
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 , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be FinSequence of REAL ;
let b be Element of X , c be Element of Y ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> | C ;
let R be non empty multMagma , I be non empty non empty non empty doubleLoopStr ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be | mamaid id ;
let N be non empty thesis for the be non empty Subset of M ;
let R be RelStr with finite finite finite finite finite for C be finite finite finite RelStr ;
let n , k be Nat ;
let P , Q be RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not [ a , I ] ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` ;
redefine func S --> T -> such that S is such ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F1 is finite ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> [: 6 , 7 :] ;
set i1 = i + 1 ;
assume that a1 = b1 and b1 = c1 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec | Z ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( y - x ) ;
1 <= len g1 + 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 & 1 in dom D2 ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be functor of on , S ;
cluster m * n -> square ;
let kk be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive & R is transitive ;
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 ^ <* p *> ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom f1 /\ dom f2 ;
assume [ X , p ] in C ;
BQ c= XQ & BQ c= XQ ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & m in dom g2 ;
let a be Element of R ;
t `2 in dom ( e2 | X ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , M be Element of S ;
i . y in rng i ;
R^1 c= dom f & dom g c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T = S , S = T ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , V be Subset of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n2 + 1 < n2 + 1 ;
n1 < n1 + 1 & n2 + 1 < n2 + 1 ;
cluster [: T , X :] -> \overline W ;
[ 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 g ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume that n in dom h2 and n + 1 in dom h2 ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
let i be Element of NAT ;
Support u = Support p \/ { x } ;
assume X is complete thesis ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n2 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L5 ^ L5 ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let seq1 be real-valued sequence of NAT ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being \tt being being being being being being being being being being being being being being being being being being being being being being being Element of X ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n + m ) = n ;
\cal ] c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be PartFunc of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( L2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f | dom f-129 ) ;
set nn8 = n + j ;
let D7 be non empty set , f be PartFunc of D , REAL ;
let K be add-associative non empty addLoopStr , M be Matrix of K ;
assume f opp = f & h opp = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b are_maximal distance 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 ) ;
cluster -> nees[
not u in { ag } ;
the carrier of f c= B \/ { x }
reconsider z = x as VECTOR of V ;
cluster the Str of L -> \rangle ;
r (#) H is convergent ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( x in { y } ) & r-35 in { x } ;
let x , y be Element of X ;
let A , I be such such that A is such ;
[ y , z ] in [: O , O :] ;
} = dom Macro i .= dom Macro i ;
rng Sgm A = A ;
q |- \! such that 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 & LIN o , a , b ;
p . 2 = Z |^ Y ;
( k - 1 ) * ( D1 - D2 ) = {} ;
n + 1 + 1 <= len g ;
a in [: \mathbb N ( ) , D ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G ( ) , { x } :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 ;
p0 <= sup M1 & M1 <= sup M2 ;
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = 0 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
( seq " ) is non-zero & ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
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 addLoopStr , S be non empty Subset of IT ;
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 Subset of B ;
let L be non empty reflexive RelStr , X be Subset of L ;
R is reflexive & R is transitive ;
E , g |= the_right_argument_of ( H ) ;
dom G `2 /. y = a ;
( 1 / 4 ) * ( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of [: F , G :] , y be Element of F ;
D [ P-6 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng fl c= [: NAT , NAT :] ;
j `2 + 1 in dom ( s1 . f ) ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = [: A , B :] +* {} .= [: A , B :] ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
holds holds holds holds holds Y = b-a ;
assume that the distance of V , Q is , v ;
let a be Element of op ( V ) ;
let s be Element of PH ( ) ;
let Px be non empty thesis RelStr , P be Subset of X ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= v . ( v . j ) ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in dom ( JumpParts T ) & x in dom ( JumpParts T ) ;
let h be Morphism of c , a ;
Y c= [: R , the_rank_of Y :] & Y c= [: R , the_rank_of Y :] ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , 7 , 8 = Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q1 <> q2 ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & g2 is one-to-one ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( id R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 & v2 in W2 ;
assume func support L misses rng G ;
let L be lower-bounded antisymmetric RelStr , X be Subset of L ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , i be Element of I ;
0 <= Arg a & Arg b < 2 * PI ;
o9 , a9 // o9 , y & o9 , c9 // o9 , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & 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 & 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 support ( f ) ;
conv @ S c= conv A & conv @ S c= conv @ S ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! l ( ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 , W2 are_well field W1 & R is well field W2 ;
assume x in the carrier of R & y in the carrier of R ;
dom nn = Seg n & dom nn = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in T ;
assume that that that card I c= J and \rm Reloc ( J , I ) c= K ;
Im ( lim ( seq , n ) ) = 0 ;
( sin . x ) ^2 <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0
t3 . n = t3 . n & t3 . n = s . n ;
dom ( ( dom } ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W1 . x ;
y in W .vertices() \/ W .vertices() ;
k9 <= len ( v | ( k -' 1 ) ) ;
x * a \equiv y * a . mod m ;
proj2 .: S c= proj2 .: P & proj2 .: P c= proj2 .: P ;
h . p4 = g2 . I & h . p2 = g2 . I ;
\vert Y. .| = ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f . rr1 in rng f & rr2 in rng f ;
i + 1 + 1-1 <= len f - 1 ;
rng F = rng ( F | n ) .= rng ( F | n ) ;
mode be well unital non empty multMagma is commutative associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x1 . o in { x } ;
the carrier of support ( m ) c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
( s ^\ k1 ) is lower & ( s ^\ k1 ) is lower ;
len ( F | ( len F -' 1 ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , r 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 be Element of be Element of be \langle T *> ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total Function of J , J ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : ( a - 1 ) / a = 1 ;
assume that ca c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o9 , b , b9 & LIN o9 , a9 , c9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FX2 be non empty thesis ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x , pp = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty be strict be \langle non empty be strict be | of A ;
rng c `2 misses rng ( e `2 ) \/ rng ( e `2 ) ;
z is Element of gr { x } & z is Element of gr { y } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * ( cot + cot ) ) ;
the component of Q c= UBD A & Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 + ( f ^ ) ) ;
pred f = u means : : : a * f = a * u ;
for n holds P1 [ ( \mathop { } ) ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
gcd ( n1 , n2 , n2 , n3 ) = 1 & gcd ( n1 , n2 , n3 , n3 , n3 ) = 1 ;
set oo = a * 0. ( INT , o ) ;
seq . n < |. r1 .| & seq . n < x0 ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x2 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
aa , ag , ag , ah , bh , bh , bh be set ;
assume that Y = { 1 } and s = <* 1 *> ;
If1 . x = f . x .= 0 .= 0 ;
W3 .first() = W3 . 1 & W3 .first() = W3 . 2 ;
cluster trivial -> finite for M -connected finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B |^ n implies A , B |^ n are_relative_prime
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - sn ) / ( 1 - sn ) ;
f1 is_as as as as as <= rng f2 & f2 is non empty ;
f `2 / |. f .| <= q `2 / |. q .| ;
h is_\HM { implies Cage ( C , n ) /. 1 = Cage ( C , n ) /. 1 ;
b `2 / |. p .| <= p `2 / |. p .| ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( f , g ) ) ;
