thesis ;
thesis ;
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thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from of o , o ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , e be set ;
let G be _Graph , e be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
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 = as Set ;
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 T ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> directed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= \subseteq len \kern1pt s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , M be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , X be set ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G ( ) c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
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 <= i9 & i9 <= len f ;
1 <= i9 & i9 <= len f ;
pp c= cos .: { 0 } ;
1 <= i9 & i9 <= len G ;
1 <= i9 & i9 <= len G ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
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 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of continuous X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Set C = or C c= f ;
x4 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 , f be Function ;
assume P [ n ] ;
assume union S is independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT & f is one-to-one
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be ManySortedSet of I ;
b ` c= b9 ` & b ` c= b ;
assume not x in NAT ;
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 & sin . x > 0 ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i1 < i2 ;
a * h in a * H ;
p , q in Y ;
Observe : sqrt I is left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
a\hbox { $ n $ } , n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a // 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_differentiable_in x0 ;
assume O is symmetric & O is 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 , P3 = P +* I ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } is_>=_than c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ` ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ^ <* 0 *> ) ;
h2 . a = y ;
P [ n + 1 ] ;
Observe : G * F is pre. ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co /\ L~ 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 ;
let N be non empty <* the \cal of M *> ;
let R be RelStr with finite finite A) ;
let n , k be Nat ;
let P , Q be be Let ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` ` ` ` ` ;
Observe S --> T is such that S is such that T is .] ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : x in A6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec /\ dom sec ;
assume [ x , y ] in R ;
set d = ( x / y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom f1 /\ dom f2 ;
1 in dom D2 & 1 in dom D2 ;
( p `2 ) ^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 ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be functor of on ;
cluster m * n -> square ;
let k9 be Nat , n be Nat ;
i -' 1 > m ;
R is transitive implies R is transitive
set F = <* u , w *> ;
pp c= P3 & P3 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is real ;
i <= len f2 + len g2 ;
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 ;
B9 c= ( ( X \/ Y ) \ { 0 } ) ;
n2 <= ( 2 * n ) / ( 2 * n ) ;
A /\ [: P , Q :] c= A `
cluster x -valued -> x -valued for Function ;
let Q be Subset-Family of S , A be Subset of T ;
assume n in dom g2 & n in dom g2 ;
let a be Element of R ;
t `2 in dom ( e2 . i ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty set ;
i . y in rng i ;
REAL c= dom f & f is continuous ;
f . x in rng f ;
mt <= ( r / 2 ) ^2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= Nseq . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S \to T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & f . x = x ;
n1 < n1 + 1 + 1 ;
n1 < n1 + 1 + 1 ;
cluster [: X , Y :] -> 8 ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S . i ) ;
b = upper_bound dom f & b = upper_bound 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 = ( a + 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u c= Support p ;
assume X is complete holds X is complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n2 + 1 ;
let x be Element of REAL , r be Real ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 / 2 ;
len ( L ^ W ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let r8 be real-valued sequence of REAL , r be Real ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in C := ] -] ( 0 , A ) ;
let i be set ;
n - 1 = n-1 ;
len ( n |-> 0 ) = n ;
\mathop { \rm lim } ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , F be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E -tuples_on omega ;
let B1 be Basis of x , B ;
L3 /\ L2 = {} ;
L1 /\ LSeg ( f , k ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f | -1-129 ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume that f `2 = f and h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) ` is open ;
assume a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster nInt for dist of N ;
not u in { ag } ;
the carrier of f c= B \/ C ;
reconsider z = x as VECTOR of V ;
cluster bounded for non empty RelStr ;
r (#) H is non-zero ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , x be set ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ex y being Element of : y in : x <= y & y <= x } ;
let x , y be Element of X ;
let A , I be such that I is such that I is such ;
[ y , z ] in [: O , O :] ;
( that that that card Macro i ) = 1 ;
rng Sgm A = A ;
q |- p \! q implies q |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z to_power Y ;
( D D ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: NAT , { 0 } :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x ` , y ` in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative for multMagma ;
x in support ( ( support t ) --> 0 ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= ( len y9 ) `2 ;
assume p divides b1 + b2 & b1 divides b2 ;
M . x0 <= upper_bound ( M1 * M2 ) ;
assume x in ( W-min X ) `1 ;
j in dom ( z ^ <* x *> ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= IC Comput ( P3 , s3 , k ) ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G , uH = Vertices G , uH = Vertices G , uH = Vertices H , uH = Vertices H , uH
seq " (#) ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
h-4 c= h-14 ( x , y ) ;
]. 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 zero Nat , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , D be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= ( g . ( the_left_argument_of H ) ) ;
dom G `2 /. y = a ;
( 1 / 4 ) ^2 >= - r ;
G . p0 in rng G & G . O in rng G ;
let x be Element of FB , y be Element of FB ;
D [ P-6 , 0 ] ;
z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng f\mathbb R c= NAT & rng f\mathbb R is finite ;
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 & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* ( A +* {} ) ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
( h | [. a , b .[ ) = b-a ;
assume that the distance of V , Q and v in Q ;
let a be Element of ^ ( V ) ;
let s be Element of Pz ;
let Py be non empty RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = the carrier of SCM R ;
consider b being element such that b in B ;
set BM = BCS ( K , n ) ;
l <= IC ( F . j ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in ( JumpParts T ) . ( T . i ) ;
let h be Morphism of c , a ;
Y c= [: { \bf R } , { \bf R } :] ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 9 :] is set ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[. a , b .] = [. a , b .] ;
