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 ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is 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 ;
Y c= Z ;
A // M ;
let U , S , U , U , S , U , U , U , S , U ,
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G , H ;
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 , Y ;
let C , a , b , c , d ;
x _|_ p ;
o is monotone ;
let X , Y ;
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 z from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , 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 Real ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in 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 = - 1 ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_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 not F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , b be Int-Location ;
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 in dom f ;
cluster uparrow x -> in ;
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 ;
\cap \cap /\ /\ /\ /\ B >= Assume - s ;
G . y <> 0 ;
let X be RealNormSpace , Y be RealNormSpace ;
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 Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded upper-bounded ;
rng f = Y ;
( G \/ { x } ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
and still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp `1 c= PI ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , g , h ;
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 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
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 c= C ;
x9 is increasing & x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
Gseq is non-decreasing ;
Gseq is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be MSAlgebra over S ;
assume P [ n ] ;
assume union S is finite independent ;
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 , f be Function of A , S ;
b ` ` c= b9 ` ;
assume not x in REAL ;
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 sin . x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
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 transitive ;
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 , P ;
d , c // a , b ;
let t , u be set ;
let X be set with non-empty ManySortedSet of I ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , Y be non empty SubSpace of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable of f , g , h be FinSequence ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom mnn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] ] ;
let R be non empty multMagma , a , b be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co /\ 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 for \mathop { \rm for H being ] Subgroup of M ;
let R be RelStr with finite finite finite \vert ;
let n , k be Nat ;
let P , Q be V ;
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 not I does not ] ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v - u ) / 2 ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> * ;
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume that F1 <> F2 and F1 <> F2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> ( A \/ B ) ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom ( sec | A ) ;
assume [ x , y ] in R ;
set d = x / y ;
1 <= len g1 + 1 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 /\ dom D1 ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
( for i being Element of NAT st i in dom i holds i <= i ) implies ( i <= j )
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be e in the carrier of 4 ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q & I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len f2 - 1 ;
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 ;
BX c= ( X3 \/ X3 ) ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster -> $ -valued for Function ;
let Q be Subset-Family of S , P be Subset of Q ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom e2 /\ dom e2 ;
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 ;
REAL 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 , w be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S , T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , V be non empty Subset of K ;
let i be Element of NAT , f be Function ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n1 + 1 < n1 + 1 ;
n1 < n1 + 1 & n1 + 1 < n1 + 1 ;
cluster 1. ( X , Y ) -> & X is r2 ;
[ y2 , 2 ] `1 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S\cdot ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 /\ dom h2 ;
w + 1 = ma + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p \/ { x } ;
assume X is complete thesis ;
assume f = g & p = q ;
n1 <= n1 + 1 & n1 + 1 <= n1 + 1 ;
let x be Element of REAL , y be Element of REAL ;
assume x in rng s2 /\ rng s2 ;
x0 < x0 + 1 & x0 < r2 ;
len Carrier ( L ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom p = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued subsequence of seq ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being being being being being being being being being being being being being being being being being being being being Element of A ;
let i be set ;
n -' 1 = n-1 - 1 ;
len ( n-27 ) = n ;
\mathop { Z , c } c= F
assume x in X or x = X ;
x is midpoint of b , c , d ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E .: omega ;
let B1 be Basis of x , B2 be Basis of x ;
Carrier ( L2 ) /\ Carrier ( L1 ) = {} ;
L1 /\ L2 = {} & L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-129 /\ rng f-129 ;
set nN = n + j ;
let D be non empty set , f be FinSequence of D ;
let K be right_zeroed non empty addLoopStr , V be Subset of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b ) `1 > 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 -> nes] for ;
not u in { ag } ;
the carrier of f c= B \/ C ;
reconsider z = x as VECTOR of V ;
cluster the carrier of L -> being
r (#) H is e non-zero ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , a be Element of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : y in : x = r ;
let x , y be Element of X ;
let A , I be |^ of X ;
[ y , z ] in [: O , O :] ;
( for i being Element of I holds Macro i . i = 1 ) implies card Macro i = 1
rng Sgm A = A & Sgm B = B ;
q |- \! such that p |- y implies q |- p ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o9 , a , b & LIN o9 , b , a ;
p . 2 = Z |^ Y ;
( for k being Nat st k in dom MD1 holds k <= len f ) implies f is convergent
n + 1 + 1 <= len g ;
a in [: [: A , A :] , D :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
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 ) + support ( t ) ) ;
assume a in [: G ( ) , G ( ) :] ;
i `2 <= len ( y-5 ) ;
assume that p divides b1 + b2 and p divides b2 ;
> upper_bound M1 & upper_bound M1 <= upper_bound M2 ;
assume x in W-min ( X ) & y in E-max ( X ) ;
j in dom ( z | k ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= IC s2 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK1 c= h-14 & hK1 c= hh2 ;
]. 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 , f be Function of IT , IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x `2 = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive transitive & R is transitive ;
E , g |= the_right_argument_of H implies E , g |= the_right_argument_of H
dom G `2 /. y = a ;
1 / 4 >= - r / 4 ;
G . p0 in rng G & p in G . p2 ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom ( id B ) /\ 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 H ;
rng fset c= [: the carrier of G , the carrier of G :] ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of REAL ;
f . z1 in dom h & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) & N = ( A +* {} ) ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n1 in rng g ;
M . ( F . 0 ) in REAL ;
holds holds holds ( - a ) = b-a ;
assume the distance of V , Q is_V , v ;
let a be Element of op ( V ) ;
let s be Element of PP , v be Element of V ;
let PP be non empty the RelStr of L ;
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 BM = BCS ( K , n ) ;
l <= ( v . j ) `1 ;
assume x in downarrow [ s , t ] ;
x `2 `2 in uparrow t & x `2 in uparrow t ;
x in ( JumpParts T ) \/ ( JumpParts T ) ;
let h be Morphism of c , a ;
Y c= 1. ( K , the_rank_of Y ) ;
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x1 <> x2 ;
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 be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> non empty for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 in P ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , v be Element of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( S * R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 ;
assume that the carrier of L misses rng G ;
let L be lower-bounded antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , y be Element of M ;
0 <= Arg a & Arg a < 2 * PI ;
o9 , a9 // o9 , y & o9 , c9 // x , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) /\ dom f ;
rng F c= ( product f ) |^ X
assume that D2 . k in rng D and D2 . k in rng D1 ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv A & conv @ S c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! ( l , P ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\HM { field W1 , field W2 } ;