p2 in [: N . p1 , N . p2 :] & p2 in [: N . p1 , N . p2 :] ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def1 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , A3 , A3 , len A1 ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in 4 ( p , SC ) & c in { p , SC } ;
then S is -> atomic ;
Cl Int [#] T = [#] T .= [#] T ;
f12 | A2 = f2 | A2 .= f2 | A2 .= f2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 is conjunctive ;
1_ 1 c= ( ( t * ( p - 1 ) ) * ( p - 1 ) ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vFinSequence = ( v /. n ) * ( v /. n ) ;
r = 0. ( REAL-NS n ) & ||. 0. ( REAL-NS n ) - g .|| < r ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W \rm as Element of ( len W ) -tuples_on ( W ) ) ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t16 does not destroy b1 & t16 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id 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 , x3 ] -> pair for element ;
downarrow a /\ downarrow t is Ideal of T ;
let X be set , NAT , { 0 } , { 1 } , { 2 , 3 } ;
rng f = \langle \rm \rm <* S , X *> , X *> ;
let p be Element of B , s be Element of the connectives of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & i = q2 ;
assume gR in the right & gR in the right of g ;
let A1 , A2 be Point of S , B1 , B2 be Point of T ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , X' = Y as non empty Subset of Tlet X ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the thesis ) -valued ;
n1 <= i2 + len g2 & n2 + len g2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
that ( there ( - 1 ) * p ) ^2 = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng D2 & rng M5 c= rng D1 ;
for p being Real st p in Z holds p >= a
( the carrier of X ) = proj1 * f & ( the carrier of Y ) c= dom f ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p , M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P , g ) , g . ( mod P , g ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being strict Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Element of B ) . n & x in ( the Element of B ) . n ;
len } in Seg ( len f2 ) & len ( f1 ^ f2 ) = len f1 + len f2 ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , B be Basis of T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , u , v is_collinear and u , v , w , w is_collinear ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , S = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , b1 , c1 & LIN a1 , c1 , c1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 . O ) `2 ) ^2 / ( 1 + ( g2 . O ) `2 ) ^2 ) <= 1 ;
p in LSeg ( E . i , F . i ) ;
IB * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | Seg n ) & y = ( q | Seg n ) . x ;
Carrier ( Lxy ) misses Carrier ( Lxy ) \/ Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume that N5 = ooand o6 = oo6 and o6 = o8 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ Cj ) " { x } ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [: [. f . j , f . j .] , { 1 } :] ;
pred 0 <= x & x <= 1 implies x ^2 <= x ^2 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 = 0. TOP-REAL 2 ;
redefine func aathat ( S , T ) -> Subset of T ;
let x be Element of [: S , T :] ;
( the Arrows of F ) . ( a , b ) is one-to-one ;
|. i .| <= - - ( 2 |^ n ) & |. i .| <= - ( 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom Q ;
} * ( n + 1 ) ! > 0 * } ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ ( A1 /\ A2 ) c= A1 /\ A2 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G & y Joins X , Y , G ;
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 , A1 , A2 , A2 , C , D , E , F , J , M , N , N , M , N , N , M being set ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> being for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = q . x as Subset of V ;
x in the carrier of Lin ( A ) & y in Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and b is lower and a in { - a } ;
Int Cl A c= Cl Int Cl Int Cl A & Cl Int Cl A c= Cl Int Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1 / p2 `1
Cl Q ` = [#] ( ( T | P ) | Q ) ;
set S = the carrier of T , S = the carrier of T ;
set I8 = for f |^ n , _ { n } ;
len p - n = len ' ( p , n ) - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider nn} = nthat ( n} ) - nx as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s . f0 ) ;
let q\subseteq let qSet , qSet be Element of M ;
a9 in the carrier of S1 & b9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , i , j be Nat , p be Point of TOP-REAL 2 ;
y = ( ( f * SS ) . x ) . x ;
consider x being element such that x in be Element of implies x in be element ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = ( n , h ) `2 , i1 = ( n + 1 ) + 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , A be Subset of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 = len p2 + 1 ;
let T1 , T2 be Scott Scott Scott thesis of L , x be Element of T1 ;
then x <= y & : ex x st x c= : x in : x in y ;
set M = n -is ( m , n ) -< ( n + 1 ) -to_power ( m + 1 ) ;
reconsider i = x1 , j = x2 , i = x3 as Nat ;
rng ( the_arity_of a9 ) c= dom H & dom ( the_arity_of a9 ) = dom H ;
z1 " = z9 " & z2 " = z9 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & ( S is non empty ;
set xx = xx ^ <* Z *> ^ <* Z . ( i + 1 ) *> ;
len w1 in Seg ( len w1 + len w2 ) & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
p `1 <= Gik `1 & p `2 <= G * ( 1 , k ) `2 ;
rng ( g | X ) c= L~ ( g | X ) \/ rng ( g | X ) ;
reconsider k = i-1 * ( i + j ) + j as Nat ;
for n being Nat holds F . n is \HM { +infty } ;
reconsider xx = xx , xx = xx , xx = xx , xx = xx as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y as Element of ( REAL m ) * ;
assume i in dom ( a * p ^ q ) ;
m . ( ag ) = p . ( ag ) .= s . ( bg ) ;
a to_power ( s . m - s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = B2 \/ C1 ;
X . i = { x1 , x2 } . i .= ( X --> { x1 } ) . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
that b0 = a and b0 = b ;
FF is_closed_on t3 , Q8 & FF is_halting_on t3 , Q8 ;
set T = for X , T = for x0 holds x0 in X & x0 in X ;
Int Cl Int R c= Int Cl R & Int Cl Int R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng F[: F , G :] = { F . x , F . y } ;
G-23 " { c } c= B \/ S ;
f[#] A reduces X , X & X c= dom f ;
set RQ = the Element of P , RQ = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , i be Element of NAT ;
reconsider pp = u , pp = v as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | P ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 and f1 . n = f1 . n ;
reconsider T = b * N as Element of G / ( N , X ) ;
len Pt <= len Pt & len Pt <= len Pt ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A + B ) & [ i , j ] in Indices A ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of ( Carrier A ) & f . x = ( Carrier A ) . x ;
assume s1 = sqrt ( 2 * p ) - ( p |^ 2 ) ;
pred a > 1 & b > 0 implies a to_power b > 1 ;
let A , B , C be Subset of [: Ik , Ik , Ik :] ;
reconsider X0 = X , Y0 = Y , X0 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , r 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 of the carrier of T , the carrier of T ;
Q [ e-14 \/ { v-14 } , f . v-5 , f . v-5 , f . vm , f . em , f . em , f . em , f . em , f . em , f . e
g \circlearrowleft ( W-min L~ z ) = z implies ( g /. 1 ) .. z < ( g /. len g ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\rrangle ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan + cot ) `| Z ) . x in dom ( sec + cot ) ;
i2 = ( f /. len f ) & i2 = ( f /. len f ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G & a = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of F_Complex ;
dom g2 = the carrier of ( I[01] ) | ( the carrier of I[01] ) .= the carrier of I[01] ;
dom f2 = the carrier of ( I[01] ) & rng 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 & a1 . n < x0 + r ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
SFinSequence @ = ( S . g ) @ .= S . g ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom ( Initialized p ) & intloc 0 in dom ( Initialized p ) ;
i1 does not destroy ( i3 , i3 ) & I does not destroy ( b3 , b ) ;
arccos r + arccos r = ( PI / 2 ) + 0 ;
for x st x in Z holds f2 * ( f1 + f2 ) is_differentiable_in x ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j + 1 ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( ( T . u ) at ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( ( 1 - ( 1 - ( 1 - ( 1 - 2 ) ) ) ) ) ;
p2 `1 - x1 > - g & p1 `2 - x1 < p2 `2 - x1 ;
|. r1 - `2 .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of ( Seg 8 ) -tuples_on ( Seg 8 ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .( ) = D0W .( k + 1 ) ;
i1 = ma + n & i2 = K & i1 = K & i2 = L ;
f . a [= f . ( f |^ O1 "\/" f . a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( 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~ ( R4 ^ ( len R4 -' 1 ) ) ;
set h = the continuous Function of X , R , f = the Function of X , R ;
set A = { L . ( k9 . n ) where k is Element of NAT : k in dom L } ;
for H st H is atomic holds P7 [ H ] ;
set b' = S5 ^\ ( i + 1 ) , S|. s . i - s . i .| ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / n + 1 ) < ( 1 / s ) " ;
l `1 = [ dom l , cod l ] `1 .= [ dom l , cod l ] `1 ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , x be Element of CQC-WFF ( Al ( ) ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f1 ) ;