cluster -> sqrt closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q2 is_collinear ;
dom M1 = Seg n & len M2 = n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) ;
rng g2 c= dom ( W ^ <* x *> ) ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( the InternalRel of R ) * ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > 0 ;
u1 + v1 in W2 + W3 & v2 in W1 + W2 ;
assume not L misses rng G ;
let L be lower-bounded antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI & 2 < PI ;
o , a9 // o , y & o , a // o , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D . k in rng D ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & dom 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 ;
conv @ A c= conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P \lbrack l , P \lbrack l , P \lbrack l , P \lbrack l , v \rbrack ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
seq1 // field ( seq1 + seq2 ) & seq2 field ( seq2 + seq2 ) c= field ( seq1 + seq2 ) ;
assume x in the carrier of R & y in the carrier of R ;
dom ( n |-> 0 ) = Seg n & dom ( n --> 0 ) = Seg n ;
s4 misses s2 & s4 misses s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an ( ) ;
assume that that that function I c= J and function I c= K and I c= K ;
Im ( ( lim seq ) - ( lim seq ) ) = 0 ;
( sin . x ) ^2 <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies sin * ( sin + cos ) is_differentiable_on Z
t3 . n = t3 . n .= s . n ;
dom ( element * F ) c= dom F ;
seq1 . x = seq2 . x .= seq2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v ^ w ) ;
x * a divides y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( f . I ) `1 ;
G * ( i , j ) `1 = U /. 1 .= G * ( i , j ) `1 ;
f . rr1 in rng f & rr1 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F . n ) .= rng F ;
mode seq is well unital associative associative associative non empty multMagma ;
[ x , y ] in A ~ \ { a } ;
x1 . o in L2 . o ;
the carrier of support } , B are_equipotent ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ ( k1 + 1 ) is lower & seq ^\ k1 is lower ;
len ( F ^ <* x *> ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , r 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 \HM { the / of T : not contradiction } ;
cluster directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j1 in K . j1 ;
Observe that J => y is total and J is $ -defined ;
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 : Def1 : ( a / a ) = 1 ;
assume that succ a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 or LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty <> TopSpace , f be Function ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x , pp = y , pp = z as Subset of REAL m ;
let A , B , C be Element of R ;
Observe non empty for D_Space ;
rng c `1 misses rng ( ee e ) ;
z is Element of gr { x } ;
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 / ( f . x ) ) ;
pred f = u means : Def1 : a * f = a * u ;
for n holds P1 [ n ] implies P [ n + 1 ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = q2 ;
gcd ( n1 , n2 ) = 1 & gcd ( n2 , n2 ) = 1 ;
set o9 = ( 2 * PI ) / ( 2 * PI ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 , h , f , g , h , i , j ;
assume that Y = { 1 } and s = <* 1 *> ;
IS1 . x = f . x .= 0 ;
W3 .last() = W3 . 1 .= ( W . 1 ) ;
cluster trivial -> trivial for M -connected _Graph ;
reconsider u = u as Element of Bags X ( ) ;
A in B ^ ) implies A , B are_equipotent
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 ) ^2 / ( |. q .| ) ^2 ;
f1 is_as such that f2 iff f1 . 1 = f2 . len f2 ;
( f /. i ) `2 <= ( q `2 ) ^2 ;
h is_\HM { the } \HM { Go-board ( C , n ) ;
( b `2 ) ^2 / ( b `1 ) ^2 <= ( p `1 ) ^2 / ( p `2 ) ^2 ;
let f , g be for X , Y be set ;
S * ( k , k ) <> 0. K ;
x in dom ( f - g ) ;
p2 in Nc . p1 & p2 in Nc . p2 ;
len ( ( H ) . i ) < len ( H ) ;
F ( A ) = [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def1 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng q2 c= dom C1 ;
A1 , L , A2 , A3 , A3 , A3 be set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in element ( p , Sy ) & b in dom ( p , Sy ) ;
then S is atomic & P-2 [ S ] ;
Cl Int ( [#] T ) = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of V ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in M ;
( 1_ G ) c= ( Q * t ) / ( ( _ 1 ) * r ) ;
0 * a = 0. R .= a * 0 ;
A |^ 2 = A |^ 2 .= A |^ 2 ;
set vFinSequence = ( v /. n ) `1 , vq = v /. n ;
r = 0. ( \langle \cal E , \Vert * \Vert *> ) ;
( f . p4 ) `1 >= 0 ;
len W = len ( W -] ) & len W = len ( W -] ;
f /* ( s * G ) is divergent_to+infty & lim ( f /* ( s * G ) ) = 0 ;
consider l being Nat such that m = F . l ;
t8 / ( a + 1 ) does not destroy ( b1 , b2 , T ) ;
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 ] -> non pair for set ;
downarrow a /\ downarrow t is Ideal of T ;
let X be non empty set , N be non empty set ;
rng f = element ( S , X ) ;
let p be Element of B , x be the sort Element 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 & p2 = q2 or p1 = q2 ;
assume gR in the right & FR is right ;
let A1 , A2 be Point of S , x be Point of S ;
x in h " P /\ [#] T1 & x in [#] T1 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Xsuch that X = { y } and X is non empty Subset of Tsuch that X is non empty ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 + len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
as Element of NAT & len ( ( - 1 ) * p ) gcd p = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x <= x ) implies ( ( x - b ) (#) ( x - b ) ) `| Z ) = f
rng ( M * ( D , j ) ) c= rng ( D2 , j ) ;
for p being Real st p in Z holds p >= a
( cn ) * ( f . p ) = proj1 . p ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path ( M , n ) ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h divides gg . ( ( mod P ) + 1 ) ;
reconsider i1 = i-1 , i2 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , a be Real ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i , i9 = j , i9 = i as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( ( the Sorts of B ) . n ) . n ;
len reconsider len \kern1pt f2 in Seg ( len f2 + 1 ) ;
pp1 c= the topology of T & p2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Point of T2 ;
G * ( B * A ) = ( id o1 ) * ( id o2 ) ;
assume that p , u are_not zero and u , v , w thesis ;
[ z , z ] in union rng ( F | ( Seg n ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , C = $1 .. S , D = $1 .. S , D = $1 .. S , D = $1 .. S , E = $1 .. S , F = $1 .. S , N
LIN a1 , a3 , b1 & LIN a1 , b1 , b2 ;
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 ;
p in LSeg ( E . i , F . i ) ;
In * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
qk . x in rng ( qk ^ <* x *> ) ;
Carrier ( gLet ) misses Carrier ( Lj ) ` ;
consider c being element such that [ a , c ] in G ;
assume that for o9 being Element of Nreal holds o9 = o9 & o9 = i & o9 = j ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C ) " C ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 implies x ^2 <= x ^2 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 <= 0 ;
Observe aa] is non empty ;
let x be Element of S ~ ;
cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster cluster id -> one-to-one for Morphism of F , G ;
|. i .| <= - ( - 2 |^ n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = the carrier of I[01] ;
! * ( n + 1 ) ! > 0 * <i> ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // b3 , b2 & a3 , a4 // b3 , b2 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G implies x = y & y = z ;
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= ( r . n ) `1 ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A2 :] & dom d2 = A2 ;