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = 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 implies [ a , y ] in implies a = b ;
assume that len Sgm I c= J and len Sgm J c= K ;
Im ( lim seq , x0 ) = 0 ;
( ( sin - cos ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies sin + cos is_differentiable_on Z & for x st x in Z holds sin + cos . x = 1 / ( cos . x ) ^2
t3 . n = t3 . n .= s . n ;
dom ( ( - x ) (#) F ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | k ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g . I ;
G6 = ( U /. 1 ) `1 .= G * ( i , j ) `1 ;
f . rs1 in rng f & rf . rs1 in rng f ;
i + 1 + 1-1 <= len f - 1 ;
rng F = rng FF2 & rng FF2 = rng FF2 ;
mode st the multF of A is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of [ [ m , n ] , B ] c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F | k ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , a be Real ;
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 Element of be Element of 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 NAT -defined Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 .= G . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def2 : ( a / a ) = 1 ;
assume that not ( a c= b & 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 , b9 , a9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let FF2 be non empty element , f be non empty FinSequence ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine func strict non empty <* an , b *> -> strict , normal ;
rng c `1 misses rng em `1 & rng c `2 misses rng em `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 * ( f1 + f2 ) ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) /\ dom ( 1 / 2 ) ;
pred f = u means : Def3 : a * f = a * u ;
for n holds P1 [ ( being Nat ) ] ;
{ 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 p = p1 ;
gcd ( n1 , n2 ) = 1 & gcd ( n1 , n2 ) = 1 ;
set oi = a * ( mj - 1 ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c be Nat st P [ c ] ;
set g = { n to_power 1 where n is Element of NAT : n >= 1 } ;
k = a or k = b or k = c ;
aa , ag , bg , bh , bh , p ;
assume Y = { 1 } & s = <* 1 *> ;
Is1 . x = f . x .= 0 .= Is2 . x ;
W3 .last() = W3 . 1 .= W2 .last() .= W2 .last() ;
cluster trivial -> finite for Walk of G ;
reconsider u = u , v = v as Element of Bags X ;
A in B @ implies A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that A , B are_that B , A are_that A , B are_that A ,
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_P is_\HM { len f2 , f , i + 1 } ;
f `2 <= q `2 & q `2 <= q `2 or f . q = q `2 ;
h is_the carrier of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= p `2 or b `2 >= p `2 ;
let f , g be s1 Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - g ) ) ;
p2 in NN & p2 in NN & p2 in NN ;
len ( the_right_argument_of H ) < len ( H ) ;
F [ A , FF . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def3 : 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 is_collinear & A1 , L , A3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C & p in C & q in C ;
then S is atomic ;
Cl ( Int [#] T ) = [#] T .= [#] T ;
f12 | A2 = f2 | ( A1 /\ A2 ) ;
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 & Y \ V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 is Subgroup of G ;
1_ G c= ( \mathop { t } ) * ( ( \mathop { t } ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v, vX = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & r = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p2 ) `2 >= 0 ;
len W = len ( W { x } ) + len ( W { x } ) ;
f /* ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 does not ^ ( W | ( k + 1 ) ) ;
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 ] -> pair for element ;
\HM { a } /\ downarrow t is Ideal of T ;
let X be N with NAT , f be Function of X , NAT ;
rng f = being Element of implies W in \ { o } ;
let p be Element of B , t be Element of the carrier 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 implies i = j or i = j
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of S ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , X, X, X, y = Y as non empty Subset of Tsuch
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n1 + 1 <= 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 ;
( as - 1 ) * ( ( - 1 ) * p ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng D2 & M5 c= rng ( D2 | ( k + 1 ) ) ;
for p being Real st p in Z holds p >= a
( for x being Point of X holds f . x = proj1 . x ) implies f is continuous
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) , g . ( mod P ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for W being Subspace of V holds W 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 ;
len that that that len } such that i in Seg len f2 and len f1 = len f2 ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Basis of T2 , f be Function of T2 , T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , u is_collinear and u , v , q is_collinear ;
[ z , z ] in union rng ( F | [: X , Y :] ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S & $1 in dom S ;
LIN a1 , a3 , b1 & LIN a1 , a3 , b1 ;
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 <= 1 & ( ( g2 ) . I ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ip * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q | k ) ;
Carrier ( Lxy ) misses Carrier ( Lxy ) \/ Carrier ( LR2 ) ;
consider c being element such that [ a , c ] in G ;
assume that N_ = o_ ( o , m ) and N_ = o_ ( o , m ) ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= FGij .: Cn & F is one-to-one ;
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 `2 - p `2 <> 0. TOP-REAL 2 ;
redefine func aa] ( S , T ) -> non empty TopSpace ;
let x be Element of [: S , T :] ;
the thesis of F ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
n * ( n + 1 ) ! > 0 * ! ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= A1 /\ A2 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} & dom B <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins Y , G2 implies x = y
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= seq1 . n .= ( seq1 + seq2 ) . 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 ) ;
dom d2 = [: A2 , A2 :] & dom d2 = [: A2 , A2 :] ;
0 < p / ( ||. 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 -> 1 -element 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 & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp0 = p . x , pp0 = p . y as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - b is Element of - X ;
Int Cl A c= Cl Int Cl A & Cl Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p2 `2 or p2 `2 >= p `2 & p `2 <= p2 `2 ;
Cl Q ` = [#] ( ( T | A ) | A ) ;
set S = the carrier of T , T = the carrier of S ;
set I8 = for f be FinSequence of TOP-REAL n , i be Element of NAT st i in dom f holds f /. i = f /. ( i + 1 ) ;
len p -' n = len p - n .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = ni - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len s-26 ;
let q\subseteq let q_ 1 , q_ 2 be State of M ;
a in the carrier of S1 & b in the carrier of S1 ;
c1 /. n1 = c1 . n1 .= c2 /. n1 .= c2 /. n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * S ) * ( x , y ) ) . i ;
consider x being element such that x in be N -carrier of A ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = ( n , i ) `1 , i1 = ( n , i ) `1 , i2 = ( n , i ) `1 , i2 = ( n + 1 ) `1 ;
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = x / 2 , n = x / 2 as Element of REAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 <= len p1 + 1 ;
let T1 , T2 be Scott n -topological s of L ;
then x <= y & ( x c= y & y c= x ) ;
set M = n -| m , N = n -| m ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng ( the_arity_of o ) c= dom H & rng ( the_arity_of o ) c= dom ( the_arity_of o ) ;
z1 " = ( z " ) * ( z * ( z * ( z * y ) ) ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ L <> {} & w in L ;
set x-10 = xx ^ <* Z *> , xq = xx ^ <* Z *> ;
len w1 in Seg len w1 & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) / ( |. a .| ) ;
p `1 <= Gik `1 & p `1 <= G * ( 1 , k ) `1 ;
rng ( g ) c= L~ ( g ) \/ rng ( g ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n be Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y as Element of ( the carrier of X ) * ;
assume i in dom ( a * p ^ q ) ;
m . ag = p . ag .= p . cg ;
a to_power ( s . m ) - 1 / ( s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 \/ C2 = {} ;
X . i = { x1 , x2 } . i .= { x1 } ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
- ( 0. R ) = a & b-0 = b ;
FF is_closed_on t2 , Q1 & I is_halting_on t2 , Q1 ;
set T = -> { the InInInof X , x0 } ;
Int Cl ( Int R ) c= Int R & Int Cl ( Int R ) c= Cl Int R ;
consider y being Element of L such that c . y = x ;
rng F{} = { F{} . x } .= { F{} . x } ;
G-23 G \ { c } c= B \/ S ;
f\rm Relation of [: X , Y :] , X & f\rm \cap [: Y , X :] c= X ;
set RQ = the Point of P , RQ = the carrier of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Element of NAT ;
reconsider p`2 = u , pj = v as Element of ( TOP-REAL n ) | P ;