p2 in rng ( f /^ ( p1 -' 1 ) ) & p2 - 1 < p2 - 1 ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= indx ( D2 , D1 , j1 ) ;
assume x in ( ( ( ( ( ( ( ( ( K /\ K0 ) \/ ( K /\ K0 ) ) ) ) \/ ( K /\ K0 ) ) ) ) /\ K0 ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . O ) `2 <= 1 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , ( TOP-REAL 2 ) | P , ( TOP-REAL 2 ) | P ;
k1 -' k2 = k1 - k2 .= k1 - k2 .= k1 + k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= dom f /\ ]. x0 , x0 + r .[ ;
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `1 , K ) = - ( 1_ K , 1 ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of ordinal st a = Sum A ;
Cl ( union HH ) = union ( ( Cl H ) /\ ( Cl H ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + len t2 ;
v-29 = v + w |-- v + A9 & v-29 = v + A9 ;
v ( ) <> DataLoc ( t0 . GBP , 3 ) & v ( ) . GBP = s ( ) . GBP ;
g . s = sup ( d " { s } ) .= s ;
( \dot y ) . s = s . ( \dot y ) . s ;
{ s : s < t } in INT & t = {} ;
s ` \ s = s ` \ 0. X .= ( s ` \ 0. X ) \ ( s ` \ 0. X ) ;
defpred P [ Nat ] means B + $1 in A & not $1 in B ;
( 339 + 1 ) ! = 3339 ! * ( 339 + 1 ) ;
U . succ A = ( T . ( U , A ) ) . ( U , A ) ;
reconsider y = y as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT , the carrier of G ;
set f = ( S , U ) \mathop = S , F = S S . U , G = S . U , F = S . U , G = S . U , F = S . U , G = S . U , F = S . U , G =
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | P , ( TOP-REAL n ) | P , ( TOP-REAL n ) | P ;
( 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 ) * , R1 , R2 be Element of REAL n ;
reconsider l = 0. ( Lin A ) , r = 0. ( Lin A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + |. s .| + a ;
consider y being Element of S such that z <= y and y in X ;
a be being being being being being being being set ;
||. x9 - g9 - g .|| < r2 - 0 & ||. x - g .|| < s ;
b9 , a9 // b9 , c9 & b9 , c9 // a9 , c9 & a9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k2 + 1 ;
( p `2 / |. p .| - sn ) >= 0 ;
( q `2 / |. q .| - sn ) < 0 ;
E-max C in right_cell ( Rmax ( C , 1 ) , 1 ) /\ L~ ( Rmax ( C , 1 ) ) ;
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 `2 , a `2 // a `2 , b `2 or p `2 , a `2 // b `2 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is bijective ;
F | ( N2 , S ) = CircleMap * ( F | [: N2 , S :] ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( s , r ) ;
the carrier of (0). V = { 0. V , 0. V } .= { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] .= dom ( cos | [. - 1 , 1 .] ) ;
assume that Re ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 as 0 string of S2 , t2 = I . {} as string of S2 ;
reconsider x9 = seq , y9 = ( seq ^\ n ) . m as sequence of REAL n ;
assume that C meets L~ go and C meets L~ pion1 and x in L~ pion1 and x in L~ co ;
- ( 1 / ( n + 1 ) ) < F . n - x ;
set d1 = let f1 , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( x -' 00 ) = 2 |^ ( 2 * 100 ) - 1 ;
dom vb2 = Seg ( len db2 ) .= dom ( db2 ) ;
set x1 = - k2 + |. k2 .| + |. k1 .| + |. k2 .| + |. k2 .| ;
assume for n being Element of X holds 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 ( L ) + Carrier ( L2 ) ) c= I2 & Carrier ( L1 ) \/ Carrier ( L2 ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal S of {} ;
Z c= dom ( ( - sin * ( f1 + f2 ) ) ^ ) /\ dom ( ( - sin * ( f1 + f2 ) ) ^ ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 - |. q .| / 2 ;
ConsecutiveSet2 ( B , succ B ) c= ConsecutiveSet2 ( A , succ ( d , L ) ) ;
E = dom ( L . m ) & L . n is_measurable_on E & E is measurable & L . m c= E ;
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 ss2 = P . IC s2 .= ( I . IC s2 ) .= ( I . IC s2 ) ;
pred x > 0 means : : : 1 / x = x to_power ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [: [. p , q .] , p :] ;
b , c are_connected & - C , - C + - C + ( - C , - C + - C ) + ( - C , - C + - C ) + ( - C + - C ) + ( - C + - C ) + ( - C + - C ) ;
assume f = id the carrier of OO & f is continuous Function of [: O , O :] , O ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { v } ) ;
reconsider g = f " as Function of U2 , U1 , f " ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of G_ ( k , X ) ;
|. - x .| = - ( - x ) .= x - x .= - x ;
set S = ) ( x , y , c ) ;
Fib n * ( 5 * Fib n ) - 1 >= 4 * ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = ( - i ) * ( i qua Nat ) .= ( - i ) * ( i qua Nat ) ;
Indices M1 = [: Seg n , Seg n :] & dom M2 = Seg n & dom M2 = Seg n ;
Line ( S\mathopen { i , j } , j ) = S\mathopen { j , i } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| (#) h ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 ^ b2 & y = ( a1 ^ <* x2 *> ) ^ b1 ;
ME is_closed_on IExec ( I , P , s ) , P & ME is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f\rangle . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= y1 + z1 ;
f, f . a = felement . a & v in InputVertices S & [ v , f . a ] in InputVertices S ;
p `1 <= ( E-max C ) `1 & p `2 <= ( E-max C ) `2 ;
set R8 = Cage ( C , n ) :- E8 , R7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `2 >= ( E-max C ) `2 or p `2 = ( E-max C ) `2 ;
consider p such that p = pp and s1 < p and p < s2 and p < s2 ;
|. ( f /* ( s * F ) ) . l - ( G * F ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim s1 = x0 & lim s1 = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V & rng B = 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 in dom ( ( 1 / 2 ) (#) ( 1 / 2 ) ) ;
for L being complete LATTICE for o being Element of L holds L , L are_isomorphic implies L is isomorphic
[ gi , gj ] in Ii \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g in dom ( f1 ^ ) ;
reconsider y = ( a ` ) / ( F ` ) , y = ( a ` ) / ( F ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) (#) f ) ) . c <= h . c ;
set G3 = the \HM { of G , v , w be Vertex of G , v , w be Vertex of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / p / p & s1 . m < s / p ;
for x be element st x in ( ( for u be element st u in ( ( t - u ) * t ) ) holds x in ( ( t - u ) * t )
P = the carrier of ( ( TOP-REAL n ) | Px0 ) .= ( TOP-REAL n ) | Px0 ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p10 ) and p1 in LSeg ( p1 , p10 ) /\ LSeg ( p1 , p00 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of bounded bounded Function of X , Y , h be Function of X , Z ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | ( Seg m ) = idseq ( m ) | ( Seg n ) & m <= n ;
H * ( g " * a ) in the right * the right * N .= the right * the right * N .= N ;
x in dom ( ( cos * sin ) `| Z ) & x in dom ( ( cos * sin ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i2 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q2 , p , P , p1 , p2 ;
attr B is bounded means : Def1 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + - n & n + - n < len p ;
attr a <> 0. K means for M st the_rank_of M = the_rank_of ( a * M ) holds the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ dom /\ /\ dom ' ( i , m ) and I = len } + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ] ;
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = P2 ;
set cv = 3 / ( a , b , c ) , cv = 2 / ( a , b , c ) ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 - |. v2 - v1 .| ) / ( 2 * ( 1 - r ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) /\ dom f4 .= dom f /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) F ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq as Point of ( TOP-REAL n1 ) | K1 , ( TOP-REAL n1 ) | K1 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom *> <* *> & ( Frege ( ( Frege ( A . o ) ) . o ) ) = rng ( ( Frege ( A . o ) ) . o ) ;
for I being non degenerated doubleLoopStr holds the carrier of I is commutative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* J ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & S2 . 0 in [. a , b .] ;
v . ( lpp . i ) = ( v *' lpp ) . i .= v . i ;
consider n being element such that n in NAT and x = ( sn | n ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in dom F2 ;
{} ( X , 0 , x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , F ) = { E } ;
j + ( 2 * kk ) + m1 > j + ( 2 * kk ) + ( 2 * kk ) ;
{ s , t } on A3 & { s , t } on B3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , F ) ;
mg . ( HT ( mg , T ) ) = 0. L .= mg . ( HT ( mg , T ) ) ;
then H1 , H2 are_<* H1 , H2 , H *> & card H1 , card H2 -> finite ;
( N-min L~ ff ) .. ff > 1 & ( N-min L~ ff ) .. ff > 1 implies ( N-min L~ ff ) .. ff > 1
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | L~ g ) & x2 in [#] ( ( TOP-REAL 2 ) | L~ g ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of T ;
DigA ( t-23 , z9 ) is Element of k ( ) -tuples_on k ( ) ;
I \overline 22is k2 & I is k2 & I is k2 implies I is k2 ;
[: ua , { ua } :] = { [ a , ua ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W3 ;
for y st y in rng F ex n st y = a |^ n & a |^ n in H ;
dom ( ( g (#) ( ( id V \dot \to C ) | K ) ) | K ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) . s ) & x in ( ( ( the Sorts of U0 ) . s ) . s ) ;
ex x being element st x in ( ( ( and O ) \/ A ) . s ) ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) /\ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p10 } /\ LSeg ( p1 , p2 ) ;
( b + bholds a - b ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x be element st x in X ex u be element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in G ;