0 < ( p / ||. z .|| + 1 ) / ( 1 + 1 ) ;
e . ( mm + 1 ) <= e . ( mm + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
- +infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM { \HM { 0 } } -valued for OperSymbol of X ;
let U1 , U2 be non-empty MSAlgebra over S , x be set ;
Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & y 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 lower and a in b ;
Int Cl ( A ) c= Cl Int Cl ( A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 / ( p2 `1 ) ^2 <= ( p2 `1 ) ^2 / ( p2 `2 ) ^2 ;
Cl ( Q ` ) = [#] ( ( T | A ) ` ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ( _ n ) |^ n , I8 = ( f |^ n ) ;
len p -' n = len ( - n ) - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n6 , n6 = n6 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s ) ;
let q\mathopen { - } q , q\mathopen { - } p } , q\mathopen { - q } , q - p , s - p , q - p , s - p , s - p , q - p , s - q - p ,
a1 , a2 in the carrier of S1 & a2 in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c2 /. n2 = c2 . n2 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * S8 ) . x .= ( f * S8 ) . x ;
consider x being element such that x in an -set A ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( n , h ) `1 , h = ( n , i ) `1 , i = ( n , j ) `1 , j = ( n , i ) `1 , i = ( n , j ) `1 , j = ( n
h2 . ( j + 1 ) in rng h2 /\ rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x - 1 ) / 2 , n = ( x - 1 ) / 2 as Element of REAL ;
let U1 , U2 be non-empty MSAlgebra over U0 , B be Subset of U0 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 + 1 < len p1 ;
let T1 , T2 be Scott topological \mathclose of L , x be Point of L ;
then x <= y & \mathbin { \rm mod } x c= Comput ( y , x , y ) ;
set M = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 , k = x3 , l = x4 as Nat ;
rng ( ( the_arity_of o ) . a ) c= dom H & ( the_arity_of o ) . a in dom ( H * a ) ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) ;
x0 - r / 2 in L /\ dom f /\ dom f ;
then w is that rng w /\ rng w <> {} & rng w /\ S <> {} ;
set x-10 = xZ ^ <* Z *> ^ <* Z *> ^ ( x , Z ^ <* Z *> ) ;
len w1 in Seg ( len w1 + len w2 ) & len w2 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 / ( 1 + ( p `1 / p `2 ) ^2 ) <= ( G * ( 1 , 1 ) ) `1 ;
rng ( g ) c= L~ ( g ) \/ L~ ( g ) ;
reconsider k = i-1 * ( l + j ) , j = l + j as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 , z9 = w as VECTOR of M ;
dom ( f | X ) = X /\ dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a (#) p ^ q ) ;
m . ( ag . k ) = p . ( ag . k ) ;
a to_power ( s . m ) - n / ( s . n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 & C2 \/ C2 = C1 \/ C2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
that \cal R = a and b-0 R = b ;
F8 is_closed_on t1 , Q1 & Q8 c= Q8 ;
set T = k1 -such that X = { 0 , x0 } ;
Int Cl ( Cl Cl Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng Flen x = { Flen F } & Fy . x = { F . x } ;
G " { c } c= B \/ S \/ S \/ S ;
f[#] ( X ) is Relation & X c= [: X , X :] ;
set Rz = the Element of P , Rz = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pp = u , pp = v , \overline = w , pp = w , pp = y , \overline = z , pp = w , pp = z , N = w as Element of TOP-REAL n ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G |^ N ) ;
len Pt <= len P-35 & len Pt <= len P-35 ;
x " in the carrier of [: A1 , A2 :] & x in the carrier of [: A1 , A2 :] ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Point of REAL i ;
rng f = the carrier of ( ( Carrier A ) * ( i , j ) ) ;
assume s1 = sqrt ( 2 * ( p `1 ) - r ) ;
pred a > 1 & b > 0 & a to_power b > 1 ;
let A , B , C be lines Subset of [: I , I :] ;
reconsider X0 = X , Y0 = Y , Y1 = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Real ;
r (#) ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 , t-3 be Relation of T ;
Q [ e-14 \/ { v-5 } , f ] ;
g \circlearrowleft ( ( W-min L~ z ) .. z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vrelational ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x ;
( 7 / ( 1 + e ) ) ^2 > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan * cot ) `| Z ) . x in dom ( sec * cot ) ;
i2 = ( f /. len f ) + ( f /. 1 ) .= ( f /. 1 ) ;
X1 = X2 \/ ( X1 \ X2 ) & X2 is open ;
[. a , b , 1_ G .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of COMPLEX ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= X ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n - r / 2 ;
|. ( f /* s ) . k - ( G . k ) .| < r ;
len ( Line ( A , i ) ) = width A ;
SFinSequence = ( S . g ) ^2 .= ( S . g ) ^2 ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized Initialized ( p +* I ) ;
i1 , i2 , a3 , a4 , a5 , a5 , 8 , 8 , 9 , 8 , 8 , 8 , 9 , 8 , 8 , 8 , 8 , 9 be element ;
( arccos r + 1 ) * ( arccos r ) = ( cos r ) ^2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( 1 - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [: X \to Omega Y , [#] Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( TRUE T ) at ( C , u ) = TRUE & ( T . ( u , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
( p2 `1 ) ^2 - x1 > - g / 2 / 2 / 2 / 2 / 2 ;
|. r1 - `2 .| = |. a1 .| * |. thesis .| ;
reconsider S-14 y = 8 , Sp2 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .2 ( n ) = D0W .2 ( n ) + 1 ;
i1 = ( a + n ) + n & i2 = ( a + n ) + ( n + 1 ) ;
f . a [= f . ( f ^ O1 ) "\/" f . ( a "\/" b ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , ( F . n ) | E ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R /. ( len R + 1 ) ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( k9 . n ) : n >= k } ;
for H st H is atomic holds P7 [ H ]
set b8 = S5 . ( i + 1 ) , S8 = S5 . ( i + 1 ) , S8 = F8 . ( i + 1 ) , b = b . ( i + 1 ) , c = c . ( i + 1 ) , d =
Hom ( a , b ) c= Hom ( a `1 , b `2 ) ;
( 1 / ( n + 1 ) ) < ( 1 / s ) " ;
( l ) `1 = [ [ dom l , cod l ] , cod l ] `1 .= ( l . 1 ) `1 ;
y +* ( i , y /. i ) in dom g ;
let p be Element of QC-WFF ( Al ( ) ) , x be Element of D ( ) ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f2 - f3 ) ;
p2 in rng ( f /^ ( i + 1 ) ) & p2 in rng ( f /^ ( i + 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( ( ( TOP-REAL 2 ) ) \/ ( ( TOP-REAL 2 ) | K1 ) ) ;
- 1 <= ( ( f2 ) . O ) `2 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | K1 , R^1 ;
k1 -' k2 = k1 - k2 + k2 .= k1 - k2 + k2 .= k1 - k2 + k2 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ /\ dom f ;
sgn ( p `1 , K ) = - 1 .= ( - 1 ) * ( K ) ;
consider u being Nat such that b = p |^ y * u ;
ex A being subset of T st a = Sum A & A in F ;
Cl ( union ( H ) ) = union ( ( Cl H ) \ ( Cl H ) ) ;
len t = len t1 + len t2 .= len t1 + len t2 .= len t2 + len t1 ;
v = v + w |-- v + v |-- ( A , B ) ;
v ( ) <> DataLoc ( ( t . GBP ) , 3 ) ;
g . s = upper_bound ( d " { s } ) .= s . s ;
( \dot { y } ) . s = s . ( \dot { y } . s ) ;
{ s : s < t } in INT implies t = {} ;
s ` \ s = s ` \ ( s ` \ s ) .= ( 0. X ) ` ;
defpred P [ Nat ] means B + $1 in A & C + $1 in B ;
( 3\langle 1\langle *> + 1 ) ! = 331111\langle + 1 *> ;
( 1_ A ) `1 = ( 1_ A ) `1 .= ( 1_ A ) `1 .= ( 1_ A ) `1 ;
reconsider y = y , z = z as Element of ( len x ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * ( i2 , j2 ) ;
reconsider p = Y | Seg k , q = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) \mathop { {} } , z = S . {} , F = S . {} , G = S . {} , C = { {} } , N = { {} } , N = { {} } , N = { {} } , N =
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL n ) | ( P ) ;
( \mathopen { M M M . [ n + i , 'not' A ] ) `1 <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Real ;