g . x in dom f & x in dom g implies g . x = f . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of carr ( G ) ;
len PM <= len P-35 & len PM <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( ( A * B ) @ ) ;
for m be Nat holds Re ( F . m ) is simple Function of S , S
f . x = a . i .= a1 . k ;
let f be PartFunc of REAL i , REAL , g be PartFunc of REAL , REAL ;
rng f = the carrier of ( ( Carrier A ) * ( i , j ) ) ;
assume that s1 = sqrt ( 2 / p ) - r / 2 ;
pred a > 1 & b > 0 implies a to_power b > 1 ;
let A , B , C be Subset of Ik , D be Subset of Ik ;
reconsider X0 = X , Y0 = Y as RealNormSpace ;
let f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of n , k + 1 ;
Q [ e-14 \/ { v-5 } , f . v-5 , f . v-5 ] ;
g \circlearrowleft W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len g ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vLet ;
- f . 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 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= f . x2 ;
( ( tan - sec ) `| Z ) . x = ( sec - sec ) . x ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & [. a , b , 1_ G .] = { b } ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & rng g2 = 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 - ( G /* s ) . m .| < r ;
len Line ( A , i ) = width A .= width A ;
Sv .: ( g .: ( f .: ( g .: ( f .: ( g .: ( f , g ) ) ) ) ) ) = f ;
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 := i2 does not ` not ` not contradiction & ( f does not contradiction or f not f is not contradiction ) ;
arccos r + arccos r = PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & diff ( f2 , x ) = 1 / ( x + 1 )
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( ( T . u ) at ( C , u ) ) . x = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in the carrier of I[01] ;
p2 `1 - x1 > - g & p2 `1 - x1 < - g ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 k = 8 as Element of ( Seg 8 ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .( ) = D0W .2 + 1 ;
i1 = ma + n & i2 = K1 & i1 = K1 or i1 = i2 ;
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 or L~ R4 meets L~ R4 \/ L~ R5 ;
set h = the continuous Function of X , R , g = the continuous Function of X , R ;
set A = { L . ( k . n ) : n <= k } ;
for H st H is atomic holds P [ H ] ;
set bA = S5 ^\ ( iA + i ) , bA = S5 ^\ ( i + j ) ;
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 +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ) , x be Element of D ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ 1 ) & p2 in rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= indx ( D2 , D1 , j1 ) ;
assume x in ( ( ( ( ( ( ( ( ( K ) \/ D ) \/ ( K ) ) /\ D ) ) /\ / ( 1 + 1 ) ) ) /\ ( ( ( K ) \/ D ) /\ D ) ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . I ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 .= k1 - k2 + 1 .= k1 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 .[ ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of T st a = Sum A & A is normal ;
Cl ( union HB ) = union ( ( Cl H ) \/ ( Cl H ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t1 + len t2 ;
v-29 = v + w |-- v + ( w |-- ( x , p ) ) ;
v <> DataLoc ( t0 . GBP , 3 ) & v <> DataLoc ( t0 . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot , y ) . s = s . ( \dot , y ) ;
{ s : s < t } in REAL implies t = {} or t = {}
s ` \ s = s ` \ 0. X .= s ` \ ( s ` \ s ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in B ;
( 339 + 1 ) ! = 3315 * ( 339 + 1 ) ;
<% U %> = <% U %> & <% U %> = <% U %> ;
reconsider y = y as Element of COMPLEX * , a be Element of COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 <= len p ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) \mathop { \rm \hbox { - } of 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 , g be Function of I[01] , TOP-REAL n ;
( ( SAT M ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n + 1 ) -tuples_on REAL , a be Element of REAL ;
reconsider l = (0). V , r = 0. V , s = 0. V ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a L L L L L 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. x9 - gg .|| < r2 & ||. g - g .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 + 1 & k2 + 1 = k2 + 1 ;
( p `2 / |. p .| - sn ) / ( 1 - sn ) >= 0 ;
( q `2 / |. q .| - sn ) / ( 1 + sn ) < 0 ;
E-max C in right_cell ( Ri , 1 , G ) /\ L~ Cage ( C , n ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( lim G ) ;
LIN b , a , c or LIN b , c , a ;
p , a // a , b or p , a // b , a ;
g . n = a * Sum fk1 .= f . n ;
consider f being Subset of X such that e = f and f is strict ;
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 ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of (0). V = { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = t11 as 0 string of S2 ;
reconsider x9 = seq . n , y9 = g . n as sequence of REAL n ;
assume that C meets L~ go and C meets L~ pion1 and not E-max L~ Cage ( C , n ) meets L~ pion1 ;
- ( 1 / 2 ) < F . n . x - ( 1 / 2 ) ;
set d1 = \bf dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d1 = dist ( x1 , z2 ) ;
2 |^ ( 2 |^ 00 -' 1 ) = 2 |^ ( 2 |^ 100 ) - 2 ;
dom vp2 = Seg len d6 .= dom ( ( len d6 ) |-> d ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - k2 + 1 ;
assume for n be Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L1 + L2 ) ) c= I2 & the carrier of ( Carrier ( L1 + 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 \Rightarrow of {} ;
Z c= dom ( ( - sin * f1 ) / ( f1 + f2 ) ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 + r / 2 ;
A c= \bf \ ( A , st B c= A & A c= B implies A , B , C is_collinear
E = dom Carrier ( L ) & ( L is_measurable_on E implies L (#) F is_measurable_on E )
C to_power ( A + B ) = C to_power B * C to_power A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= ( I . IC ss2 ) ;
pred x > 0 means : Def2 : 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 + d + c + d + c + d + c + d + c + d + e + e + f ;
assume that f = id the carrier of OS and f is Function of OS , OS ;
consider v such that v <> 0. V & f . v = L * v ;
let l be
reconsider g = f " as Function of U2 , U1 , f " of U1 ;
A1 in the carrier of G_ ( k , X ) & A2 in the carrier of G ;
|. - x .| = - ( - x ) .= x - x .= - x ;
set S = is A -\frac ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * be / 4 ;
vseq /. ( k + 1 ) = v . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - i ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S, j ) = St . j .= St . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * h ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = a1 ^ b1 ;
MI is_closed_on IExec ( I , P , s ) , P & MI 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 , b ;
f, st . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) ;
f' . a = f{ a } & v in InputVertices S & v in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `2 <= ( E-max C ) `2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R8 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `2 >= ( E-max C ) `2 ;
consider p such that p = p9 and s1 < p /. i and p in L~ f ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 , x0 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f /. len f = f /. 1
dom ( Proj ( i , n ) * s ) = REAL m .= dom s ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V & dom ( B * A ) = the carrier of V ;
consider r such that r _|_ a and r \not _|_ 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 in the carrier of I[01] ;
for L being complete LATTICE holds ( for x being Element of L holds x in rng <* A , B *> ) implies L is complete
[ gi , gj ] in [: Ii \ Ij , Ij :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r in dom f1 /\ dom f2 holds f1 /. r = r * ( f1 /. x0 ) ;
reconsider y = ( a " ) * ( ( a " ) * ( b " ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) . c ) <= h . c ;
set G3 = the Vertex of G , { v } , { v } = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , x0 be Point of REAL-NS m ;
|. s1 . m - p .| / |. m - p .| < d / ( p - q ) ;
for x being element st x in ( ( for u being element st u in ( ( t \ u ) holds x in A ) ) holds x in ( ( t \ u ) \ A )
P = the carrier of ( ( TOP-REAL n ) | PP ) .= the carrier of ( ( TOP-REAL n ) | D ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ L1 ;
( 0. X \ x ) to_power ( m * k + 1 ) = 0. X ;
let g be Element of Hom ( cod f , dom f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the carrier of X , Y ;
set h = Hom ( a , g (*) f ) ;
then ( idseq n ) | ( Seg m ) = idseq m & m <= n implies m <= n & n <= m ;
H * ( g " * a ) in the carrier of H & g * ( g " * a ) in the carrier of H ;
x in dom ( ( cos - sin ) / ( cos - sin ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p , P , p1 , p2 ;
attr B is closed means : Def3 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + - n ;
attr a <> 0. K means : Def2 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb thesis and I = len } + j and k = len ( I ) + j ;
consider x1 such that z in x1 and x1 in [: P , Q :] and x1 <> x2 ;
for n ex r being Element of REAL st X [ n , r ] & r <= n
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = P2 ;
set cv = 3 / ( 2 * ( a , b , c ) ) , cv = 4 / ( 2 * ( a , b , c ) ) ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( arccot ) ) ;