( the sequence of ( ( the carrier of X ) --> ( the carrier of Y ) ) ) ) . ( being Element of ( the carrier of X ) --> ( the carrier of Y ) ) ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` .= g | E-4 ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 , i1 = x4 , i2 = x4 as Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ) ` ;
assume ex n0 being Element of NAT st f to_power n0 is such & f to_power n0 is such ;
Seg len ( ( ( ( ( f2 ) | Seg len ( f2 ) ) ) ^ <* p *> ) ) = dom ( ( ( f2 ) | Seg len ( f2 ) ) ) ) ;
( Complement ( A . m ) ) . n c= ( Complement ( A . n ) ) . m ;
f1 . p = pp & g1 . p = d & g1 . p = c & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) .= FinS ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( n + 1 ) <= ( r2 |^ n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( G ) c= dom f ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W3 is Subspace of W3 and W3 is Subspace of W3 ;
||. t-15 . x - t-15 . x .|| = lim ||. xx - x .|| .= ||. xx - x .|| .= ||. xx - x .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( p `2 ) ^2 - 1 <= ( thesis `2 ) ^2 - 1 ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( Sphere ( p , r ) ) ;
set N8 = N-min L~ Cage ( C , n ) , N7 = N-min L~ Cage ( C , n ) , N8 = N-min L~ Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
attr a <> 0 , b means : Def1 : ( A \+\ B ) c= ( A Y. a ) \+\ ( B Y. a ) ;
then f is_\mathbin { \frac 2 } (#) pdiff1 ( f , 1 ) , 3 (#) pdiff1 ( f , 3 ) , 3 (#) pdiff1 ( f , 3 ) , 3 ) ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c > 0 and a > 0 ;
w1 , w2 in Lin { w1 , w2 , w1 } & w2 in Lin { w1 , w2 , w2 } ;
p2 /. IC s-7 = p2 . IC s-7 .= p2 . IC sU .= p2 . IC sU .= ( I I ) . IC sU ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind ( T-10 | b ) ;
[ a , A ] in the carrier of Line ( A9 , b ) & [ a , A ] in the carrier of Line ( A9 , b ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) ) . z = FALSE .= TRUE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi as Element of ( S , U ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) . ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( (0). V1 ) /\ ( the carrier of V1 ) .= { 0. V1 } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and for k be Nat st k in dom P2 holds P2 . k = F ( k ) ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s in dom ( f | X ) ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) = c /. ( |[ a , c ]| ) .= c /. ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t1 = p3 as Term of C , V , f be Term of C , V ;
( 1 - 2 ) * ( 1 - 2 ) in the carrier of [. 1 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 + D ) `2 .= C * ( p1 `2 + D , p1 `2 + D ) `2 ;
R . b \rm \hbox { - } = 2 * - b .= 2 * b - b .= b ;
consider \leq 1 such that B = - 1 * ] + ( 1 - 0 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a9 , b9 ) ) .= dom ( ( the Sorts of A ) * ( a9 , b9 ) ) ;
[ P . U7 , P . 7 ] in => ( T7 \/ T8 \/ { P . 7 } ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as /. ( len z -' 1 ) , ( L~ z ) , ( L~ z ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( 0 , 1 ) = 1 & 0 / ( 0 , 1 ) = 0 ;
assume x in the left & x in the left & y in the left & x in the left & y in the carrier of G ;
consider M being strict strict Subgroup of A9 such that a = M and T is strict Subgroup of M ;
for x st x in Z holds ( ( ( #Z n ) * f ) `| Z ) . x <> 0 & ( ( #Z n ) * f ) `| Z ) . x = f . x
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + len W3 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian LinearOperator of X , Y ;
( - i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is \langle s1 , F *> and F in the O of s2 and F in the O of s1 and F in the O of s2 ;
( ( ( ( x - y ) / ( x - y ) ) * ( x - y ) ) ) * ( x - y ) = gcd ( x , y ) ;
for u be element st u in Bags n holds ( p `2 + m ) . u = p . u
for B be 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 = tree ( p ) \/ W1 , W3 = tree ( 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 W1 /\ W2 c= the carrier of W1 /\ W2 ;
( 1 - a ) * id a = ( 1 - a ) * id a .= ( 1 - a ) * id a .= ( 1 - a ) * id a ;
( ( X --> f ) . x ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( 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 + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = c1 . c2 ;
ex P st a1 on P & a2 on P & b on P & c on P & c on P ;
reconsider gf = g opp * f opp , hg = h opp * g opp as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in V ;
n in { i where i is Nat : i < n0 + 1 & i < n + 1 } ;
F * ( i , j ) `2 >= ( F * ( m , k ) ) `2 & F * ( i , j ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn * |. p .| & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( ConsecutiveSet ( A , O1 ) ) * ( T . O1 ) ) *' ;
set I1 = Macro AddTo ( a , intloc 0 ) , I2 = SubFrom ( a , intloc 0 ) , I2 = SubFrom ( a , intloc 0 ) , I3 = goto 2 , I4 = goto 3 , I5 = goto 3 , I6 = goto 3 , I6 = goto 3 , I6 = goto 4 ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L1 & the carrier of L1 c= the carrier of L2 ;
consider xx be Element of GF ( p ) such that xx |^ 2 = a and xx |^ 2 = a ;
reconsider ee = ee , fe = fe , ff = ff as Element of D * ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) . x0 ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A = succ $1 & A = succ $1 ;
the left & - g = the left & ( for x being Element of X holds g . x = the left & g . x = - g ) implies f is left & g is left & g is left ;
reconsider pp = x , pp = y , pp = z , pp = w , pp = y , pp = z as Point of Euclid 2 ;
consider g3 such that g3 = y and x <= ex g3 being Element of X st x = g3 & ex x0 st x0 < x0 & x0 < x0 & x0 < x0 + r ;
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 y2 .= len x2 + len y2 .= len x2 + len y2 ;
for x be element st x in X holds x in the set of the set of positive & f . x = ( the set of n0 ) . x ;
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} & LSeg ( p1 , p11 ) /\ LSeg ( p1 , p11 ) = {} ;
func such such that ( X ) = [: the carrier of X , the carrier of Y :] & [: the carrier of X , the carrier of Y :] = [: the carrier of X , the carrier of Y :] ;
len ( -> ( non empty Subset of ( TOP-REAL 2 ) | ( len C -' 1 ) ) ) <= len ( C | ( len C -' 1 ) ) ;
attr K is with_a , a , b , c be Element of K , i be Nat , a be Element of K ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is NAT & f . x = f . y ;
IC Comput ( P-6 , s\rm J , k ) in dom ( PJ ) & IC Comput ( PJ , sJ , k ) in dom ( PJ ) ;
pred q < s & r < s implies ]. r , s .] \not c= ]. p , q .] & s <= q ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> id the carrier' of S2 , the carrier' of S2 , the carrier' of S2 , the carrier' of S2 = the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( #Z n ) * ( arccot ) ) `| Z ) ) /\ dom ( ( #Z n ) * ( arccot ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f \/ L~ f /\ L~ f .= { ( GoB f ) * ( i , 1 ) } ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 .= ( Cage ( C , n ) /. i ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f - len f + len f - len f + 1 - len f + 1 - len f ;
for n ex x st x in N & x in N1 & h . n = x- x0 & x in N1 ;
set s0 = ( ( a , I , p , s ) +* ( i , I ) ) . 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 xx being set such that x in xx and xx in V1 and xx in V1 and x = f . xx ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - ( len p ) ) ;
g + h = gg + hf1 & A1 + h = g + h & A2 + h = g + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) implies f / x = f / y & f . x = f . y ;
assume that 1 < p and ( 1 - p ) * q + ( 1 - p ) * q = 1 and 0 <= a and a <= b ;
F* ( f , Sum M ) = rpoly ( 1 , the carrier of F_Complex ) *' *' t + 0. F_Complex .= 0. F_Complex ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = {} implies A = {} ;
( N-min X ) `1 <= ( ( ( N-min X ) `1 ) / 2 ) `1 & ( ( ( N-min X ) `2 ) / 2 ) `2 <= ( ( ( N-min X ) `2 ) / 2 ) `2 ;
for c being Element of the *> of the bound variables of A , a being Element of the free of A holds c <> a ;
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies a = b & b = 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y = x ` \ y ;
mode BCK-algebra of i , j , m , n , m , n be Nat , i , j , m be Element of NAT ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n - 0 .| < ( e / 2 ) / ( 2 |^ n ) ;
( - q `1 ) ^2 / ( - q `2 ) ^2 <= ( - q `1 ) ^2 / ( - q `2 ) ^2 / ( - q `2 ) ^2 ;
set A = ( 2 / b-a ) ;
for x , y being set st x in R" R holds x , y are_\hbox { x , y } ;
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in ( \rm \rm \rm \rm \rm \rm holds s in ( \rm \rm \rm \rm \rm \rm carrier } ( f ) ) ) ;
for S being non empty non void non void holds S is connected holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & degree ( z `2 ) = 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and for n holds seq . ( n + k ) < x0 + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = np1 '&' ( M . x qua Element of ( the carrier of X ) --> { 0 } ) , n-15 = ( M . x ) --> TRUE ;
f " V in ' ( X ) & f " V in D & f " ( the carrier of X ) in D & f " X in the topology of ( the carrier of Y ) ;
rng ( ( a sequence c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being s of G1 such that y `1 = y and dom y `1 = WWG and y `2 = WG ;
dom ( 1 / f ) /\ ]. x0 - r , x0 .[ c= ]. x0 - r , x0 .[ /\ ]. x0 , x0 + r .[ ;