reconsider l = (0). ( V ) , r = 0. ( A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a is ' of 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x - g ) . x - g . x .|| < r2 / 2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 + 1 implies k1 + 1 = k2 + 1
( p `2 / |. p .| - sn ) ^2 >= 0 ;
( q `2 / |. q .| - sn ) ^2 / ( 1 + sn ) ^2 < 0 ;
( E-max C ) `1 in ( GoB Cage ( C , n ) ) * ( 1 , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) ) ;
LIN b , a , a or LIN b , c , a ;
p `2 , a `2 // a `2 , b `2 or p `2 , a `2 // b `2 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is being being being set ;
F | ( N2 , S ) = CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( id Z ) (#) cos ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im ( seq ) is summable and Im ( seq ) = 0 ;
||. ( ( vseq . n ) - ( vseq . m ) ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as 0 -started string of S2 , t2 = ( 0 , 1 ) = ( 0 , 1 ) --> 1 ;
reconsider x9 = seq . n , y9 = seq . ( n + 1 ) as sequence of REAL n ;
assume that that that C meets L~ go and L~ Cage ( C , n ) meets L~ go and C c= L~ go and L~ Cage ( C , n ) c= L~ go ;
- ( ( - 1 ) / ( 1 - x ) ) < F . n - ( x / ( 1 - x ) ) ;
set d1 = being element , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , z2 ) ;
2 to_power ( 2 - 1 ) = 2 to_power ( 2 - 1 ) - 1 ;
dom ( v ^ <* d *> ) = Seg ( len ( v ^ <* d *> ) ) ;
set x1 = - ( k2 + 1 ) , x2 = - ( k2 + 1 ) , x3 = - ( k2 + 1 ) , x4 = - ( k2 + 1 ) ;
assume for n being Element of X holds 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and Tbeing . i <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of Carrier ( LF2 ) c= ( I \/ { x } ) ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal w.r.t. over S ;
Z c= dom ( ( id Z ) (#) ( f1 + f2 ) ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 / ( 1 + ( q `1 / q `2 ) ^2 ) ;
ConsecutiveSet2 ( q , succ ( d ) ) c= ConsecutiveSet2 ( A , succ ( d ) ) ;
E = dom Carrier ( f ) & Carrier ( f ) c= E & Carrier ( f ) c= E ;
C to_power ( A + B ) = C to_power ( B + A )
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
pred x > 0 means : Def1 : ( 1 / x ) ^2 = x ^2 / ( 1 - x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c are_connected & - C , - C + ( - C , - C ) + ( - C , - C ) + ( - C , - C ) + ( - C , - C ) + ( - C ) + ( - C , - C ) + ( - C ) +
assume f = id the carrier of O1 & g is Function of the carrier of O2 , the carrier of O2 ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) --> 0. K ) ;
reconsider g = f " as Function of ( U2 ) , ( ( U2 ) * f ) ;
A1 in the carrier of ( ( G . k ) `1 ) & A2 in the carrier of ( ( G . k ) `2 ) ;
|. - x .| = - ( x + - x ) .= x + - x .= x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * ( sqrt 5 ) ) / ( 2 * ( sqrt 5 ) ) >= 4 * ;
( v /. ( k + 1 ) ) `1 = ( v . ( k + 1 ) ) `1 ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M2 = n & width M1 = n ;
Line ( S\mathopen { - } j } , j ) = S\mathopen { - j } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| is_integrable_on dom ( Re f ) & ( ( Re f ) (#) h ) is_integrable_on S ;
assume x = ( a1 ^ <* x1 *> ) ^ <* x2 *> ^ b1 ;
MB is_closed_on IExec ( I , P , s ) , P & I is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
f.: ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) ;
flim . a = flim . a & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 / ( p `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 / ( p `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E8 = Cage ( C , n ) ;
( p `1 ) ^2 / ( p `1 ) ^2 >= ( ( E-max C ) `1 ) / ( p `1 ) ^2 ;
consider p such that p = pp and s1 < p and p `1 <= i and i <= len f ;
|. ( f /* ( s * F ) ) . l - G . l .| < 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 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng s c= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on ( the carrier of V ) \ { {} } ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 , B1 = the carrier of X1 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in dom ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) ) (#) ( ( 1 / 2 ) ) ) ) ;
for L being complete LATTICE holds <* <* \mathclose ( L ) , L *> , L *> *> is isomorphic ;
[ gi , gj ] in Ii \ Ij \ Ij ;
set S2 = \mathop { \rm is non empty set , x , y , c be set ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 . x0 = f2 . x0 ;
reconsider y = ( a ` ) / ( a ` ) , z = ( a ` ) / ( a ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 ;
( min ( g , ( 1 - r ) ) ) . c <= h . c ;
set G2 = the subgraph of G , s3 = the subgraph of G , e = the Vertex of G , S = the set of G , T = the set of G , S = the set of G , T = the carrier of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| / p < d / p / p / p / q ;
for x being element st x in ( 1 - u ) holds x in ( 1 - u )
P = the carrier of ( ( TOP-REAL n ) | P ) .= P ;
assume that p11 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , Z ) ;
2 * a * b + ( 2 * c * d ) <= 2 * ( C1 * C2 ) ;
let f , g , h be Point of the complex normed space of X , Y , Z be set ;
set h = Hom ( a , g (*) f ) ;
then ( idseq ( n ) ) | Seg m = ( idseq ( m ) ) | Seg m & m <= n ;
H * ( g " * a ) in the carrier of H * ( g " * a ) ;
x in dom ( ( id Z ) (#) ( ( id Z ) ^ ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is BDD of A means : Def1 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 in union rng $2 & $2 in union rng $2 ;
n + - n < len ( p ^ <* n *> ) + ( - n ) ;
pred a <> 0. K means : Def1 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb Z and I = ( len \/ \/ { j } ) + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ]
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) ;
set cv = 3 / ( a , b , c ) / ( a , b , c ) ;
conv @ W c= union ( F .: ( E " ( W " ( W ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ;
r3 <= s0 + ( r0 - ( r0 / 2 ) ) / 2 + ( r0 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - ( v2 - v2 ) ) ) ) ) ) ) / 2 ) ) ;
dom ( f (#) f4 ) = dom f /\ dom ( f (#) f3 ) .= dom f /\ dom ( f (#) f3 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg ( k + 1 ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq , gq = gq , gr = g1 as Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( L . J . x ) = ( I * L ) . x ;
y in dom st st y in dom <* *> holds ( Frege ( Frege ( A . o ) ) ) . y = ( Frege ( A . o ) ) . y ;
for I being non degenerated integral for I being commutative commutative commutative commutative commutative commutative distributive non empty doubleLoopStr holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , s3 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P1 , s1 , k ) ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( l-13 . i ) = ( v *' ( l-13 . i ) ) . i ;
consider n being element such that n in NAT and x = ( sn " ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F2 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } \/ { F } ;
j + ( 2 * ( k9 + 1 ) ) > j + ( 2 * ( 2 * ( k9 + 1 ) ) ) ;
{ s , t } on A3 & { s , t } on B2 implies { s , t } on B2
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n4 , n4 , n4 ) ;
( mg ) . HT ( mg , T ) = 0. L & ( mg ) . HT ( mg , T ) = 0. L ;
then H1 , H2 are_that card H1 , H & card H1 , H are_equipotent implies card H1 , H \kern1pt = H ;
( ( ( ( N-min L~ f ) /. 1 ) .. f ) .. f > 1 & ( ( ( N-min L~ f ) /. 1 ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , the carrier of T ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on ( k + 2 ) ) ;
I \mathop { \rm k2 , k2 } = da & I \mathop { \rm k2 , k2 } = k2 ;
uz ~ = { [ a , uz ] } & uz = { [ a , uz ] } ;
( w | p ) | ( p | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | ( w | p ) ) ) ) ) ) ) ) ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & a |^ n in F
dom ( ( g * ( ( /. ( V \dot \to C ) ) ) | K ) ) = K ;