r3 <= s0 + ( r0 / ( |. v2 - v1 .| - 1 ) ) / ( |. v2 - v1 .| - 1 ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom f1 /\ dom f2 ;
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 , gp = gp , gp = gp as Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom thesis , [: the carrier' of S , ( commute A ) . o :] ;
for I being non degenerated integral of R holds the carrier of I is commutative doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 = a * ( lim S1 ) ;
v . ( lw . i ) = ( v *' lw ) . i ;
consider n be element such that n in NAT and x = ( sn | n ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> F2 . x ;
{} ( X , 0 , x1 , x2 ) = { Ek } & card ( X , 0 , x1 , x2 ) = k ;
j + ( 2 * k9 ) + m1 > j + ( 2 * k9 ) + m1 ;
{ s , t } on A3 & { s , t } on A3 & { s , t } on A3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n1 ) & n1 >= len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n1 ) ;
mg . ( HT ( mg , T ) ) = 0. L .= g . ( HT ( mg , T ) ) ;
then H1 , H2 are_M & Cl H1 , Cl H2 are_) & the carrier of H1 , the carrier of H1 G ;
( N-min L~ f ) .. ( f | ( len f -' 1 ) ) > 1 & ( N-min L~ f ) .. ( f | ( len f -' 1 ) ) > 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 , x be Point of S ;
DigA ( ti1 , z ) is Element of k -tuples_on ( the carrier of K ) ;
I is da 2) & I is k2 & I is k2 implies I is k2 * k1 * k2 = k2 * k2
[: u , { u9 } :] = { [ a , u9 ] } & [: u , { u9 } :] = { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W1 ;
for y st y in rng F ex n st y = a |^ n & a |^ n = b
dom ( ( g * ( f , C ) ) | K ) = K ;
ex x being element st x in ( ( let U0 ) \/ A ) . s & x in ( ( the Sorts of U0 ) \/ B ) . s ;
ex x being element st x in ( ( and ( the carrier of OO ) \/ A ) . s ) ;
f . x in the carrier of [. - r , r .] & f . x = |[ - r , r ]| ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p01 } /\ LSeg ( p1 , p2 ) ;
( b + bs ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the sequence of ( ( the carrier of X ) --> the carrier of X ) ) . ( x - x0 ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( ( TOP-REAL 2 ) | K1 ) and q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | EK1 ` = g | EK1 ` .= g | EK1 ` .= g | EK1 ` ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 , j1 = x4 , j2 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is min & f to_power n0 is min ;
Seg len ( ( ( len f2 ) |-> d ) ^ ( ( len f1 ) |-> e ) ) = dom ( ( len f2 ) |-> d ) ;
( Complement ( A ) ) . m c= ( Complement ( A ) ) . n ;
f1 . p = p9 & g1 . ( p ^ q ) = d & f1 . ( p ^ q ) = p ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( ( r to_power n ) / ( r to_power n ) ) <= ( r2 to_power n ) / ( r to_power n ) ;
Sum F-12 = Sum f & dom F-12 = dom g & dom F-12 = dom g ;
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 W1 is Subspace of W3 and W2 is Subspace of W3 ;
||. ( t . x ) .|| = lim ||. ( x9 - y9 ) .|| .= ||. ( x9 - y9 ) .|| ;
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 + ( - ( p `2 / |. p .| - sn ) ) ^2 ) <= ( - ( - ( p `2 / |. p .| - sn ) ) ^2 ) ;
g | Sphere ( p , r ) = id E & g | Sphere ( p , r ) = id E ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
pred a <> 0 means : Def2 : ( A \+\ B ) \+\ a = ( A carrier a ) \+\ ( B be set ) ;
then f is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w1 in Lin { w2 , w1 } ;
p2 /. IC s-7 = p2 . IC ss2 .= p2 . IC ss2 .= p2 . IC ss2 ;
ind ( T-10 | b ) = ind b .= ind B .= ind B .= ind ( Tp | b ) ;
[ a , A ] in the K of K & [ a , A ] in the K of K ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o1 ) ;
( a , CompF ( PA , G ) ) . z = FALSE & ( a , CompF ( PA , G ) ) . z = TRUE ;
reconsider phi = phi /. 11 , phi = phi /. 2 , phi = phi /. 3 as Element of ( I , D ) ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = x ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of (0). V1 ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 . 0 = P2 . 1 ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s in dom ( f | X ) ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) = c /. ( |[ a , c ]| ) .= c /. ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as term of C , V ;
1 / 2 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in the carrier of [. 1 / 2 , 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 * p1 `2 .= C * p1 `2 + D * p1 `2 ;
R . b - b ` = 2 * - b .= 2 * a-b .= b ;
consider of [ - 1 , 0 * C + ( - 1 ) * A , 0 * C + 0 * A + 0 * C + 0 * C ;
dom g = dom ( ( the Sorts of A ) * a9 ) & dom g = dom ( ( the Sorts of A ) * b9 ) ;
[ P . ( l ) , P . ( l ) ] in => ( ( P . l ) , ( P . l ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) as non empty Subset of ( TOP-REAL 2 ) | ( L~ z ) ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) & y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the right of g or x = the right of g ;
consider M being strict Subgroup of AS such that a = M and T is strict Subgroup of M ;
for x st x in Z holds ( ( ( #Z 2 ) * f ) + ( #Z 2 ) * f ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W2 + m ;
reconsider h1 = ( vseq . n ) - ( t - p ) as Lipschitzian LinearOperator of X , Y ;
( - ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is_for s1 , F , s2 , F and F in the { of s2 : F in the { of s2 : F in the carrier of s2 } ;
( ( ( ( for x , y st x in Y ) holds x <> y ) & y <> x ) implies ( ( x , y ) --> ( x , y ) ) = 1
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u + m
for B being Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = W \/ W1 , W3 = W \/ W2 ;
x in { X where X is Ideal of L .: : X in the carrier of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b being Element of A holds id a * id a = id ( a + b ) ) implies a = b
( ( X --> f ) . x ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set cos = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - 2 |^ ( n -' m ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f1 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = { c2 } ;
ex P st a1 on P & a2 on P & b on P & c on P & a , b on P ;
reconsider gf = g `1 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in W ;
n in { i where i is Nat : i < n0 + 1 & i <= n + 1 } ;
F * ( i , j ) `2 >= ( F * ( m , k ) ) `2 & F * ( i , k ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn & p `2 >= 0 } ;
( / ( A , succ O1 ) ) = ( ( / ( A , O1 ) ) * ( A , O1 ) ) ;
set Is1 = Macro SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = SubFrom ( a , intloc 0 ) , Is2 = goto 2 , Is2 = goto 3 , Is2 = goto 3 ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the carrier of L1 ) /\ the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 3 = b ;
reconsider eM = e4 , fN = f-5 , fN = f-5 as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. <= M . ( n + 1 ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and n <= m ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A = ( succ A ) + ( succ $1 ) ;
the left of - g = the left of g & the carrier of - g = the carrier of g ;
reconsider pM = x , pM = y , pM = z , pM = w as Point of TOP-REAL 2 ;
consider g3 such that p4 = y and x <= g3 and ex x st x in dom f & y <= x & x <= y ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & r <= n
len ( x2 ^ y2 ) = len x2 + len y2 .= len x2 + len y2 .= len x2 + len y2 ;
for x being element st x in X holds x in the set of the set of positive Real & x in the carrier of n0 n & x <> 0. TOP-REAL n
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = {} & LSeg ( p1 , p01 ) /\ LSeg ( p1 , p01 ) = {} ;
func such that thesis for X being set holds X in [: \mathop { h } , { h } :] implies X in [: X , { h } :]
len ( { o } /. ( len Co ) ) <= len ( Co ) + len ( Co ) - 1 ;
attr K is with_a , a , b , c , i , a , b , c , d ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in { [ o , the carrier of S ] } ;
for x st x in X ex y st x c= y & y in X & y is \rm of f & y is a of f . x
IC Comput ( P-6 , sd , k ) in dom ( sJ ) & IC Comput ( Pd , sd , k ) in dom ( PJ ) ;
pred q < s means : Def2 : for r st r < s holds ]. r , s .[ \not c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def3 : for x being Element of the carrier' of S2 holds it . x = x ;
set yxy = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( ( - 1 ) (#) ( arccot ) ) * ( arccot ) ) `| Z ) ;
ri2 in Int cell ( GoB f , i , GoB f ) \ L~ f & ri2 in Int cell ( GoB f , i , GoB f ) ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + len f1 - len f & i - len f <= len f - len f + len f ;
for n ex x st x in N & x in N1 & h . n = x- x0 & h . n > 0
set ss0 = ( \mathop { a , I , p , s ) . i , sx0 = ( \mathop { a , I , p , s ) . i ;
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 or p ( k ) . 0 = 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in Vd and x9 in { x } and x9 in { x } ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( - len p ) ;
g + h = gg + hg1 & 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 ) means : Def3 : f = f | y ;
assume that 1 < p and 1 / p + 1 / q = 1 and 0 <= a and 0 <= b and b <= 1 ;
F* ( f , M ) = rpoly ( 1 , M ) *' t + ( 0. L L ) .= z ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = X
( ( N-min X ) `1 ) ^2 <= ( ( ( E-max X ) `1 ) ^2 + ( ( E-max X ) `2 ) ^2 ) ;
for c being Element of the st c <> a holds c <> a & c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= 0 .= s2 . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & x \ y = 0. X ;