as Element of f2 ( i , j , n , r ) , f = ( i , j , n , - r ) ;
v ^ ( ( n-3 |-> 0 ) ^ ( nc1 ^ ( nc2 ^ c1 ) ) ) in Lin ( rng ( Bc2 ^ c1 ) ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i = c := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( 5 , NAT ) .= succ ( 5 , NAT ) .= NAT ;
assume that F is bbSubset-Family of X and rng p = Seg ( n + 1 ) and rng p = Seg ( n + 1 ) ;
( not LIN b , b9 , a ) & not LIN a , a9 , c & LIN a , a9 , c & a , a9 // c , a ) ;
( L1 \HM { \tt or L2 } ) \& O c= ( L1 => O ) Let ( L2 => O ) ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E st d in I holds F . d = G ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( -w ) and 0 < a and a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & for k st k in dom $1 holds $1 . k = Sum ( |. $1 .| ) ;
u = cos . ( x , y ) * v + ( cos . ( x , y ) * v ) .= cos . ( x , y ) * v .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| \bullet p , {} , id ( the Sorts of A ) ] means p = id ( the Sorts of A ) ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is ininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h . ( k + 1 ) & h . l1 <= g . ( k + 1 ) } ;
vol ( ( G . n ) vol ( E . n ) ) <= ( Partial_Sums ( ( G . n ) vol ( E . n ) ) ) vol ( E . n ) ;
f . y = x .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 , j2 %> , n ) = { i1 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , succ i2 , j2 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p10 ) = { p1 , p1 } /\ LSeg ( p1 , p01 ) .= { p1 , p1 } ;
Product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in [: Z , Z :] ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) .= Exec ( i , s1 ) ;
W-bound Qs2 <= q1 `1 & q1 `1 <= E-bound Qs2 & W-bound Qs2 <= E-bound Qs2 & W-bound Qs2 <= E-bound Qs2 & W-bound Qs2 <= E-bound Qs2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) -' 1 ) & f /. ( ( i1 + len g -' 1 ) -' 1 ) = f /. i2 ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) ) / ( x. 0 , a ) / ( x. 4 , a ) / ( x. 0 , a ) ;
len ( ( P ^ ) ) in dom ( ( P ^ ) ) & len ( ( P ^ ) ) = len ( ( P ^ ) ) + len ( ( P ^ ) ) ;
A |^ mn c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a } ;
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) - v .| & ||. v .|| = |. v .| + |. v .| ;
for phi holds phi in X implies not phi in X & phi in X & phi in X & phi in X ;
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & c = c ;
the_arity_of ( a , b , c ) = <* Hom ( b , c ) , Hom ( a , b ) *> .= <* Hom ( a , b ) , Hom ( c , d ) *> ;
consider f1 be Function of the carrier of X , R^1 such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & b1 = b1 & b2 = b2 & b1 = b2 & b1 = b3 & b2 = b3 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r , r ]| .|| /. 1 .= <* r *> . 1 .= <* r *> . 1 .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . n = C-25 . m and C . m = C-25 . n ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) * ( K * |. h .| ) <= p0 + ( K * |. h .| ) * ( K * |. h .| ) ;
attr F is commutative means : Def1 : for b being Element of X holds F \hbox { b } _ f = f . b ;
p = - ( - ( p0 + 0. TOP-REAL 2 ) ) + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= p0 + 0. TOP-REAL 2 .= p0 + 0. TOP-REAL 2 .= p0 ;
consider z1 such that b , x3 , x3 is_collinear and o , x1 , x1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear ;
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg q + ( 2 * PI * i ) and i in dom ( ( 2 * PI * i ) ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = { x } and rng g = { x } ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 misses P2 and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 ;
attr F is associative means : Def1 : F .: ( f , g ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `2 or x in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and ( P [ k2 ] ) and ( P [ k2 ] ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n & seq is convergent & lim seq = x0 & seq is convergent & lim seq = x0
F1 . [ ( id a ) , [ 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 ) . ( i + 1 ) ) and ( ( dom F ) . ( i + 1 ) ) = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y ;
cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( ( ( Mx2Tran J ) , B||. v .|| ) ) . ( Y. , j ) ) . ( Y. , j ) ;
- 1 / ( m , n ) = mmD | n .= mmD | n .= mmD .= ( - 1_ ( m , n ) ) ) * ( Det M ) .= ( Det M ) ;
attr for x be set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( Ex ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k , F . k ) c= Cl RightComp Cage ( C , k + 1 ) /\ RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k \ a ;
k -inth -ininin-in-inin-in= ( ( commute ( k + 1 ) ) . k ) .= ( ( commute ( k + 1 ) ) . i ) . i .= ( ( ( the Sorts of A ) * ( k + 1 ) ) ) . i ;
for s being State of [: A , B :] holds Following ( s , n . 0 + ( n + 2 ) * ( n + 1 ) ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is Z & for x st x in Z holds f1 . x = 1 & f2 . x <> 0 ;
support ( support ( support n ) \/ support ( support ( support m ) ) ) c= support ( support ( support ( n ) ) ) \/ support ( support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of C , ( the carrier' of C ) * the Arity of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi ( succ b1 ) . a = g . a & phi ( b ) . a = f . a & phi ( a ) . a = f . a ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 } \/ { x2 , x3 , x4 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 /\ ( U1 "\/" U2 ) c= the Sorts of U2 ;
( - ( 2 * a * ( b - a ) ) + b ) ^2 - delta ( a , b , c ) ^2 > 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 and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = <* a *> ;
Z = dom ( ( exp_R (#) ( arccot + arccot ) ) `| Z ) /\ dom ( ( arccot + arccot ) `| Z ) .= dom ( ( exp_R + arccot ) `| Z ) ;
sum ( f , SD1 ) is convergent & lim ( ( f | S ) . n ) = integral ( f , SD1 ) & lim ( ( f | S ) . n ) = integral ( f , SD1 ) ;
( X . ( ( a => f ) => ( g => x9 ) ) ) => ( x9 => ( g => x9 ) ) in
len ( M2 * M3 ) = n & width ( M3 ~ * M2 ~ ) = n & len ( M2 ~ * M3 ) = n & width ( M2 ~ * M3 ) = n ;
attr X1 union X2 is open SubSpace of X means : : : for X1 being SubSpace of X , X2 being SubSpace of X1 st X1 is open & X2 is open holds X1 is open SubSpace of X2 ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L } & X "\/" { Top L } = { Top L }
reconsider f-1= ( F . b ) `2 , f-129 = ( F . b ) `2 , f-129 = ( F . b ) `2 as Function of ( the carrier of M ) , M ;
consider w being FinSequence of I such that the InitS of M = ( the InitS of M ) -{ s ^ w } ^ w and the InitS of M , q ^ w ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in D & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = C & 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 C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o-21 = o `1 as Element of ( TS ( ( the Sorts of A ) * ( the Arity of S ) ) ) . ( o . {} ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , 0 , 0 } , x2 + 0 , 0 , 0 , 1 *> .= x1 + ( 0 * x2 + 0 ) .= x1 + x2 ;
EE " . 1 = ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 .= ( ( E qua Function ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of 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 / ( |. M .| + 1 ) ) * ( s1 . l1 ) ;
LSeg ( ( C /. n ) , ( C /. ( i + 1 ) ) ) is vertical & LSeg ( C /. ( i + 1 ) , ( C /. ( i + 1 ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( g . c * f . c ) + f . c <= h . c * ( ( - g ) . 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 \HM { the } \HM { set } , ColVec2Mx b = ( ColVec2Mx b ) \ ( ColVec2Mx b ) and len f = width A and width ColVec2Mx b = width A and len ColVec2Mx b = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & len ( - M3 ) = width M1 & width ( - M3 ) = width M2 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( TOP-REAL n ) | n ) \/ ( the InternalRel of ( TOP-REAL n ) | n )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 ;
attr a <> 0 & b <> 0 & Arg a = Arg b implies Arg ( - a ) = Arg ( - b ) & Arg ( - b ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the } \HM { of } f , a , b ) & not c in dom ( f | ]. a , b .[ )
assume that V1 is linearly-independent and V2 is linearly-independent and V1 = { v + u : v in V1 & u in V1 & v in V1 } and V1 = V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in M
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg ( card dk1 ) .= dom ( ( Pk1 ) " * Sk1 ) .= dom ( ( Pk1 ) " * Sk1 ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b and s2 . n <= c ;
h2 " . n = h2 . n " & 0 < - ( 1 / ( ( ( - 1 ) |^ n ) * ( ( - 1 ) |^ n ) ) ) & 0 < ( - ( ( - 1 ) |^ n ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. seq1 . m - seq2 . m .|| .= ||. seq1 . m - seq1 . m .|| .= ||. seq1 . m - seq1 . m .|| .= ||. seq1 . m - seq1 . m .|| .= ||. seq1 .|| . m - seq1 . m .= ( ||. seq1 .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ G ) * v & - w = 0. ( G ) & - v = 0. ( G ) ;
sup ( k .: D ) = sup ( ( k .: D ) .: ( k .: D ) ) .= k . ( sup D ) .= k . sup D .= k . sup D ;
A |^ ( k , l , .. A ) = ( A |^ ( n , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R , I being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds support ( a * b ) = support ( a ) + support ( b ) & support ( a * b ) = support ( a ) + support ( b )