ex x being element st x in ( ( ( ( ( ( ( ( U0 ) ) \/ A ) ) \/ B ) ) . s ) ;
ex x being element st x in ( ( ( ( ( ( ( ( O2 O ) \/ A ) ) . s ) ) . s ) ) . s ;
f . x in the carrier of [. - r , r .] & f . x in the carrier of [. - r , r .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p11 } /\ LSeg ( p1 , p2 ) ;
( b + be ) / 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 P [ z ] ;
( the sequence of ( ( the sequence of X ) * , ( the carrier of X ) * ) ) . ( 2 * e ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | E-4 ` = g | E-4 ` .= g | E-4 ` ` .= g | E-4 ` ` ` ` ` ` ` ` ` .= g | E-4 ` ;
reconsider i1 = x1 , i2 = x2 , z = x3 , n = x4 , m = x4 , n = 6 , n = 7 , m = 8 , n = 8 + 1 , m = 8 + 1 ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f to_power n0 is L & f to_power n0 is L ;
Seg ( len ( ( f ^ g ) | ( i + 1 ) ) ) = dom ( ( f ^ g ) | ( i + 1 ) ) ;
( Complement ( A * B ) ) . m c= ( Complement ( A * B ) ) . n ;
f1 . p = p9 & g1 . p = d & g2 . p = d & g2 . p = d & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) to_power n ) / ( 2 * n ) <= ( ( r2 ) to_power n ) ;
Sum ( F-12 ) = Sum f & dom ( Fc ) = 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 /\ W2 is Subspace of W1 and W2 /\ W3 is Subspace of W2 ;
||. ( t-15 . x ) - ( x - y ) .|| = lim ||. ( ( x - y ) * ( x - y ) ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) ^2 + ( p `2 ) ^2 ) <= ( ( - 1 ) ^2 ) / ( ( p `2 ) ^2 ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( { p } ) ;
set N8 = ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable implies the TopStruct of T is countable
width ( B |-> 0. K ) = Line ( B , i ) .= Line ( B , i ) .= B * ( i , i ) ;
pred a <> 0 means : Def1 : ( A Let B ) Y. = ( A Y. ) \diffsym ( B Y. ) ;
then f is_differentiable on pdiff1 ( f , 1 ) & 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 b <> 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC Comput ( p2 , s , k ) = p2 . IC Comput ( p2 , s , k ) .= ( IC Comput ( p2 , s , k ) ) ;
ind ( T-10 | b ) = ind b .= ind b .= ind b .= ind b ;
[ a , A ] in the carrier of Line ( 2 , 2 ) & [ a , A ] in the carrier of Line ( 2 , 2 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o2 , o1 ) = ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) ) . z = FALSE & ( a 'imp' CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi /. 2 = phi /. 2 , phi = phi /. 2 , phi = I /. 3 , phi = I /. 4 , phi = I /. 3 , phi = I /. 4 , phi = I /. 5 , phi = I /. 4 , phi = I /. 5 , phi = I /. 5 , phi = I /. 4 ,
( len s1 - ( len s2 - 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 ~ & f22 = the carrier' of A ~ ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and q = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= the carrier of (0). V1 .= the carrier of V1 ;
consider P2 being FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 . 1 = q ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s in dom ( f | X ) ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> ^ <* p *> ;
c / ( |[ b , c ]| ) = c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p2 , t1 = p3 , t2 = p2 , t2 = p3 , t2 = p2 , t1 = p2 , t2 = p3 , t2 = p2 , t2 = p3 , t2 = p4 ( C ) ;
( 1 - ( 2 * x ) ) / ( 1 - ( 2 * x ) ) in the carrier of [. 0 , 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 * ( p2 `2 ) ;
R . b / b ^2 = 2 * \cal b .= 2 * b / b .= b ;
consider over REAL such that B = ( - 1 ) * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , ( the Sorts of A ) * ( b , ( the Sorts of A ) * ( a , ( the Sorts of A ) * ( b , ( the Sorts of A ) * ( b , ( the Sorts of A ) * ( a , ( the Sorts of A )
[ P . ( l ) , P . ( l + 1 ) ] in => ( T . ( k + 1 ) ) ;
set s2 = Initialize s , s3 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z - i1 + 1 as Nat ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume that x in the left & x in the right & y in the right & x in the right or x = the right & y in the carrier of g ;
consider M being strict Subspace of A9 such that a = M and T is the Sorts of M ;
for x st x in Z holds ( ( ( id Z ) (#) f ) `| Z ) . x <> 0
len ( seq1 + seq2 ) + len ( seq2 + m ) = 1 + len ( seq1 + seq2 ) .= len seq1 + len seq2 ;
reconsider h1 = ( vseq . n ) - ( tf1 . n ) , tf2 = ( vseq . n ) - ( tf2 . n ) as Lipschitzian Lipschitzian from X , Y ;
( ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is for s1 , s2 st F in the |= of s2 & F in the |= of s2 holds s1 <= s2 ;
( ( for x , y being Element of X ) holds ( x , y ) * ( x , z ) = gcd ( x , y ) * ( x , z )
for u being element st u in Bags n holds ( p `2 + m ) . u = p . u ;
for B be Subset of u-5 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = ( \in dom p ) \/ ( dom q ) ;
x in { X where X is Ideal of L : X is Ideal of L |^ ( L + 1 ) } ;
the carrier of ( W1 /\ W2 ) c= the carrier of ( W1 /\ W2 ) & the carrier of ( W1 /\ W2 ) c= the carrier of ( W1 /\ W2 ) ;
( for a , b being Element of L holds a * ( a + b ) ) = ( 1 / ( a + b ) ) * ( a + b )
( ( X --> f ) . x ) = ( X --> f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( being / ( n -' m ) ) * ( 2 |^ ( n -' m ) ) + 1 ) ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 . r } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & b2 on P & b1 on P & b2 on P & b2 on P & b2 on P & b1 <> b2 ;
reconsider gf = g `2 * f `2 , hf = h `2 * g `2 , hf = 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 v2 in V ;
n in { i where i is Nat : i < ( n + 1 ) + 1 & i < ( n + 1 ) + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 / |. p .| >= cn & p `2 >= 0 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ConsecutiveSet ( A , O1 ) ) ;
set I1 = in dom [ AddTo ( a , intloc 0 ) , intloc 0 ] , I2 = [ a , intloc 0 ] , I2 = [ a , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1
X c= ( the carrier of L1 ) ~ & the carrier of L1 c= the carrier of L2 implies the carrier of L1 c= the carrier of L2
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 2 = b ;
reconsider ez = ez , fw = f-5 , fw = f-5 , fw = f-5 , fw = f-5 , fw = f-5 , fw = f-5 , fw = f/ ( 2 * ( 2 * ( 2 * ( 2 * ( 1 + 1 ) ) ) ) ) , ew = e / ( 2 * ( 2 * ( 2 * ( 2 * ( 2
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl ( f . O ) ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U1 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) ;
defpred P [ Nat ] means A + ( succ $1 ) = succ A + ( succ $1 ) & A = succ $1 + ( succ $1 ) ;
the left & ( - g ) * ( - g ) = the left & ( - g ) * ( - g ) = the right of g ;
reconsider pp = x , pp = y , pp = z , pp = z , \overline = x , pp = y , pp = z , pp = x , \overline = y , pp = z , N = y , N = z , S = x , T = y , N = z , S = y , T = z , N = z , S = z , T = y , T = z , S
consider g2 such that g2 = y and x <= g2 and x <= x and g2 <= x and g2 <= x and x <= g2 and g2 <= x ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 + len y2 .= len x2 + len y2 + len y2 .= len x2 + len y2 + len y2 ;
for x being element st x in X holds x in the set of the set of f & f . x = ( the positive div ( n + 1 ) ) * ( f . x )
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} or LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func that ) -> set equals ( \mathop { \rm <* hhhhh.| ) . ( id X ) ;
len ( ( CR /. 1 ) ) <= len ( ( \mathbb C ) /. 1 ) & ( ( \mathbb C ) /. 1 ) = len ( ( \mathbb C ) ^ <* 0 *> ) ;
pred K is bounded means : Def1 : a <> 0. K & v . ( a |^ i ) = a * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and o in ( the carrier of S ) ;
for x st x in X ex y st x c= y & y in X & y is element & f . x = f . y
IC Comput ( P-6 , k ) in dom ( ( n + k ) .--> ( n + k ) ) ;
pred q < s & r < s & s < q implies ]. p , q .] /\ ]. p , s .] c= ]. p , q .]