mode BCK-algebra of i , j , m , n , m , n , m , m , n be BCK-algebra of i , j , m , n , m , m ;
set x2 = |( Re y , Im ( y ) )| ;
[ y , x ] in dom u5 & ( y , x ) `1 = g . y & ( y , x ) `2 = x ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & ]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < e / 2 ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `2 / |. q .| - cn ) ) ^2 ;
set A = 2 / b-a ;
for x , y being set st x in Rf " { x } & x , y are_\hbox { x } , R holds x , y are_
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = b * ( M * G ) . $1 ;
for s being element holds s in -> element iff s in -> Element of \rm \rm \rm \rm \rm Carrier ( f ) \/ \rm \rm \rm \rm \rm \rm \rm \rm Carrier ( g )
for S being non empty non void holds S is connected holds S is connected iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = nw '&' ( M . x qua Element of BOOLEAN ) , nw = ( M . x qua Element of BOOLEAN ) ;
f " V in ' ( X ) & f " V in D . ( the carrier of S , p ) & f " V in D . ( the carrier of S , p ) ;
rng ( ( a is :] set ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being & and y `1 = y and dom y `1 = WWthesis & y `2 = WWg ;
dom ( ( 1 / f ) (#) ( f ^ ) ) c= ]. - \infty , x0 .[ /\ dom ( f ^ ) ;
as Morphism of i , j , n , r be Element of j , - r , r be Element of REAL ;
v ^ ( n-3 |-> 0 ) in Lin ( ( B | c1 ) ^ <* ( B | c2 ) ^ <* ( B | c2 ) ^ <* 0 *> *> ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k1 = b . k2 & k2 = b . k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= ( NAT .--> succ i1 ) . NAT .= u ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c ;
( L1 C C C ) \& O c= ( L1 C C ) => ( L2 C ) & ( L1 , O ) on O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( /. v ) = b * ( J - w ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & |. Sum $1 .| <= Sum |. $1 .| ;
u = cos . ( x , y ) * x + ( cos 2 ) . ( x , y ) * y .= v + ( 2 * x ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| , {} , id ( the Sorts of A ) ] means p is not bound implies P [ p ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is inininand X c= Y ;
|. 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 & h <= g & l1 <= h } ;
( Partial_Sums ( G . n ) vol ) . k <= ( Partial_Sums ( ( G . n ) vol ) ) . k ;
f . y = x .= x * 1_ L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 %> , k ) = { i1 , succ i1 } .= { i1 , succ i1 } .= { succ i1 , succ i1 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p01 ) = { p1 } /\ LSeg ( p1 , p01 ) .= { p1 } /\ LSeg ( p1 , p01 ) ;
product ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) in ( ( the carrier of I-15 ) +* ( i , { 1 } ) ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ;
W-bound Qs2 <= q1 `1 & q `1 <= 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 + 1 ) ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 4 , a ) |= H / ( x. 4 , a ) / ( x. 4 , a ) ;
len ( ( P ^ ) ^ ( Q ^ ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) ;
A |^ ( n , m ) c= A |^ ( m , n ) & A |^ ( k , m ) c= A |^ ( k , l ) ;
( TOP-REAL n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p1 . n1 and p1 . n1 = p2 . 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 , s2 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s2 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id ( the carrier of V ) ) .| & ||. v .|| = 0
for phi holds not phi in X implies not phi in X & not phi in X & phi in X & phi in X
rng ( Sgm dom fp ) c= dom ( ( Sgm dom fp ) | dom ( Sgm dom ( Sgm dom ( Sgm dom ( Sgm dom dom ( Sgm dom ( Sgm dom ( Sgm dom dom ( Sgm dom dom ( q | dom q ) ) ) ) ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & b = c ;
the_arity_of o = <* o , b , c *> & the_arity_of o = <* o , b , c *> & the_result_sort_of o = <* o , b , c *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 . 0 = f . 1 ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b1 or a1 = b2 & a2 = b2 & a3 = b3 or a1 = b1 & a2 = b2 ;
D2 . ( indx ( D2 , D1 , n1 + 1 ) + 1 ) = D1 . ( n1 + 1 ) .= D1 . ( n1 + 1 ) ;
f . ( |[ r , r ]| ) = |[ |[ r , r ]| `1 .= <* r *> . 1 .= <* r *> . 1 .= x ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . n = C-25 . m and for m be Nat st m <= n holds C-25 . m = C . m ;
consider d be 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 .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def2 : for b being Element of X holds F -Sum { b } = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) .= 1 * p0 + 0. TOP-REAL 2 .= p + 0. TOP-REAL 2 .= p + 0. TOP-REAL 2 .= p + 0. TOP-REAL 2 .= p + 0. TOP-REAL 2 .= p + 0. TOP-REAL 2 ;
consider z1 such that b , x3 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 ;
consider i such that Arg ( Rotate ( s ) ) = s + Arg q + ( 2 * PI * i ) and Arg ( Rotate ( s ) ) = i ;
consider g such that g is one-to-one and dom g = card f and rng g = f . x and for x st x in dom f holds g . x = f . x ;
assume that A = P2 \/ Q2 and P2 <> {} and P2 <> {} and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 and P2 misses Q2 ;
attr F is associative means : Def3 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and k = P-2 . k2 and k <= k2 + 1 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
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 ( ( the Sorts of F ) . z ) and ( ( the Sorts of F ) . z ) . y = 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 & G * ( 0 , 1 ) `2 <= s } ;
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 ( J , BZ , BZ ) ) . ( \mathbb Z ) . ( \mathbb Z ) ;
- 1 = mm (#) D | n .= mm (#) D .= ( - m ) (#) ( ( - m ) (#) D ) .= Det M ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & ( for x being set st x in dom g holds g . x is nonnegative ) ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' All ( a , A , G ) , B , G ) '<' Ex ( 'not' All ( a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E . k0 , F . k ) c= RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k \ a ;
k -inininininin-in1 = ( commute IB ) . k .= ( ( commute IB ) . k ) .= ( ( commute IB ) . k ) .= ( ( commute IB ) . k ) ;
for s being State of AS holds Following ( s , n ) . 0 + ( n + 2 ) * n + ( n + 2 ) * n is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( thesis ) \/ support ( n ) c= support ( ( n ) + ( m ) ) \/ support ( ( n ) + ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * , the carrier' of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( g . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } \/ { x4 } .= { x1 } \/ { x2 , x3 , x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 ;
( - ( 2 * a * ( b / 2 ) ) + b ^2 - delta ( a , b , c ) ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] holds W00 = W ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and for x being Element of S holds x in ( the Arity of S ) . o ;
Z = dom ( ( exp_R * ( arccot ) ) / ( f1 + #Z 2 ) ) /\ dom ( ( exp_R * f1 ) / ( f1 + #Z 2 ) ) ;
sum ( f , SS1 ) is convergent & lim ( \HM { x } , SS2 ) = integral ( f , SS1 ) ;
( X ( ) ) => ( g => ( x9 => x9 ) ) => ( x9 => ( x9 => x9 ) ) in is Element of \rm + ( the carrier of L ) ;
len ( M2 * M3 ) = n & width ( M3 ~ * M2 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 union X2 is open SubSpace of X means : Def3 : X1 , X2 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated implies X1 union X2 misses X2 ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1= F1 . ( b `2 ) , f-1= F2 . ( b `2 ) , f-1= F2 . ( b `2 ) , f-1= F2 . ( b `2 ) , f-1= F2 . ( b `2 ) as Function of M , M ;
consider w being FinSequence of I such that the InitS of M is_ststst' <* s *> ^ w ^ w , q ^ w ^ w ^ w ^ w , q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 .= ( g . a ) |^ 0 ;
for i be Nat st i in dom f ex z be Element of L st f . i = rpoly ( 1 , z ) & z in rng f
ex L being Subset of X st Carrier L = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C1 ) c= the carrier' of C1 ;
reconsider oY = o `2 -tree p , oY = o `1 . {} as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , \dots , 0 } , x2 + x3 *> .= x1 + <* \underbrace { 0 , \dots , 0 } *> *> .= x1 + <* x1 *> ;
Ef " . 1 = ( Ef qua Function ) " . 1 .= ( ( 1 - 2 ) * ( 1 - 2 ) ) " . 1 .= ( 1 - 2 ) * ( 1 - 2 ) .= 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U0 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < 1 / ( |. M .| + 1 ) ;
LSeg ( ( Lower_Seq ( C , n ) ) * ( i , ( n + 1 ) ) , ( ( Lower_Seq ( C , n ) ) * ( i + 1 , j + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ( for i st i in the set of A \HM { b } holds f . i = ( i - 1 ) * ( i , j ) ) and ( i - 1 ) in dom A holds f . i = ( i - 1 ) * ( i , j ) ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M2 & width ( - M1 ) = width M2 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the carrier of n ) --> ( the carrier of 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 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg b ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the topology of a , b ) & not c in { a }
assume that V1 is closed & V2 is closed and V = { v + u : v in V1 & u in V2 & v in V2 } ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * x1 + ( 1 - z ) * x2 in M