consider A9 being countable set such that r is Element of CQC-WFF ( Al ) & A9 is ( A ) ` & A9 is ( A ) ` & A9 is ( A ) ` ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] , [ y2 , y2 ] } c= [: { x1 , y1 } , { y2 , y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime implies for Nat st not ( for p being prime Nat holds p divides m & p divides n ) & not ( p divides n & p divides n ) & not ( p divides n implies 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 \ a <= c holds a \+\ b <= c & c <= d
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and b in { x } ;
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 . 5 , W . 7 , G & e Joins W . 3 , W . 7 , G ;
( ( r (#) f ) | X ) . ( 2 * n ) . x = ( r (#) delta ( f ) ) . ( 2 * n ) . x + ( ( n + 1 ) * h ) . ( x + ( n + 1 ) * h ) ;
j + 1 = ( len h11 + 1 ) - len h11 + 2 .= i + 1 - len h11 + 2 - 1 .= i + 2 - 1 .= i + 2 - 1 ;
( S *' ) . f = S *' . ( ( S *' ) . f ) .= S . ( ( S *' ) . f ) .= S . ( f , f ) .= S . ( f , f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L1 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is } means : : : for p , q st p in R & q <> q holds ex P st P \/ Q = P & p in P & q in P ;
dom product ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) ;
upper_bound ( proj2 .: ( Upper_Arc C /\ /\ /\ /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r ;
i * fN - fN = i * fN - ( i * yN ) .= i * ( fN - ( i * yN ) ) .= i * ( f . ( i + 1 ) ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in C and g2 in C ;
func d |^ n -> Nat means : : : : for d being Nat st d |^ n divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n ;
f\in f . [ 0 , t ] .= ( - P ) . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 ) ^2 / ( q `2 ) ^2 <= ( q `1 ) ^2 / ( q `2 ) ^2 / ( q `2 ) ^2 / ( q `1 ) ^2 / ( q `2 ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h0 . ( i + 1 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and [ o , x2 ] in the carrier' of S ;
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 .|| .= ||. h . n .|| ;
( ( - ( #Z n ) ) * ( exp_R - exp_R ) ) . x = f . x - ( exp_R - exp_R ) . x .= ( ( - exp_R ) * ( exp_R - exp_R ) ) . x .= ( ( - exp_R ) * ( exp_R - exp_R ) ) . x ;
attr r = F .: ( p , q ) means : : : for i st i in dom r holds r . i = min ( len p , len q ) ;
( r\mathbin { 2 } ) ^2 + ( r8 - r8 ) ^2 <= ( r ^2 + ( r ^2 + 1 ) ^2 ) + ( r ^2 + 1 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det ( M @ ) = Sum ( ( ( Det M ) @ ) @ )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * v = 1 * v & a " * v = 1 & a " * v = 1 ;
p . ( j - 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p . ( j -' 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) (#) ( h ^\ n ) ) " . $1 - ( ( R /* ( h ^\ n ) ) " ) . $1 ;
assume that the carrier of H1 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 & ( |. 2 to_power ( n + 1 ) + h .| ) = n + 1 & ( n + 1 ) <= n + 1 implies ( n + 1 ) <= ( n + 1 ) & ( n + 1 ) <= ( n + 1 ) & ( n + 1 ) <= n + 1 ;
( O = 0 & O = 1 & O = 1 & O = 2 & O = 3 & O = 1 & O = 2 & O = 3 ) & O = 4 & O = 5 & O = 5 & O = 6 & O = 7 ;
F1 .: ( dom ( F1 /\ dom ( F | ( n + 2 ) ) ) ) = F1 .: ( ( n + 1 ) /\ dom ( F | ( n + 2 ) ) ) ) .= { f /. ( n + 2 ) } .= { f /. n } ;
attr b <> 0 & d <> 0 & b <> d & ( a = d & b = e implies a = b & b = e ) & ( a = be implies a = be - be ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v be Element of L st g /. i = u * a * v & u in B & v in C & v in C ;
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( i + 1 ) in dom ( s1 . i ) and f . ( i + 1 ) <> s1 . i ) and not ( i + 1 ) in dom ( s1 . i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] & [ s2 , t2 ] , [ s3 , t2 ] & [ s3 , t2 ] , [ s3 , t2 ] ;
then H is negative & H is non negative & H is non empty & H is non empty & H is non negative & H is non negative implies H is non negative ;
attr f1 is total means : Def1 : f1 is total & f2 is total & ( for c st c in dom f1 holds f1 . c = f1 . c ) & ( for c holds f1 . c = c * f2 . c ) & ( for c st c in dom f1 holds f1 . c = c * f1 . c ) implies f1 is total & f2 is total ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & ( z1 in W2 & not z1 in W1 & z2 in W2 & not z1 in W1 & z1 in W2 ) ;
p = 1 * p .= a " * a * p * p .= a " * ( b * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ k ) <= K & upper_bound rng ( seq1 ^\ k ) <= K
C meets ( L~ go \/ L~ pion1 ) or C meets ( L~ co \/ L~ co ) or C meets ( L~ co \/ L~ co ) or C meets L~ co \/ L~ co or C meets L~ co \/ L~ co ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K * K to_power k ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ' ) +* ( E .--> F ' A ) +* ( A .--> J ) +* ( A .--> A ' ) +* ( A .--> A ' ) +* ( A .--> A ' ) +* ( A .--> A ' ) +* ( A .--> A ' ) +* ( A .--> A ' ) ) ;
|. ( ( ( ( ( ( ( H . n ) || A ) ) || A ) ) . k - ( ( ( ( H . n ) || A ) . k ) || A ) . k ) .| <= e * ( ( ( ( ( ( ( ( ( H . n ) || A ) . k ) ) . k ) - ( ( ( H . n ) || A ) . k ) ) ) ) ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2 , x2
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) cos ) , A ) = 0 and ( ( exp_R (#) cos ) , A ) . x = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( ( Sgm Y ) " ) * p " & p `2 " = ( ( Sgm Y ) " ) * p " ;
for x , y st x in A holds |. 1 / ( f . x ) - 1 / ( f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) - sn ) .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) - sn .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f | X is compact holds rng ( f | X ) is compact & rng ( f | X ) c= dom ( f | X )
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and for k be Nat st k in n1 holds Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 , u1 , v1 , u1 , v1 , v2 , u2 , v2 , u2 , v2 , u2 , v2 , u2 , v2 , u1 , u2 , v2 , u2 , v2 , u1 , u2 , v2 , u2 , v2 , u2 , v2 , u1 , u2 , v2 , u2 , v2 , u1 , u2 , v2 , u2 , v2 , v2 , u2 , v2 , v2 , u2 , v2 , v2 , w , v2 , v2 , w , v2 , v2 , u2 , v2 , u2 , v2 , v2 , u2 , v2 , u2 , v2
for G being Group , A , B being non empty Subset of G , N being normal 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 to_power 0 , ( f to_power ( k + 1 ) ) (#) ( f to_power ( k + 1 ) ) (#) ( f to_power ( k + 1 ) ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= width ( f2 , b1 , b2 ) .= width ( ( f2 , b1 , b2 ) * ( i , j ) ) .= width ( ( f2 , b1 , b2 ) * ( i , j ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & for x st x in ]. - PI / 2 , PI / 2 .[ holds f " { 0 } = ]. - 1 , 1 .[ & for x st x in ]. - PI / 2 , PI / 2 .[ holds f . x = - 1 ;
assume that X is closed and a in X and a in X and y in a |^ f and x in { { n , x } } \/ y and x in a ;
Z = dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( arctan + arccot ) ) " { 0 } ) .= dom ( ( #Z 2 ) * ( ( #Z 2 ) * ( arctan + exp_R ) ) " { 0 } ) ;
func [: V , l :] -> Subset of V means : - 1 <= l & l <= len l & 1 <= k & k <= len l & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is_9 , M holds c is_\in N iff c is_<= M
for s being Element of NAT holds ( ( for v being Element of C\mathop ( C\mathop ( v , C\mathop ( C\mathop ( C\mathop ( C\mathop ( C\mathop ( v , C\mathop ( C\mathop ( C\mathop ( C\mathop ( CC\mathop ( v , C\mathop ( C\mathop ( CC\mathop ( v , C\mathop ( C\mathop ( CC\mathop ( CC\mathop ( CC\mathop ( v ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) . s = ( ( ( C\mathop ( C\mathop ( C\mathop ( CC\mathop ( C\mathop ( C\mathop
then z /. 1 = N-min L~ z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( N-min L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Nat ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and for x st x in Z holds f . x = x & f . x > 0 ;
for R being add-associative right_zeroed right_complementable commutative distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Subset of R , J being Subset of I st I = J + J holds ( I *' J ) *' ( I *' J ) c= I *' J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + y2 ) .= Seg len ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( y2 + y2 ) .= dom ( y2 + z2 ) ;
for S being Functor of C , B for c being Object of C holds card S . c = id ( ( Obj S ) . c ) & ( id C ) . c = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f6 & for x st x in f6 holds holds holds \rrangle in \rrangle & { f . x , f . x } = { f . x , f . x } ;
a in Free ( H / ( x. 4 , x. 0 ) ) '&' ( H2 / ( x. 4 , x. 0 ) ) & a in Free ( x. 0 , x. 0 ) '&' ( x. 0 , x. 0 ) ;
for C1 , C2 being v1 , f , g being Function of C1 , C2 st `1 = ( C2 ) . ( C1 , C2 ) holds f = g & f = g implies f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ ( W-bound L~ co ) .= W-bound L~ co \/ E-bound L~ co .= W-bound L~ co \/ W-bound L~ co .= W-bound L~ co \/ W-bound L~ co .= W-bound L~ co \/ W-bound L~ co ;
assume that u = <* x0 , y0 , z0 *> and f partial u , z0 & SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z0 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t = x . {} & ( t . {} ) `2 = s & ( t . {} ) `2 = s ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , 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 & b >= y ;
func Class R -> Subset-Family of R means : : : for A being Subset of R holds A in it iff ex a being Element of R st a = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( \HM { the } \HM { vertices } ) . $1 ) `1 ) c= G ) & ( ( ( \HM { the } \HM { vertices } ) . $1 ) `2 ) c= ( ( the carrier' of G ) . $1 ) `2 ;