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in { ( F . c ) `1 , 3 } ;
func the ResultSort of S2 -> Function means : Def1 : for x being set st x in the carrier' of S2 holds it . x = id ( the carrier' of S2 ) ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( id Z ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) ;
r-7 in Int cell ( f , i , ( GoB f ) * ( i , j ) + ( GoB f ) * ( i + 1 , j ) ) \ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) ^2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + ( len f -' 1 ) - len f + ( len f -' 1 ) + ( len f -' 1 ) ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - r ) & h . n > 0 ;
set s0 = ( ( a , I , p , s ) +* ( i , I ) ) . i ;
p ( ) . k = 1 or p ( ) . 0 = - 1 or p ( ) . 0 = 1 or p . 1 = 0 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x = [ x9 , y9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( : len p + k ) ;
g + h = gg + h1 & A1 : h + 1 = g + h & h + 1 = h + h ;
L1 is distributive & L2 is distributive implies L1 ~ is distributive & L2 is distributive & L1 ~ is distributive & L2 is distributive & L2 is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive & L is distributive
pred x in rng f & y in rng ( f /^ x ) implies f . x = f . y & f . y = f . y ;
assume that 1 < p and ( 1 + p ) ^2 + ( 1 + q ) ^2 = 1 and 0 <= a and a <= b ;
FM * ( f , <* <* 0. F_Complex *> ) = rpoly ( 1 , <* 0. F_Complex *> ) *' + t .= z ;
for X being set , A being Subset of X , B being Subset of X holds A ` = {} implies A = B
( ( ( ( ( ( ( ( ( ( ( X ) ) ) ) ) ) ) ) ) `1 ) <= ( ( ( ( ( ( ( ( ( ( ( ( ( ( X ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `1 ;
for c being Element of the Sorts of A , a being Element of the bound qua Element of the Sorts of A holds c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) implies b >= 0 & a >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , n , m , m , n , m , n , m , m , n , m , n , m , n , m , m , n , m , n , m , n , m , m , n , m , n , m , n , m , m is BCK-algebra ;
set x2 = |( Re ( y - x ) , Im ( y - x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A & divset ( D , k ) c= A ;
0 <= ( delta ( S2 ) . n ) & |. \delta ( S2 ) . n .| < ( e / 2 ) / 2 ;
( - ( q `1 / q `2 ) ^2 ) / ( 1 + ( q `1 / q `2 ) ^2 ) <= ( - ( q `1 / q `2 ) ) / ( 1 + ( q `2 / q `1 ) ^2 ) ;
set A = ( 2 / ( b-a ) ) ;
for x , y being set st x in R-1 holds x , y are_\hbox { - } ;
deffunc FF2 ( Nat ) = b . ( ( M . $1 ) * ( G . $1 ) ) * ( M . $1 ) ;
for s being element holds s in -> Element of -> Element of -> Element of -> Element of -> Element of REAL : s in -> Element of contradiction \/ { s } ;
for S being non empty non void non empty non void holds S is connected iff S is connected
max ( ( degree ( z ) ) / ( 1 + ( degree ( z ) ) ) , ( degree ( z ) ) / ( 1 + ( degree ( z ) ) ) / ( 1 + ( degree ( z ) ) ) ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s . ( n + k ) ;
Lin ( A /\ B ) is Subspace of Lin ( A /\ B ) & Lin ( A /\ B ) is Subspace of Lin ( B ) ;
set n-15 = ( n-13 ) '&' ( M . x qua Element of BOOLEAN ) , n-15 = ( M . x ) ^2 , n-15 = ( M . x ) ^2 , n-15 = ( M . x ) ^2 ;
f " V in such that V in [ X , p ] and f " V in D ( the carrier of X , p ) . X ;
rng ( ( a , b ) } \mathbin { { + } \cdot } ( 1 , b ) ) c= { a , c , b } ;
consider y being /* subgraph of G1 such that y ` = y and dom y ` = WWR and dom y = WWR ;
dom ( 1 / ( f . 0 ) ) /\ ]. - r , x0 .[ c= ]. - r , x0 .[ ;
as Element of as Matrix of ( i , j , n , r ) , K , r , s , n , - r ;
v ^ ( ( n-3 |-> 0 ) ^ ( ( ( B ^ ( ( B ^ ( C ^ ( C ^ D ) ) ) ) ) ) ) ) in Lin ( ( ( B ^ ( C ^ ( C ^ D ) ) ) ) ) ;
ex a , k1 , k2 st i = a /. k1 & i = b /. k2 & k2 = c /. k2 & k2 = c . k2 & k2 = c . k2 ;
t . NAT = ( NAT .--> ( i1 + 1 ) ) . NAT .= ( i1 + 1 ) .= ( i1 + 1 ) .= ( i1 + 1 ) ;
assume that F is bbfamily and rng p = Seg ( n + 1 ) and dom p = Seg ( n + 1 ) ;
not LIN b , b9 , a & not LIN b , a , c & LIN a , b , c & LIN a , b , c ;
( L1 \HM { L } \HM { O } ) \& O c= ( L1 \HM { O } ) \HM { O } ) \HM { O } : O in ( L2 . O ) }
consider F be ManySortedSet of E such that for d be Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( v - u ) = b * ( -w - a ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ |. p .| , {} ] & P [ p , id ( the Sorts of A ) ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is finite and X is finite and ininP [ X ] ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= g & g . l1 <= h . l1 } ;
( Partial_Sums ( G . n ) ) . k <= ( Partial_Sums ( G . n ) ) . ( ( G . n ) * ( G . n ) ) ;
f . y = x .= x * 1. L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 %> , ( n + 1 ) ) = { i1 , ( n + 1 ) } .= { i1 , ( n + 1 ) } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } /\ LSeg ( p1 , p2 ) .= { p1 } ;
Product ( ( ( the support of I-15 ) +* ( i , { 1 } ) ) ) in Z1 ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ;
( W-bound Q1 ) / ( 1 + ( W-bound Q1 ) / ( 1 + ( W-bound Q1 ) ) / ( 1 + ( W-bound Q1 ) ) ^2 ) <= ( W-bound Q1 ) / ( 1 + ( W-bound Q1 ) / ( 1 + ( W-bound Q1 ) ) / ( 1 + ( W-bound Q1 ) ) ^2 ) ;
f /. i2 <> f /. ( ( i1 + len g ) -' 1 ) & f /. ( ( i1 + 1 ) -' 1 ) <> f /. ( ( i1 + 1 ) -' 1 ) ;
M , v / ( x. 3 , x. 4 ) / ( x. 0 , x. 4 ) / ( x. 0 , x. 0 ) / ( x. 4 , x. 0 ) |= H ;
len ( ( P ^ ) ^ ( Q ^ ) ) in dom ( ( P ^ ) ^ ( Q ^ ) ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) ;
R |^ n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and 1 = p1 . n1 and 1 <= n1 and n1 <= len p1 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 in Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( seq_id ( v ) ) . ( v ) .| & ||. v .|| = ||. ( seq_id ( v ) ) . ( v ) .||
for phi st phi in X holds phi in X & not phi in X & not phi in X & not phi in X & not phi in X ;
rng ( ( Sgm dom ( f | ( dom ( f | ( dom ( f | ( dom f \ f . i ) ) ) ) ) ) ) ) c= dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom ( f | ( dom f \ ( dom ( f | ( dom f \ ( dom f \ ( f . i ) ) ) ) ) ) ) ) ) ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c ;
( the_arity_of ( a , b , c ) ) = <* ( ( b , c ) --> ( a , b ) ) , ( ( a , b ) --> ( b , c ) ) *> ;
consider f1 being Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f . 0 = r ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & a3 = b3 & a4 = b2 & a5 = b3 & a5 = b2 & 8 = b3 & 8 = b3 ;
D2 . indx ( D2 , D1 , n1 ) = D1 . ( indx ( D2 , D1 , n1 ) + 1 ) .= D2 . ( indx ( D2 , D1 , n1 ) + 1 ) ;
f . ( |. |[ r , r ]| .| ) = |. |[ r , r ]| .| /. 1 .= <* r *> /. 1 .= x ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . m = C-25 . m & C-25 . n = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative & for b being Element of X holds F \hbox { b } = f . b & f . b = f . b ;