rng ( ( Pk1 qua Function ) " * Sk1 ) = Seg ( card dk1 ) .= Seg ( card dk1 ) .= Seg ( card ( dom ( Pk1 ) ) ) ;
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 ;
h2 " . n = h2 . n " & 0 < - ( 1 / ( h2 . n ) ) & 0 < - ( 1 / ( h2 . n ) ) ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. seq1 .|| . m .= ( ||. seq1 .|| ) . m .= ( ||. seq1 .|| ) . m .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= s2 . b .= s2 . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ G ) * w & - w = ( - 1_ G ) * v ;
sup ( ( k .: D ) ) = sup ( ( k .: D ) ) .= k . ( sup D ) .= k . ( sup D ) .= k . ( sup D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ n ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R , J being Subset of R st I + J + K = ( I + J ) + K holds J + K = L
( f . p ) `1 = ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) = ( [ a * b ] + ( a * b ) + ( a * b )
consider A5 being countable Nat such that r is Element of CQC-WFF ( Al ) & A5 is ( Al ) -\hbox { A } & ( not ( ex x being Element of NAT st x in A & x in A ) & ( not x in A ) implies x in A ) ;
for X being non empty addLoopStr , 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 ] } c= [: { x1 , y1 } , { x2 } :] \/ [: { y2 } , { y2 } :] ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime means : Def2 : for p being prime Nat holds p divides m & not p divides n & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and ( F (#) g ) . x = ( F (#) g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( H ] (#) ( f , h ) . n ) . x = ( h (#) delta ( h ) ) . ( 2 * n + h . n ) ;
j + 1 = len h11 + 2 + 1 .= i + 1 - len h11 + 2 - 1 + 1 .= i + 1 - len h11 + 2 - 1 + 1 ;
( *' ( S *' ) ) . f = *' ( S . ( opp f ) ) .= S . ( ( opp f ) . f ) .= S . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L1 ) and for k st k in dom L1 holds L1 . k = L2 . k ;
attr R is max means : Def2 : for p , q st p in R & p <> q holds ex P st P is_an_arc_of p , q & P c= R & P c= R ;
dom product ( X --> f ) = meet ( ( X --> f ) . 0 ) .= meet ( X --> f ) .= meet ( ( X --> f ) . 0 ) .= dom f /\ dom f .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ E-bound C ) ) <= upper_bound ( proj2 .: ( C /\ E-bound C ) ) & upper_bound ( proj2 .: ( C /\ E-bound C ) ) <= upper_bound ( proj2 .: ( C /\ E-bound C ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - ps2 .| < r
i * fN - fN = i * fN - ( i * yN - i * yN ) .= i * ( fN - i * fN ) - i * fN ;
consider f being Function such that dom f = 2 -tuples_on X & 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 union C and g2 in C ;
func d |^ n -> Nat means : Def3 : d |^ it divides n & d |^ it divides n & d |^ it divides n & d divides n & it divides n ;
f\in . [ 0 , t ] = f . [ 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 or t = M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / 2 ;
( q `1 / q `2 ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) <= ( q `2 / q `2 ) ^2 / ( 1 + ( q `2 / q `1 ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h0 . ( i + 1 + 1 -' len h11 + 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 & a >= b & b <= a
||. h1 .|| . n = ||. h1 . n .|| .= |. h .| . n .= |. h .| . n .= |. h .| . n .= |. h .| . n .= |. h .| . n ;
( ( ( - 1 ) (#) ( #Z 2 ) ) `| Z ) . x = f . x - ( #Z 2 ) . x .= ( - 1 ) * ( #Z 2 ) . x .= ( - 1 ) * ( #Z 2 ) . x .= ( - 1 ) * ( ( #Z 2 ) . x ) .= ( - 1 ) * ( ( #Z 2 ) * ( #Z 2 ) ) . x ;
pred r = F .: ( p , q ) means : Def2 : len r = min ( len p , len q ) ;
( rmin / 2 ) ^2 + ( rmin / 2 ) ^2 <= ( r ^2 + ( r / 2 ) ^2 ) + ( r / 2 ) ^2 ;
for i be Nat , M be Matrix of n , K st i in Seg n holds Det M = Sum ( ( Det M ) * ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 / R * v & a " * ( a * v ) = 1 / R * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) * r3 ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) . $1 * ( h ^\ n ) " .= L . ( h + n ) * ( h ^\ n ) " ;
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 H1 = 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 -: h = n + 1 -: h = n + 1 -: h = n + 1 -: h = n + 1 ;
( O6 = 0 & O6 = 1 & O6 = 0 & O6 = 0 or O6 = 1 & O6 = 0 or O6 = 1 & O6 = 0 ) ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( 1 / 2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a / b ) = ( - e ) / ( b - d ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= dom f /\ D ;
for i be set st i in dom g ex u , v be Element of L , a be Element of B st g /. i = u * a * v & u in A & v in B
g `1 * P `2 * g `2 " = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) " .= g `2 * ( g `2 * P `2 ) " ;
consider i , s1 such that f . i = s1 and not ( i = s1 & not i <> s1 ) & not i <> s1 & not i = s1 or i = s2 & not i = s1 ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s2 , t2 ] are_connected implies [ s2 , t2 ] , [ s2 , t2 ] are_connected
then H is negative & H is not negative & H is not conjunctive & H is not g\mathop of H & H is not negative implies H is not an implies H is not an implies H is not negative
attr f1 is total means : Def3 : f1 (#) f2 is total & ( for c st c in dom f1 holds f1 . c = f1 . c * f2 . c ) & ( for c st c in dom f1 holds f1 . c = c * ( f2 . c ) " ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z2 in W2 -Seg ( z2 ) & not z1 in W2 -Seg ( z2 ) & not z1 in W2 -Seg ( z2 ) ;
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * b * ( b * q ) .= a " * b * q ;
for seq1 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound ( rng seq1 ) <= ( upper_bound ( rng seq1 ) ) * ( upper_bound ( rng seq ) ) ;
C meets ( L~ go \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) or C meets ( L~ pion1 \/ L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' +* ( C .--> D ) ) +* ( E .--> F ) +* ( F .--> J ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ;
|. ( ( \HM { the } \HM { lower ( n , T ) ) || A ) . k - ( ( \HM { the } \HM { lower ( n , T ) ) || A ) . k .| <= e * ( b-a ) ;
( (
{ 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 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) cos ) , A ) = 0 and integral ( ( exp_R (#) cos ) , A ) = 0 ;
p `2 `2 is Permutation of dom f1 & p `2 `1 = ( Sgm Y ) " Y * p " & p `2 `2 = ( Sgm Y ) " Y * p " ;
for x , y st x in A & y in A holds |. ( 1 / f . x ) - ( 1 / f . y ) .| <= 1 * |. f . x - f . y .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) .= |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 - sn ) ) .= |. q2 .| ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f holds f | X is continuous & for x be Point of CNS st x in dom f & x in dom f holds f | X is continuous
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 // v , v1 & u , u1 // v , v2 & u , u1 // v , v2 & u1 , v1 // v , v1 & u1 , v1 // v , v2 & u1 , v1 // v , v1 ;
for G being Group , A , B being non empty normal Subgroup of G , N being normal Subgroup of G holds ( N being normal Subgroup of G st N = N holds N is Subgroup of A * B
for s be Real st s in dom F holds F . s = integral ( R to_power 0 , ( f + g ) (#) e to_power k , ( f + g ) (#) e to_power k ) . x
width AutMt ( f1 , b1 , b2 ) = len b2 .= width ( ( len b2 ) |-> 0. K ) .= width ( ( len b2 ) |-> 0. K ) .= width ( ( len b2 ) |-> 0. K ) .= len ( ( len b2 ) |-> 0. K ) .= len ( ( len b2 ) |-> 0. K ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - 1 , 1 .[ & for x st x in ]. - 1 , 1 .[ holds f . x = - 1 / 2 * PI + 1 / 2 * PI ;
assume that X is closed w.r.t. for a st a in X and a c= X and y in a ^ { [ n , x ] } \/ y and x in a ;
Z = dom ( ( ( 1 / 2 ) (#) ( arctan + arccot ) ) / ( f1 + arccot ) ) /\ dom ( ( 1 / 2 ) (#) ( f1 + #Z 2 ) ) ;
func > ( V ) -> Subset of V equals { l . k : 1 <= k & k <= len l & l . k in V } ;
for L being non empty TopSpace , N being net of L , M being net of L st c is_be net of N for x being Point of L st x in M holds c is_inf M
for s being Element of NAT holds ( ( for v being Element of C\mathop ( C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ) ) ) ) ) . s = ( ( ( id Cv , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ( v , C\mathop ) ) ) ) . s ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( ( N-min L~ z ) .. z < ( E-max L~ z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= len p + 1 + 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 / ( sin . x ) ^2 and f . x > 0 ;
for R being add-associative right_zeroed right_complementable left distributive non empty doubleLoopStr , I being Ideal of R , J being Subset of R , I being Ideal of R , J being Ideal of R st J + J c= I 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 + z2 ) .= dom ( x2 + z2 ) .= dom ( x (#) z ) .= dom ( x (#) z ) .= dom ( x (#) z ) ;
for S being Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( Obj S ) . c ) & S . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds holds \rrangle in \mathop { f . x , f . x } & for x st x in f5 holds x in { f . x } ;
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' H2 / ( x. 4 , x. 0 ) '&' H2 / ( x. 4 , x. 0 ) '&' H2 / ( x. 4 , x. 0 ) '&' H2 / ( x. 4 , x. 0 ) ;
for C1 , C2 being is x , C2 being stable Function of C1 , C2 st ( for f being Function of C1 , C2 st f = g holds f is stable ) & ( for g being Function of C1 , C2 st g = f holds g is continuous ) holds f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ E-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ go \/ W-bound L~ co .= W-bound L~ pion1 \/ W-bound L~ co .= W-bound L~ pion1 \/ W-bound L~ co ;