assume dim ( W1 ) = 0 & dim ( W2 ) = 0 & dim ( W2 ) = 0 implies ( for i be Nat st i in dom ( W2 ) holds ( i = 0 implies ( i = 0 implies ( i = 1 implies ( i = 0 & i = 1 implies ( i = 1 implies i = 0 ) ) & ( i = 0 implies i = 0 ) ) ;
mamas . ( m . t ) = ( m . t ) . {} .= ( [ m . t , the carrier of C ] `1 ) . {} .= [ m . {} , the carrier of C ] `1 .= m . {} .= m . {} ;
d11 = xx ^ d22 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f ^ <* d22 *> ) . ( y9 , d22 ) .= d22 ^ d22 .= d22 ^ d22 .= d22 ;
consider g such that x = g and dom g = dom ( f . 0 ) and for x being element st x in dom ( f . 0 ) holds g . x in ( f . 0 ) . x ;
x + 0. F_Complex |^ ( len x ) = x + len x |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= ( x , len x ) |-> 0. F_Complex .= x ` .= x ` ;
kk -' ( kk + 1 ) + 1 in dom ( f | ( kk -' 1 ) ) & ( f /^ ( kk -' 1 + 1 ) ) . ( kk -' 1 + 1 ) = ( f | ( kk -' 1 + 1 ) ) . ( kk -' 1 + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P = LSeg ( p1 , p2 ) and P1 = LSeg ( p1 , p2 ) and P1 = LSeg ( p1 , p2 ) and P2 = LSeg ( p1 , p2 ) and P1 = LSeg ( p1 , p2 ) and P2 = LSeg ( p1 , p2 ) and P1 = LSeg ( p1 , p2 ) ;
reconsider a1 = a , b1 = b , c1 = c , c1 = p , c2 = p , c2 = p , c1 = p , c2 = q , c2 = r , c1 = s , c2 = s , c1 = q , c2 = s , c2 = p , c1 = q , c1 = s , c2 = s , c1 = q , c2 = s , c2 = s , c2 = s , c1 = q , c2 = s , c2 = s , c1 = q , c1 = s , c1 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = s , c2 = q , c1 = s , c2 = q , c2 = s , c2 = s , c1 =
reconsider set set set _ = G1 . ( t , b ) * F1 . f , F2 = G1 . ( t , b ) * F1 . f as Morphism of ( G1 * F1 ) . b , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n -P . m ) <= Integral ( M , P . n ) | dom ( P . n -P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 holds f1 . ( x , y ) = f2 . ( x , y ) and 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 ) `2 ) `1 - r / 2 ;
for G being Group , H being Subgroup of G , a being Element of H , b being Integer st a = b holds for i being Integer st i in dom b holds a |^ i = b |^ i & b |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where p0 is Point of TOP-REAL 2 : P [ p0 ] & ( ex p being Point of TOP-REAL 2 st p in P & p `2 >= 0 & p <> 0. TOP-REAL 2 } as Subset of TOP-REAL 2 ;
( ( N-bound C ) - ( S-bound C ) ) / 2 |^ m <= ( ( N-bound C ) - ( S-bound C ) ) / 2 |^ ( m + 1 ) - ( S-bound C ) / 2 |^ ( m + 1 ) ;
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 & |. Im ( F . n ) . x .| <= P . x ;
len @ ( @ @ p ) = len ( @ p ^ <* 0 *> ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ <* 1 *> ) + len <* 0 *> .= len ( @ p ^ <* 1 *> ) + len <* 1 *> .= len p + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) = m3 / ( x. 4 , m3 ) / ( x. 0 , m3 ) ;
consider r being Element of M such that M , v / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , x. 0 ) / ( x. 0 , x. 0 , x. 0 , x. 0 ) / ( x. 4 , x. 0 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 , x. 0 ,
func w1 \ w2 -> Element of Union ( G , RG ) equals ( ( ( ( ( ( G , R ) * ( i , j ) ) * ( i , j ) ) * ( i , j ) ) * ( i , j ) ) ) . ( w1 , j ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| . ( n + k ) - Partial_Sums ( |. seq .| . n ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| . n ) . ( n + k ) ) . ( n + k ) ;
set F = S the S -\rm implies F = S . 0 ;
( Partial_Sums ( seq ) . K ) + Sum ( seq ^\ ( K + 1 ) ) >= ( Partial_Sums ( seq ) . K ) + ( Partial_Sums ( seq ) . ( K + 1 ) ) . K ) + ( Partial_Sums ( seq ) . K ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x- x0 ) and L . ( x - x0 ) = L . ( x- x0 ) + R . ( x - x0 ) ;
func \HM { a , b , c , d , d , e , f , g , h , i , f , i , g , h , i , h , i , f , i , g , h , i , h , i , f , i , g , i , h h h , i , g , i , h h h , i , g , i , h ) ;
a * b ^2 + ( a * c ) ^2 + ( b * a ^2 + c * a ^2 ) >= 6 * a * a * b * c + ( c * a ^2 + c * a ^2 ) + ( c * a ^2 + c * a ^2 ) ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) ;
+ ( Q ^ <* x *> , p0 ) = ( ( ( Q ^ <* x *> ) +* ( M ^ { FALSE } , FALSE ) ) +* ( M ^ { FALSE } , FALSE ) ) +* ( ( M ^ <* x *> --> ( M ^ { FALSE } ) ) ) .= ( M ^ <* x *> ) +* ( M ^ <* FALSE *> ) ;
Sum ( F | n1 ) = r |^ n1 * Sum ( C | n1 ) .= C . n1 * ( n1 + 1 ) .= C . n1 * ( n1 + 1 ) .= C . n1 * ( n1 + 1 ) .= C . n1 * ( n1 + 1 ) .= C . n1 * ( n1 + 1 ) .= C . n1 * ( n1 + 1 ) ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( a * ( $1 + 1 ) * ( $1 + 1 ) ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) ;
the_arity_of g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= [ ( the Arity of S ) . g , ( the Arity of S ) . g ] `1 .= [ g , ( the Arity of S ) . g ] `1 .= g ;
( X ~ ) tolerates X ~ & card ( ( X ~ ) \ Y ) = card ( X ~ ) & card ( ( X ~ ) \ Y ) = card ( X ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . ( n + 1 ) holds b = N . ( s . n + 1 ) \ G . s ;
E , f |= All ( x. 2 , ( x. 2 ) -> ( x. 0 ) .--> ( x. 2 ) ) => ( x. 1 , ( x. 2 ) --> ( x. 1 , x. 2 ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) = the carrier of R2 & ( for i being Element of NAT st i in dom p holds p . i = 0. K ) & ( for i being Element of NAT st i in Seg n holds p . i = 0. K ) ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the \in of the \in of Sigma & ( the partial of f ) . k is Element of the set of REAL & ( the partial of f ) . k is Element of REAL & ( the partial of f ) . k is Element of REAL implies ( f . k = f . ( k + 1 ) ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , s . a ) .= Exec ( a3 := ( s . a ) , s . a ) .= s . a ;
card ( h1 ) . k = power ( F_Complex , ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k * Sum u .= ( ( - 1_ F_Complex ) * ( - 1_ F_Complex ) ) . k * Sum u .= ( ( - 1_ F_Complex ) *' ) . k * ( - 1_ F_Complex ) . k .= ( ( - 1_ F_Complex ) *' ) . k ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( 1 - g ) /. c .= ( f /. c ) * ( 1 - g ) /. c .= ( f (#) ( 1 - g ) ) /. c ;
len CC - len ( -> Element of NAT , CC ) = len ( C | ( len C -' 1 ) ) - len ( C /^ ( len C -' 1 ) ) .= len ( C /^ ( len C -' 1 ) ) - len ( C /^ ( len C -' 1 ) ) .= len C - len ( C /^ ( len C -' 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= X ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( $1 ) ) + ( 5 * Fib ( n ) * Fib ( $1 ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and n < k + 1 and f " { f . n } = { n + 1 } and for n holds f " { n + 1 } = { n } ;
consider vs be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( for A , B being Element of S holds ( A \/ B ) . A = Prob . A ) & ( B \/ C ) . A = Prob . ( c , ( B \/ C ) . A ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) and Q [ y ] ;
assume that A c= Z and f = ( ( #Z n ) * ( sin + cos ) ) / ( sin + cos ) and for x st x in Z holds ( ( #Z n ) * ( sin + cos ) ) . x = 1 ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 .= ( GoB f ) * ( 1 , j2 + 1 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 & len Seq q1 = len Seq q1 + len Seq q2 } .= dom Seq q1 \/ dom Seq q2 ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and for x being Element of G1 , y being Element of G2 st x = G2 & f . x = G1 & g . y = G2 holds G1 = G2 ;
func - f -> PartFunc of C , V means : : : : for c be Element of C st c in dom it holds it /. c = - f /. c & for c be Element of C st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a and {} <> a for v holds union rng L = a and for v holds L . ( union ( L , v ) ) = v and for a st v in union ( L , v ) holds L . a = v ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i |^ n ) and for i1 , i2 being Nat st i1 <> 0 & i2 = i |^ n holds sqrt p = ( i1 + i2 ) |^ n and ( n <= len f implies p = i2 ) ;
assume that not 0 in Z and Z c= dom ( ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( f1 + f2 ) ) ^ ) ) ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( f1 + f2 ) ^ ) ) . x = - 1 & ( ( 1 / 2 ) (#) ( f1 + f2 ) ^ ) . x = 1 ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) -' 2 ) , ( L~ f ) * ( Y -' 1 ) ) + ( L~ f -' 1 ) * ( ( L~ f ) -' 1 ) ) c= BDD L~ f ;
ex Q1 being open Subset of [: X , Y :] st s = Q1 & ex Q1 being Subset-Family of Y st Q c= F & ( for x being Subset of X st x in Q1 holds Q [ x , Q1 . x ] ) & ( x in Q & Q [ x ] ) & ( x in Q implies Q [ x , Q1 . x ] ) ;
gcd ( A9 , ( ( 1 , r1 , s1 , s2 , Amp ) , gcd ( A9 , r2 , Amp ) , gcd ( A9 , s2 , Amp ) , gcd ( A9 , s1 , Amp ) , gcd ( A9 , s2 , Amp ) , gcd ( A9 , s1 , Amp ) , gcd ( A9 , s2 , Amp ) , gcd ( A9 , r2 , Amp ) ) = 1 ;
R8 = ( ( ( ( ( ( ( ( ( ( s ) ) ) ) . ( 1 + 1 ) ) ) . ( m2 + 1 ) ) ) . ( m2 + 1 ) ) ) .= ( ( ( ( ( ( ( ( ( ( ( s ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) ) ) . ( m2 + 1 ) ) ) .= [ 3 , 4 + 1 ] ;
CurInstr ( P-6 , Comput ( Pmeans , m3 + m3 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p11 ) /\ LSeg ( p1 , p2 ) ) \/ ( LSeg ( p1 , p11 ) /\ LSeg ( p1 , p11 ) /\ LSeg ( p1 , p11 ) ) .= { p1 , p2 } \/ { p1 , p2 } .= { p1 , p2 } ;
func the still not bound not bound in the Sorts of [: A , A :] iff ex a , b , c st a in dom f & b in dom f & c in dom f & a = f . i & b = f . i & a = f . j & c = f . ( i + 1 ) & a = f . ( i + 1 ) ;
for a , b be Element of F_Complex st |. a .| > |. b .| & for f be Polynomial of F_Complex st f >= 1 holds f is or f is ] & f is or for b be Polynomial of F_Complex st b = 0 holds a * ( f . b ) is or f . b = 0