p = - ( r * p0 + 0. TOP-REAL 2 ) .= 1 * ( p0 `1 ) + 0 .= ( 1 - r ) * ( p2 `1 ) .= ( 1 - r ) * ( p2 `1 ) .= ( 1 - r ) * ( p2 `1 ) ;
consider z1 such that b , x3 , x3 , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> x2 ;
consider i such that Arg ( ( Rotate ( s , r ) ) . q ) = s + Arg ( ( 2 * PI * i ) ) ;
consider g such that g is one-to-one and dom g = card f and rng g = f . x and rng g c= f . x and g . x = f . x ;
assume that A = P2 \/ Q2 and Q1 <> {} and Q2 <> {} and Q2 is open and Q1 is open and Q1 is open and Q1 is open and Q1 is open and Q1 is open and Q2 is open and Q1 is open and Q1 is open and Q2 is open and Q2 is open ;
attr F is associative means : Def1 : F .: ( f , g ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m < i & i < m ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . k2 and l = P-2 . k2 and k2 <= len P-2 + 1 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & y in D2 } "\/" { p , q } ;
consider z being element such that z in dom ( ( dom F ) * ( F . z ) ) and ( ( F . z ) . z ) = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
Int cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = x ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , b2 , b1 , b2 ) ) . ( -Y. ) .= ( Mx2Tran J ) . j ;
- 1 / ( ( - 1 ) * D ) = ( ( - 1 ) (#) D ) | n .= ( ( - 1 ) (#) D ) | n .= ( - 1 ) (#) D .= ( - 1 ) (#) D ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) .= len ( f2 /. j ) ;
All ( All ( a , A , G ) , B , G ) 'imp' Ex ( All ( a , B , G ) , A , G ) ;
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k * a ;
k = ( 0 qua Nat ) -ininsucc succ ( k ) .= ( ( commute I ) . k ) . ( i + 1 ) .= ( commute I ) . ( i + 1 ) ;
for s being State of A holds Following ( s , n ) . 0 + ( n + 2 ) * ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( ( support ( n ) ) \/ support ( ( m ) ) c= support ( ( n ) ) \ { 0 } ) & ( ( m ) ) \ { 0 } c= support ( ( n ) ) \ { 0 } ;
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 /. ( b . a ) = f . ( g . a ) & phi /. ( b . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) ^ <* p *> ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 7 , 8 } = { x1 } \/ { x2 , x3 , x4 } \/ { x4 , 8 , 7 , 8 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 c= the Sorts of U1 "\/" ( U2 "\/" U2 ) ;
( - ( 2 * a ) ) ^2 + b ^2 > 0 & - ( 2 * a ) ^2 + b ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] and P [ z ] ;
assume that ( the ResultSort of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R * ( ( arccot ) ) ^2 ) ) /\ dom ( ( exp_R * ( f1 + #Z 2 ) ) ^2 ) ;
integral ( f , SS1 ) is convergent & lim ( integral ( f , SS1 ) ) = integral ( f , SS1 ) & lim ( integral ( f , SS2 ) ) = integral ( f , SS2 ) ;
( X ( ) ) => ( ( f . ( a , f . ( x9 , y9 ) ) => ( f . ( x9 , y9 ) ) ) in consider being Element of l ( ) such that x = [ x1 , y9 ] and x9 in as Element of l ( ) ;
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 \/ X2 is open & X1 , X2 are_separated & X2 , X1 are_separated implies X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated ;
for L being lower-bounded antisymmetric RelStr , s being non empty Element of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = ( ( F . b ) `2 ) , f-129 = ( ( F . b ) `2 ) , f29 = ( F . b ) `2 , f29 = ( F . b ) `2 , f-129 = ( F . b ) `2 as Function of ( ( M . b ) `2 ) , M . b ;
consider w being FinSequence of I such that the InitS of M , <* s *> ^ w ^ <* s *> ^ w ^ w ^ w ^ <* q *> ^ w ^ w ^ w ^ <* s *> ^ w ^ w ^ w ^ w ^ w ^ w ^ <* q *> ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ <* q
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ G .= 1_ G .= 1_ G .= 1_ G ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier L = C & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider o9 = o `2 , p = o `2 , q = p `2 , r = ( the Sorts of A ) . ( ( the Sorts of A ) . o ) as Element of TS ( ( the Sorts of A ) . o ) ;
1 * x1 + ( 0 * x2 + ( 0 * x3 ) ) + ( 0 * x3 ) = x1 + ( 0 * x2 + ( 0 * x3 ) ) .= x1 + ( 0 * x2 ) .= x1 ;
Ez " . 1 = ( ( Ez ) " ) . 1 .= ( ( ( 1 - 2 ) * ( 1 - 2 ) ) * ( 1 - 2 ) ) .= ( 1 - 2 ) * ( 1 - 2 ) ;
reconsider u1 = the carrier of ( U1 ) /\ ( U1 "\/" U2 ) , v1 = the carrier of ( U1 ) /\ ( U1 "\/" U2 ) as non empty Subset of ( ( U1 ) "\/" ( U1 "\/" U2 ) ) ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l1 + 1 ) ) .| < ( 1 / ( M . ( M . ( l1 + 1 ) ) ) ) ;
LSeg ( ( ( Cage ( C , n ) ) /. ( i + 1 ) ) , ( ( Cage ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- ( x - x0 ) ) + R /. ( x- ( x - x0 ) ) ;
g . c * ( - ( g . c ) * f . c ) + f . c <= h . c * ( - ( g . c ) ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of \HM { the } \HM { set , b is non empty and len ( ColVec2Mx b ) = width A and width ( f * b ) = width A ;
len ( - ( M1 + M2 ) ) = len M1 & width ( - ( M1 + M2 ) ) = width M1 & width ( - ( M1 + M2 ) ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of [: Z , the carrier of Z :]
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg ( - a ) & Arg ( - b ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( ( the open of a ) --> b ) & not c in Intersection ( ( the open of a ) --> b )
assume that V1 is closed and V2 is closed and V2 is closed and V = { v + u : v in V1 & u in V1 & u in V1 & v in V1 } and V1 is closed and V2 is closed and V2 is closed and V2 is closed and V2 is closed and V2 is closed and V2 is closed and V2 is closed ;
z * x1 + ( 1 / ( 1 - r ) ) * x2 in M & z * ( x1 + ( 1 / ( 1 - r ) ) * x2 ) in N ;
rng ( ( PS1 qua Function ) " ) = Seg ( card ( SS1 " ) ) .= Seg ( card ( SS1 " ) ) .= Seg ( card ( SS1 " ) ) .= Seg ( card ( SS1 " ) ) ;
consider s2 being complex 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 < h2 . n & 0 < ( 1 + ( 1 + ( 1 + n ) ) ) / 2 ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. ( seq1 . m ) .|| .= ||. ( seq1 . m ) .|| .= ||. ( seq1 . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) .: D ) .= sup ( ( k .: D ) ) ;
A |^ ( k , l ) = ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) .= ( A |^ ( n , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) = ( ( a + b ) div ( a * b ) ) + ( ( a * b ) div ( a * b ) )
consider A9 being countable set such that r is countable & ( for A being Subset of Al holds A is closed implies A is closed ) & ( A is closed implies A is closed implies A is closed ) & A is closed implies A is closed implies A is closed ;
for X be non empty addLoopStr , M , N be Subset of X , x , y being Point of X st y in M & x + y in N holds x + y in N + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] , [ y1 , y2 ] } \/ { [ x1 , y2 ] , [ y1 , y2 ] }