assume that u = <* x0 , y0 , z0 *> and f is_partial u , 3 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) is_differentiable_in z0 and 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 . {} = [ x , s ] ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T ~ st a = f . x & b = f . y holds a >= b ;
func Class R -> Subset-Family of R means : Def3 : 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 \cdot ( ( the Vertex of G ) `1 ) & ( ( the Target of G ) `2 ) `2 c= G * ( ( the Vertex of G ) `2 ) ;
assume that dim W1 = 0 and dim W1 = 0 and ( dim W2 = 0 implies dim W1 = 0 & V = { 0. W1 } ) and V = { 0. W1 } and V = { 0. W1 } and V = { 0. W1 } and V = { 0. W1 } and V = V ;
mama-\mathopen ( m . t ) = ( m . t ) `1 .= [ [ m . t , the carrier of C ] `1 .= [ m . t , the carrier of C ] `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= d22 ;
consider g such that x = g & dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 . x and g . x <> 0 ;
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 ` ;
( k -' k + 1 ) in dom ( f /. ( k -' 1 ) ) /\ ( len f -' k + 1 ) & ( f /. ( k -' 1 ) ) = ( f /. k ) -' ( k + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P2 and P1 = P2 and P2 = P2 and P1 = P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P1 = P2 and P2 = P2 \/ P2
reconsider a1 = a , b1 = b , c1 = c `1 , c1 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 ,
reconsider Gt1f = G1 . ( t * b ) * F1 . f as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b * F1 . 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 -P . m ) | 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 ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) `1 - G * ( i + 1 , 1 ) `1 ) , G * ( i + 1 , 1 ) `2 - 2 ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Integer , b being Integer st i in H holds a |^ i = b |^ i * a |^ 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 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } as Subset of TOP-REAL 2 ;
( ( ( ( for m st m in C ) holds ( ( for k st k in C holds k <= m ) ) / ( 2 |^ m ) ) implies ( ( ( ( ( k + 1 ) / 2 ) ) * ( 2 |^ m ) ) / ( 2 |^ m ) ) * ( 2 |^ m ) ) <= ( ( ( ( k + 1 ) / 2 ) * ( 2 |^ m ) ) / ( 2 |^ m ) ) / ( 2 |^ m )
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| . x <= P . x
len @ p = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len ( @ p ^ @ q ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 4 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m1 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m1 ) = m3 / ( x. 0 , m2 ) / ( x. 0 , m2 ) ;
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ;
func w1 \ w2 -> Element of Union ( G , R6 ) equals ( ( HK6 ( G , R ) ) . ( w1 , w2 ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= ( s | n2 ) . b2 .= s . b2 .= s . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . n + ( Partial_Sums |. seq .| ) . ( n + k ) - ( Partial_Sums |. seq .| ) . n ;
set F = S -\mathop { {} } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + ( Partial_Sums ( seq ) ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x0 ) + R . ( x - x0 ) ;
func closed -> closed Subset of TOP-REAL 2 equals ( the \HM { a , b , c , d } ) /\ ( the carrier of TOP-REAL 2 ) .= ( the \rm of TOP-REAL 2 ) /\ ( the carrier of TOP-REAL 2 ) ;
a * b ^2 + ( a * c ^2 + b * a ^2 ) + ( c * b ^2 + c * a ^2 + b * c ^2 + c * a ^2 >= 6 * a * b * c * a * c + c * a + b * c + c * a * b + c * a + d * a + d * b + c * a + d * b + c * a + d * b + c * a + d * b + c * a + d * a + c * b + c * a + d * b + c * a + c * a + d * b + c * b ^2 + c * a + d * a + d * a +
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x3 , m1 ) = v / ( x2 , m1 ) / ( x3 , m2 ) / ( x3 , m1 ) / ( x4 , m2 ) .= v / ( x3 , m1 ) / ( x4 , m2 ) ;
Assume that Nat ( Q ^ <* x *> , M ) = ( \hbox { x } , M ) +* ( ^ ( q , M ) --> FALSE ) and ( ( Q ^ <* x *> , M ) --> TRUE ) +* ( ( Q ^ <* x *> , TRUE ) --> TRUE ) = ( Q ^ <* x *> , TRUE ) --> TRUE ;
Sum FM = r |^ n1 * Sum Cz .= C . n1 * ( r |^ n1 ) .= CM . n1 * ( r |^ n1 ) .= CM . n1 * ( r |^ n1 ) .= ( CM | ( n1 + 1 ) ) . n1 ;
( 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 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) ) + b ;
the_arity_of g = ( the Arity of S ) . g .= [ [ ( the Arity of S ) . g , g ] , g ] .= [ g , g ] `1 .= g .= g ;
( [: X , Y :] |^ Z ) tolerates [: X , Y :] |^ Z & card ( ( X , Y ) |^ Z ) = card [: X , Y :] ;
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. 2 ) '&' ( x. 1 ) '&' ( x. 2 ) ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 2 ) '&' ( x. 1 ) '&' ( x. 2 ) ;
ex R2 being 1-sorted st R2 = ( p | n-11 ) . i & ( ( p | n-11 ) . i = the carrier of R2 & ( p | n-11 ) . i = the carrier of R2 ) & ( p | n-11 = p | n-11 ) ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the carrier of f & ( the partial of f ) . k is Element of the carrier of f & ( the partial of f ) . k is Element of the carrier of f & ( the "/\" of f ) . k is Element of the carrier of f & ( the Element of f ) . k is Element of the carrier of f ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) ;
card h1 = power F_Complex * ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) * u .= ( ( - 1_ F_Complex ) * u ) * u .= ( ( - 1_ F_Complex ) * u ) * u .= ( ( ( - 1_ F_Complex ) *' ) * u .= ( ( - 1_ F_Complex ) *' ) * ( ( - 1_ F_Complex ) *' ) ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( 1 / g ) /. c .= ( f (#) ( 1 / g ) ) /. c .= ( f (#) ( 1 / g ) ) /. c ;
len Cv - len ( { ( C /. 1 ) - 1 , C /. ( len C - 1 ) } ) = len Cv - len ( ( C /. 1 ) - len ( C /. 1 ) - len ( C /. 1 ) ) .= len ( ( C /. 1 ) - len ( C /. 1 ) ) + len ( C /. 1 - len ( C /. 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and f is onto and for n st n < k + 1 holds f " { f . n } = { n } ;
consider vs being Function of S , BOOLEAN such that vs = chi ( A \/ B , S ) and ( for x being Element of S holds vs . ( A \/ B ) = Prob . x ) and ( for x being Element of S holds vs . ( A \/ B ) = Prob . x ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and Z = dom f and f = ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) / ( f1 + f2 ) ) and for x st x in Z holds f1 . x = 1 & f2 . x = - 1 ;
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q2 & len Seq q1 = len Seq q2 & len Seq q2 = len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G3 and f from G1 , G2 and g = G2 & g = G2 & h = G1 and h = G2 and g = G2 and h = G2 and h = G2 and h = G1 and h = G2 and h = G2 and h = G2 and h = G2 and h = G2 and h = G3 ;
func - f -> PartFunc of C , V means : Def3 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for 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 & {} <> a for v holds v |= ( union L ) | ( union L ) | ( v | ( v | ( v v v ) ) ) iff L . a , v |= ( v | ( v | ( v v ) ) ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i / n ) and for i1 being Nat , n being Nat st n <> 0 & n <= len p & n = i holds n <= i1 & n <= i implies n <= i & i <= n or n <= i & i <= n & i <= n ;
assume that not 0 in Z and Z c= dom ( ( arccot * f1 ) / ( f1 + f2 ) ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 + f2 ) ) / ( f1 + f2 ) . x > - 1 & f1 . x < 1 ;
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( ( Y -' 1 ) * ( Y -' 1 ) + 2 ) ) \ ( ( Y -' 1 ) * ( Y -' 1 ) + 2 ) c= BDD L~ f1 & cell ( G1 , i1 , ( 2 |^ ( m -' 1 ) ) + 2 ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 ;
ex Q1 being open Subset of [: X , Y :] st s = Q1 & ex F8 being Subset-Family of [: Y , X :] st Q1 c= F & F8 is finite & ( for x being Subset of Y , y being Point of Y st x in Q1 & y in Q1 holds x in Q holds P [ x , y ] ) & Q [ y ] ;
gcd ( A9 , ( ( 1 , 2 ) * ( 1 , s1 ) + s2 ) , ( ( 1 , 2 ) * ( 1 , s2 ) + s2 ) ) = 1 / 2 * ( ( 1 , 2 ) * ( 1 , s2 ) + s2 ) .= 1 / 2 * ( ( 1 , 2 ) * ( 1 , s2 ) + s2 ) ;
R8 = ( ( j + 1 ) + 1 ) * ( ( j + 1 ) + 1 ) .= ( ( j + 1 ) + 1 ) * ( ( j + 1 ) + 1 ) .= [ 3 , ( j + 1 ) + 1 ] .= [ 3 , ( j + 1 ) + 1 ] .= [ 3 , ( j + 1 ) + 1 ] ;
CurInstr ( P-6 , Comput ( P3 , s , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s , m1 + m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= ( ( card I + card J + 3 ) ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p11 ) /\ LSeg ( p01 , p2 ) \/ LSeg ( p1 , p2 ) .= { p1 } \/ ( LSeg ( p01 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 } \/ { p2 } \/ { p2 } \/ { p2 } .= { p2 } \/ { p2 } ;
func the still of f -> Subset of the carrier of Al means : Def3 : a in it iff ex i , p st i in dom f & p = f . i & a in it & p in the carrier of f & p in the carrier of f & p in the carrier of f ;
for a , b being Element of F_Complex st |. a .| > |. b .| & f is \cap L~ f & a <> b holds a * ( - b * ( - f ) ) is in P
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) & 1 <= j & j <= len g holds j < i & i < j & j < len g implies g /. i < g /. j ;
assume that C1 , C2 are_`2 and for f being State of C1 , g being State of C2 , s1 being State of C1 , s2 being State of C2 st s1 = s2 holds f is stable & g is stable & f is stable iff g is stable & f is stable & g is stable & f is stable & g is stable & f is stable & g is stable ;