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices GoB f & [ i , j ] in Indices GoB f & [ i , j ] in Indices GoB f & f /. $1 = ( GoB f ) * ( i , j ) & [ i , j ] in Indices GoB f ;
assume that C1 , C2 are_`2 and for f being State of C1 , g being State of C2 , s1 , s2 being State of C1 , f being State of C2 , s1 , s2 being State of C2 st s1 = s2 holds s1 * f is stable & s2 * f is stable & s2 * g is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of [: T , T :] st F is open & not {} in F & A <> {} & A <> {} & A <> {} & A is open & A misses B holds card F = card ( A /\ B ) & card F = card ( A /\ B ) & card F = card ( B /\ F ) ;
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 and for k st k in dom F holds F . k = g . k and for k st k in dom F holds F . k = g . k ;
i |^ ( ( Let ( p |^ ( p |^ n ) - i ) |^ s ) ) = i |^ ( s + k ) - i |^ s .= i |^ s * i |^ k - ( i |^ k ) * ( i |^ k ) - ( i |^ k ) * 1 .= i |^ ( s * i - 1 ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and F8 . ( q . 1 ) = v1 and ( q . len q ) = v2 and rng q c= rng ( p ^ <* x *> ) and rng q c= rng ( p ^ <* x *> ) ;
defpred P [ Element of NAT ] means $1 <= len } implies ( g . $1 = ( g . $1 ) ^ ( g . $1 ) & ( g . $1 = ( g . $1 ) ^ ( g . $1 ) ) & ( g . $1 = ( g . $1 ) ^ ( g . $1 ) ) ;
for A , B being square Matrix of n , REAL for n , m being Nat st len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = a * b ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Re x ) * ( Re y ) ) , ( Im y ) * ( ( Re x ) * ( Re y ) ) , ( Im y ) * ( ( Im x ) * ( Im y ) ) , ( Im y ) * ( Im y ) ) )| ;
consider g0 be FinSequence of FQ such that f0 is continuous and rng f0 c= A and for i be Nat st i in dom ( g | k ) & 0 < i & i < len ( g | k ) & for k be Nat st k in dom ( g | k ) & k = len ( g | k ) holds ( g | k ) . ( len g ) = ( g | k ) . k ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , a9 , n2 , n3 , n3 , n3 , n3 , 7 , 8 , 7 , 8 , 8 , 8 ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 7 , 7 , 8 , 7 , 7 ,
q `1 * a <= q `1 & - q `1 * a <= q `2 & - q `2 * a <= q `1 or q `1 * a >= q `2 & q `2 * a <= q `1 & q `2 * a <= q `2 & q `2 * a <= q `2 & q `2 * a <= q `2 & q `2 * a <= q `2 & q `2 <= - q `2 * a ;
Fv . ( pp . ( len pp ) ) = Fv . ( p . ( len pp ) ) .= Fv . ( len pp + 1 ) .= vv . ( len pp + 1 ) .= vv . ( len pp + 1 ) .= vv . ( len pp + 1 ) .= v . ( len pp + 1 ) .= v . ( len pp + 1 ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ) ^ ( ( intloc 0 ) --> ( intloc 0 ) ) ) ^ ( <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ) and ( a := intloc 0 ) ^ <* halt SCM+FSA *> = <* halt SCM+FSA *> ^ <* a := intloc 0 *> ;
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D8 = the carrier of ( A1 \/ B1 ) and D8 = the carrier of ( A1 \/ B1 ) and B8 = the carrier of ( A1 \/ B1 ) and B8 = the carrier of ( A1 \/ B1 ) and B8 = the carrier of ( A1 \/ B1 ) ;
v2 . b2 = ( ( curry F2 ) * ( ( curry F2 ) * ( id B ) ) ) . b2 .= ( ( curry F2 ) * ( id B ) ) . b2 .= ( ( curry F2 ) * ( ( ( curry F2 ) * ( id B ) ) . b2 ) . b2 .= ( ( curry F2 ) * ( id B ) ) . b2 .= ( ( curry F2 ) * ( id B ) ) . b2 ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( ( card I + 2 ) , SCMPDS ) ;
ex dbeing Real st d-32 > 0 & for h being Real st h <> 0 & |. h .| < dH holds |. h .| " * ||. ( R + R1 ) /. h .|| < e / ( ( L + R1 ) /. h ) & |. h .| " * ||. ( R2 + R1 ) /. h .|| ) < e / ( ( L + R1 ) /. h ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + 1 -' 1 ) ) .= LSeg ( h /. ( i + 1 -' 1 ) , h /. ( i + 1 -' 1 ) ) .= LSeg ( h , i + 1 -' 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } ;
( ( - x ) .|. y ) = ( - ( 1 - x ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( 1 - x ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - ( 1 - x ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - ( 1 - x ) * ( x .|. y ) ) * ( x .|. y ) .= ( - ( x .|. y ) ;
0 * sqrt ( 1 - ( p `1 / p `2 ) ^2 ) = ( p `1 / sqrt ( 1 - ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 - ( p `2 / p `1 ) ^2 ) .= ( p `1 / sqrt ( 1 - ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 - ( p `2 / p `1 ) ^2 ) ;
( ( U * ( W * ( n + 1 ) ) ) ) * ( W * ( n + 1 ) ) = ( ( U * ( n + 1 ) ) ) * ( W * ( n + 1 ) ) ) * ( W * ( n + 1 ) ) .= ( U * ( n + 1 ) ) * ( W * ( n + 1 ) ) .= ( U * ( n + 1 ) ) * ( W * ( n + 1 ) ) .= W ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : : : for x be Element of REAL , f be PartFunc of REAL , REAL st x in dom it & it . x = - h & for x be Element of REAL holds it . x = ( x + h ) . x + ( h . x ) * ( h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices GoB f and [ i + 1 , j ] in Indices GoB f and f /. k = ( GoB f ) * ( i + 1 , j ) and f /. ( k + 1 ) = ( GoB f ) * ( i , j ) ;
assume that not y in Free H and x in Free H and Free ( H / ( x , y ) ) = ( Free H \ { x } ) \/ { y } and not x in Free H and not x in Free H and x in Free H and not x in Free H and x = y or x = y ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( P [ $1 ] implies ( $1 |^ ( p |^ $1 ) ) , ( $1 |^ ( p |^ $1 ) ) ) & ( $1 |^ ( p |^ $1 ) = ( $1 |^ ( p |^ $1 ) ) & ( $1 |^ ( p |^ $1 ) ) = ( $1 |^ ( p |^ $1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : - for A being Subset of X holds A in it iff for W being Subset of X st W c= X \ A & for W , Z being Subset of X st W c= X & Z c= it holds C . W = C . ( W \/ Z ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 , 2 } ;
( f " ( rng ( f | ( rng f ) ) ) ) . i = f . i " ( rng ( f | ( rng f ) ) ) .= ( f " ( rng ( f | ( rng f ) ) ) ) . i .= ( f " ( rng ( f | ( rng f ) ) ) ) . i .= ( f " ( rng ( f | ( rng f ) ) ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 ;
f . p2 = |[ ( p2 `1 ) ^2 + sqrt ( 1 - ( ( p2 `2 / p2 `1 ) ^2 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 / p2 `1 ) ^2 + ( ( p2 `2 / p2 `1 ) ^2 + ( p2 `2 / p2 `1 ) ^2 ) ]| ;
( ( ( ( a , X ) --> x ) " ) . x ) = ( ( ( ( a , X ) qua Function ) " ) . x ) " ) . x .= ( ( ( ( a , X ) " ) . x ) " ) . x .= ( ( ( ( a , X ) --> x ) " ) . x ) " .= ( ( ( a , X ) " ) . x ) " ) . x .= ( ( ( a , X ) " ) . x ) " ) . x .= ( ( ( ( a , x ) " ) . x ) " .= ( ( ( ( a , x ) " ) . x ) " .= ( ( ( a , x ) " ) . x ) " ) . x .= ( ( ( ( x ) " ) . x ) " ) . x ) " .= ( ( ( ( ( x ) ) . x ) " ) . x ) " .= ( ( ( ( ( a , x ) " ) . x ) " .= ( ( ( ( a , x ) " ) . x ) " .= ( ( ( ( ( a , x ) " )
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Point of T st p in B & A misses B & r in Cl A & p in Cl A holds ( for p being Point of T st p in A & r in B holds ( for p being Point of T st p in A holds ( for q being Point of T holds q in Cl ( A ) ) implies p in Cl ( A /\ B ) implies p in Cl ( A /\ B ) & p in Cl ( A /\ B ) & p in Cl ( A /\ B ) implies p in Cl ( A /\ B ) & p in Cl ( A /\ B ) & ( for r being Point of T implies p in Cl ( A /\ B ) & ( for r being Point of T & p in Cl ( A /\ B ) & for r being Point of T & r in Cl ( A /\ B & r in Cl ( A /\ B implies p in Cl ( A /\ B implies p in Cl ( A /\ B ) implies p in Cl ( A /\
for i , i st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . i & G2 = F . ( i + 1 ) & G1 is strict Subgroup of G & G2 is strict Subgroup of G & G1 is strict Subgroup of G holds G1 is strict Subgroup of G & G2 is strict Subgroup of G
for x st x in Z holds ( ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) . x ) = ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) . x + ( ( #Z 2 ) * ( arccot ) ) . x ) / ( ( ( #Z 2 ) * ( arccot ) ) . x ) ^2
synonym f is right continuous means : Def2 : x0 in dom ( f /* a ) & for a st rng a c= dom f & a in dom f & for n st n <= m holds f . n = ( f . n ) . x0 & for m st m <= n holds f . m = ( f /* a ) . x0 + ( f /* a ) . x0 ;
then X1 , X2 are_separated & X1 misses X2 or X1 , X2 are_separated & X2 , X2 are_separated & X1 misses X2 & X1 misses X2 & X2 misses X1 & X1 misses X2 & X2 misses X2 implies X1 misses X2 & X2 misses X1 & X1 misses X2 & X2 misses X1 & X1 misses X2 & X2 misses X2 & X1 misses X2 & X2 misses X2 implies X1 misses X2 & X2 misses X1 & X1 misses X2 & X2 misses X2 ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) + R . ( x - x0 )
( p2 `1 ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) >= ( ( p3 `1 ) ^2 + ( p3 `2 ) ^2 ) * sqrt ( 1 + ( p3 `2 / p3 `1 ) ^2 ) + ( ( p3 `2 ) ^2 / sqrt ( 1 + ( p3 `1 / p3 `1 ) ^2 ) ) ;
( ( 1 / t1 ) (#) ( ||. f1 .|| ) to_power n ) . x = ( ( 1 / t1 ) (#) ( ||. f2 .|| ) to_power m ) . x & ( ( 1 / t1 ) (#) ( ||. f1 .|| ) to_power n ) . x = ( ( 1 / t1 ) (#) ( ||. f2 .|| ) to_power n ) . x & ( ( 1 / t1 ) (#) ( ||. f2 .|| ) to_power n ) . x = ( ( 1 / t1 ) * ( ( 1 / t1 ) to_power n ) . x ) ;
assume for x holds f . x = ( ( sin + cos ) (#) ( cos * sin ) ) . x & x + h in dom ( ( sin + cos ) (#) ( sin * sin ) ) & x - h in dom ( ( sin + cos ) (#) ( sin * sin ) ) & x - h in dom ( ( sin + cos ) (#) ( sin * sin ) ) & x - h in dom ( ( sin + cos ) (#) ( sin * sin ) ) ;
consider X-23 being Subset of [: Y , X :] , Y1 being Subset of X such that Y1 = [: Xf1 , Y1 :] and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open and Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) + b where d is Element of GF ( p ) : [ d , 1 ] in R & [ d , 1 ] in R & d in { d , 1 } } ;
( ( W-bound D ) * ( i1 - 1 ) ) / ( 2 |^ ( m -' n ) ) * ( i - 1 ) = ( ( W-bound D ) * ( i - 1 ) ) / ( 2 |^ ( m -' n ) ) .= ( ( W-bound D ) * ( i - 1 ) ) / ( 2 |^ ( m -' n ) ) .= ( ( W-bound D ) * ( i - 1 ) ) / ( 2 |^ ( m -' n ) ) ;