( ||. f .|| | X ) . c = ||. f .|| /. c .= ||. f /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. f .|| /. c ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^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 for A , B being Subset of T st A in F & B <> {} & A <> B & A misses B holds card F = i & card F = i & card A = i & card B = i & card A = i & card B = i & card A = i & card B = i & card A = i & card B = i & card A = i & card B = i & card B = i implies card A = i & card B = i & card B = i & card A = i & card B = i & card A = i & card B = i & card B = i & card A = i & card B = i & card B = i & card A = i
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 . ( F . k , G . k ) ;
i |^ ( \mathop { \rm mod } n ) - i |^ s = i |^ ( s + k ) - i |^ s .= i |^ s * i |^ k - i |^ s .= i |^ ( s + k ) - i |^ s * 1 - i |^ s .= i |^ ( s + k ) - i |^ s ;
consider q being oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 and ( F . ( q . len q ) ) `1 = v2 and rng q c= rng ( p ^ q ) and p = ( p ^ q ) . 1 ;
defpred P [ Element of NAT ] means $1 <= len ( g , Z , I ) implies ( ( g , Z , I ) +* ( i , m ) ) . $1 = ( ( ( f , Z , I ) +* ( i , m ) ) +* ( i , m ) ) . $1 + ( ( g , Z , I ) +* ( i , m ) ) . $1 ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width A & 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 = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( Re ( x , y ) , ( Re ( x , y ) ) )| - ( ( Im ( x , y ) ) ) ^2 + ( ( Im ( x , y ) ) ^2 + ( ( Im ( x , y ) ) ^2 + ( Im ( x , y ) ) ^2 ) ;
consider g2 be FinSequence of Ff such that g2 is continuous & rng g2 c= A & g2 . 1 = x1 and g2 . len g2 = x2 and for k st k in dom g2 & k <> len g2 holds g2 . k = x1 and g2 . k = x2 and g2 . k = x2 and g2 . k = x2 and g2 . k = x3 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , a9 , n3 , n3 , n3 , a9 , n3 , n3 , a9 ) = crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , a9 ) ;
( q `1 ) ^2 * a <= ( q `1 ) ^2 * a & ( - q `1 ) ^2 * a <= ( q `1 ) ^2 * a or ( q `1 ) ^2 * a >= ( q `1 ) ^2 * a or ( q `1 ) ^2 * a >= ( q `1 ) ^2 * a ;
Fp . ( len pp ) = Fp . ( p . ( len p ) ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p + 1 ) .= v5 /. ( len p ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( k1 --> intloc 0 ) ) ^ ( ( k + 1 ) --> SubFrom ( a , intloc 0 ) ) ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *> ^ <* halt SCM+FSA *>
consider B8 being Subset of B1 , y8 being Function of B1 , A1 such that B8 is finite and D8 = { 0 , 1 , 0 } and for x , y being Element of B1 , z being Element of A1 st x in B1 & y in B2 & z in B1 holds [ x , y ] in \frac { 0 , 1 , 0 } ;
v2 . b2 = ( curry F2 ) * ( ( curry F2 ) . g ) .= ( ( curry F2 ) . g ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( dom F2 ) . g ) ) . b ) ) id b ) ) id b ) ) . b2 ) ) .= ( ( ( ( ( ( ( ( ( ( ( ( the Morphism of B ) . g ) . b ) ) id b ) ) ) ) ) ) ) ) ) . b2 .= ( ( ( ( ( a , b ) . b ) ) . b2 ) . b2 ) .= ( ( ( ( ( ( ( a , b ) . b2 ) ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) .= ( ( ( ( ( ( ( ( ( ( ( a , b ) . b2 ) . b2 ) . b2 ) . b2 ) . b2
dom IExec ( [: I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 & |. h .| < K holds |. h .| " * ||. ( R2 + R1 ) /. h .|| < e / 4 * ||. ( R2 + R1 ) /. h .||
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( h , i + i1 -' 1 ) .= LSeg ( h , i + i1 -' 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 & LE q , p1 , P , p1 , p2 } ;
( ( - x ) .|. y ) = ( - ( 1 / |. y .| ) ) * ( x .|. y ) .= ( - ( 1 / |. y .| ) ) * ( x .|. y ) .= ( x .|. ( - y ) ) * ( x .|. y ) .= ( x .|. ( - y ) ) * ( x .|. y ) .= ( x .|. ( - y ) ) * ( x .|. y ) .= ( x .|. ( - y ) ) * ( x .|. y ) .= ( x .|. ( - y ) ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U + W ) * ( ( - p ) * ( - ( p ) ) ) = ( ( U + W ) * ( - p ) ) * ( - ( p ) * ( - ( p ) ) ) .= ( U + W ) * ( - p ) .= ( U + W ) * ( - p ) .= ( U + W ) * ( - p ) .= ( W + W ) * ( - p ) .= ( W + W ) * ( - p ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def3 : dom it = - h & for x st x in dom h holds it . x = ( x + h ) . x + h . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i + 1 , j ) and f /. ( k + 1 ) = G * ( i , j ) ;
assume that not y in Free H and not x in ( Free H ) and not ( x in Free H ) and not x in { x } and not x in Free H and not x in ( Free H ) and not x in ( Free H ) and not x in ( Free H ) and not x in ( Free H ) and not x in ( Free H ) and not x in ( Free H ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means P [ $2 , $2 , $2 ] & ( $1 = p |^ $1 implies $2 = p |^ $1 ) & ( $1 = p |^ $1 implies $2 = p |^ $1 implies $2 = p |^ $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def3 : for A being Subset of X holds A in it iff for W being Subset of X , Z being Subset of Y st W c= A \ A & Z c= W holds C . W <= C . Z + C . Z ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [ S ] ^ ) ) = {} or rng ( F | ( [ S ] ^ ) ) = { 1 } or rng ( F | ( [ S ] ^ ) ) = { 2 } or rng ( F | ( [ S ] ^ ) = { 1 } or rng ( F | ( [ S ] ^ ) ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " . ( ( f . i ) " . i ) .= f . i " . i .= ( f . i ) " . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( f " ) . i .= ( 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 p2 , p1 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P1 \/ P2 and P1 = P2 and P1 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P2 \/ P2 and P1 = P2 \/ P2 and P2 = P2 \/ P2 and P1 = P2 \/ P2 and P1 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and
f . p2 = |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 / sqrt ( 1 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 / sqrt ( 1 + ( p2 `1 / p2 `1 ) ^2 ) ]| ;
( ( \mathbin { X } a ) " ) . x = ( ( \mathbin { a } X ) qua Function ) . x .= ( ( ( the carrier of X ) qua Function ) . x ) " .= ( ( ( the carrier of X ) --> a ) " ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of X ) --> a ) . x .= ( ( the carrier of 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 Real st p misses B & r in A & p in B & r in A & p in B holds ( for p being Point of T , r being Point of T st p in A & r < p holds ( for r being Point of T st r in B holds p in A ) implies ( for r being Point of T st r in B holds r in B & r in B & p in B holds r <= r & p in B & p in B & r <= r & p in B holds r <= r implies p <= r implies p in B implies p in B implies p in A & r <= r implies p in B implies p in A & r <= r implies p <= r implies p <= r implies p <= r implies for r being Point of T implies for r being Point of T implies p <= r implies p <= r implies p <= r implies for r being Point of ( A , r ex s being Point of T st r <= r & p
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G1 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G2 = F . ( i + 1 ) & G1 = F . ( i + 1 ) holds G1 is strict Subgroup of G
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( arctan - arccot ) ) `| Z ) . x = ( ( ( 1 / 2 ) (#) ( arctan - arccot ) ) `| Z ) . x ) / ( ( 1 + x ^2 ) * ( ( arctan - arccot ) / ( 1 + x ^2 ) )
synonym f is_continuous x0 means : Def2 : x0 in dom ( f /* a ) & for a st rng a c= ]. x0 , x0 + r .[ & a is convergent & lim a = x0 & rng a c= dom f & for n st n <= m holds a . n = lim ( f /* a ) & a is convergent & lim ( f /* a ) = lim ( f , x0 ) ;
then X1 , X2 are_separated & ( ex Y1 , Y2 being non empty SubSpace of X st Y1 , Y2 are_separated & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y1 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace of X2 & Y1 is SubSpace of X2 & Y2 is SubSpace of X1 & Y2 is SubSpace of X2 & Y2 is SubSpace ;
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 )
( p2 `1 ) ^2 * ( sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) ^2 >= ( ( p3 `1 ) ^2 * ( sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) ) ^2 * ( sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) ^2 ;
( ( 1 / t1 ) (#) ||. f1 .|| ) . x = ( ( 1 / t2 ) (#) ( ( 1 / t2 ) (#) ( ( 1 / t2 ) (#) ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) * ( ( 1 / t2 ) ) * ( 1 / g2 ) ) ) ) ) ) . x ) ) ^2 .= ( ( 1 / g2 ) * ( ( 1 / g2 ) * ( 1 / g2 ) ) to_power ( m ) ) . x ) ) to_power ( m ) ) ^2 + ( ( 1 / g2 ) ) . x ) ^2 .= ( ( ( 1 / g2 ) . x ) ^2 + ( ( 1 / ( ( 1 / g2 ) to_power ( m ) ) . x ) ^2 + ( ( 1 / g2 ) to_power ( m ) ) . x ) ^2 .= ( ( 1 / t2 ) to_power ( m ) ) . x ) ^2 + ( ( 1 / g2 )
assume that for x holds f . x = ( ( sin * ( cot + cot ) ) (#) ( cos + cot ) ) . x and x in dom ( ( sin * ( sin + cot ) ) (#) ( cos + cot ) ) and for x st x in dom ( ( sin + cot ) (#) ( sin + cot ) ) holds ( ( sin + cot ) (#) ( cos + cot ) ) . x = ( 1 / ( sin + cot ) ) . x ;
consider Xj1 being Subset of Y , Y1 being Subset of X such that t = [: Xj1 , 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 * d ]| + b where d , Y is Element of GF ( p ) : [ d , 1 ] in Indices GF ( p ) & [ d , 1 ] in Indices GF ( p ) & not [ d , 1 ] in Indices GF ( p ) & not d in { d , 1 } ;
( W-bound D - W-bound D ) * ( W-bound D - E-bound D ) * ( E-bound D - E-bound D ) = ( W-bound D - W-bound D ) * ( W-bound D - W-bound D ) * ( W-bound D - W-bound D ) .= ( W-bound D - W-bound D ) * ( W-bound D - W-bound D ) .= ( W-bound D - W-bound D ) * ( W-bound D - E-bound D ) .= ( W-bound D - W-bound D ) * ( W-bound D - E-bound D ) ;