h . O = |[ A * ( f . O ) `1 + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n ) divides n implies for Nat st m divides n & n is prime & m divides n holds ( m divides n ) & ( m divides n implies m divides n ) & m divides n & m divides n implies m divides n & n divides m
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( 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. 0 , y ) ) and z = H / ( x. 1 , 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 , W . 3 , G . 4 , G . 5 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 , G . 6 ,
( h (#) ' _ { h } [ f ] ) . ( 2 * n ) = ( h (#) ( f ^ ) ) . ( 2 * n ) ;
j + 1 = ( i - len h11 + 2 ) + 2 .= i + 1 - len h11 + 2 .= i + 1 - len h11 + 2 - 1 .= i + 2 - 1 ;
S ^ ( S /* ( f ^ ) ) . f = S /* ( S ^ ) .= S . ( ( f ^ ) . f ) .= S . f ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 * H ) and Sum ( L2 * H ) = Sum ( L2 * H ) ;
attr R is element means : Def1 : for p , q st p in R & q <> p ex P st P is special & p in P & q in R & P c= R ;
dom ( product ^ \ast ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) ;
sup ( ( proj2 .: ( Lower_Arc ( C ) ) /\ Lower_Arc ( C ) ) ) <= upper_bound ( ( proj2 .: ( Lower_Arc ( C ) ) /\ Vertical_Line ( w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p .| < r / 2
i * f-28 - f = i * ( f - y ) .= i * ( f - y ) .= i * ( f - y ) .= i * ( f - y ) ;
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 and g2 = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def1 : d |^ n divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) ;
f\rbrack . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . [ 2 * x , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) + ( seq . n ) ) ;
( q `1 / |. q .| - cn ) ^2 / ( 1 + cn ) ^2 <= ( q `1 / |. q .| - cn ) ^2 / ( 1 + cn ) ^2 / ( 1 + cn ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier of S } such that a = [ o , x2 ] and [ o , x2 ] in dom ( the Arity of S ) ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a
||. h1 .|| . n = ||. h1 . n .|| .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n ;
( ( - ( exp_R . x ) ) / ( 1 + x ) ) = f . x - ( exp_R . x ) / ( 1 + x ) .= ( - 1 ) / ( 1 + x ) ;
pred r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) & for i st i in dom r holds r . i = min ( p . i , r . i ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M , N being Matrix of n , K st i in Seg n & i in Seg n holds Det ( M , i ) = Sum ( Line ( M , i ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * v = 1 ;
p . ( j -' 1 ) * ( q /* r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * ( q . ( i -' 1 ) ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) " .= ( R /* ( h ^\ n ) ) " . ( ( h ^\ n ) . $1 ) ;
assume that the carrier of H2 = f .: the carrier of H1 and the carrier of H2 = f .: the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ) . o .= ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 -] & ( |. 2 to_power ( n + 1 ) + h .| ) = n + 1 + 1 .= n + 1 + 1 .= n + 1 ;
( O = 0 or O = 1 or O = 1 or O = 2 or O = 3 ) & O = 3 or O = 4 & O = 5 & O <> 5 & O <> 6 & O <> 6 & O <> 7 & O <> 6 & O <> 6 & O <> 7 ;
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= F1 .: ( ( n + 2 ) ) .= { f /. ( n + 2 ) } .= { f /. ( n + 1 ) } ;
pred b <> 0 & d <> 0 & b <> d & ( a = d ) & ( a = ( - e ) / ( b - d ) ) implies ( a = ( - e ) / ( b - d )
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & v in B & w = a * v & v = a * v
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= ( g `2 * ( g `2 * P `2 ) ) * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and f . i = s2 and not ( ex s st s = i & not ( s = 1 & s = 1 & s = 1 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] connected & [ s2 , t2 ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R implies [ s2 , t2 ] in R
then H is negative & H is non negative & H is non negative implies H is not negative -g\mathopen { H } , H ) & H is not conjunctive -g\mathopen { H } ;
attr f1 is total means : Def1 : ( 1 / 2 ) (#) ( f1 + f2 ) is total & ( 1 / 2 ) (#) ( f2 + f3 ) = f1 . c * f2 . c + f2 . c * ( f2 + f3 ) . c ;
z1 in W2 ` iff z1 = z2 & ( z1 = z2 or z1 = z2 & z2 = z1 ) & ( z1 = z2 or z1 = z2 or z1 = z2 & z2 = z1 ) & ( z1 = z2 implies z1 = z2 or z1 = z2 or z1 = z2 ) ;
p = 1 * p .= a " * a * p .= a " * a * p .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a * ( b " * q ) .= a * ( b * q ) ;
for r9 be Real_Sequence , K be Real st for n be Nat st r <= K holds seq1 . n <= ( seq . n ) ^2 & ( seq . n ) ^2 <= ( seq . n ) ^2 + ( seq . n ) ^2
( E-max C ) `1 meets ( L~ go \/ L~ pion1 ) \/ ( L~ co \/ L~ co ) or ( E-max C ) `1 meets ( L~ go \/ L~ co ) or ( E-max C ) `1 meets ( L~ co \/ L~ co ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . ( k + 1 ) ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K * ( K *
assume h = ( ( ( B .--> ( C `2 ) ) +* ( D .--> ( E .--> ( C `2 ) ) ) +* ( F .--> ( J .--> ( J .--> ( F .--> ( J .--> ( J .--> ( J .--> ( C `2 ) ) ) ) ) ) ) ) +* ( F .--> ( J .--> ( J .--> ( J .--> ( J .--> ( J .--> ( F .--> ( J .--> ( J .--> ( J .--> ( J , A ) ) ) ) ) ) ) ) ) ) ) ) ) ) +* ( F .--> ( J .--> ( J .--> ( J .--> (
|. ( ( ( ( ( ( ( ( H . n ) ) ) || ( A . k ) ) ) ) . k - ( ( ( ( ( H . n ) * ( ( ( H . k ) / ( A . k ) ) ) ) ) ) . k ) .| <= e * ( ( ( ( H . n ) * ( ( H . k ) / ( A . k ) ) ) ) ) ) ;
(
{ x1 , x1 , x1 , x1 , x2 , x3 , x4 , x4 , x5 , 8 , 8 , 7 , 8 } = { x1 , x2 , x3 , x4 } \/ { x1 , x2 , x3 , x4 } .= { x1 , x2 , x3 } ;
consider A such that A = [. 0 , 2 * PI .] and integral ( integral ( ( ( ( #Z n ) * cos ) ) , A ) = 0 and ( ( #Z n ) * sin ) . x = 0 ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) /. i & p `2 = ( Sgm Y ) /. i implies p `2 = ( Sgm Y ) /. i * ( Sgm X ) /. i
for x , y st x in A & y in A holds |. ( 1 / ( f . x ) - ( 1 / ( f . y ) ) ) .| <= 1 * |. f . x - ( 1 / ( f . y ) ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) .= ( q2 `2 ) ^2 * ( ( q2 `2 ) ^2 + ( q2 `2 ) ^2 ) .= ( q2 `2 ) ^2 ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous & f /. 1 in dom f & f /. len f in dom f holds f /. 1 in rng f & f /. len f in rng f
assume for x being Element of Y st x in EqClass ( z , CompF ( PA , G ) ) holds ( Ex ( a , PA , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k , FM . ( k + 1 ) ] ;
ex u , u1 st u <> u1 & u , u1 / ( a , v ) / ( a , v ) / ( a , v ) / ( a , v ) / ( a , u1 ) / ( a , v ) / ( a , w ) / ( a , b ) / ( a , v ) / ( a , b ) / ( a , b ) / ( a , v ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b ) / ( a , b )
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N , A ) ` * ( N , B ) ` = N ` ` * ( N ` * B )
for s be Real st s in dom F holds F . s = integral ( ( R ^2 ) (#) ( f ) ) * integral ( ( f + g ) (#) ( f + g ) ) . x
width ( AutMt ( f1 , b1 , b2 ) ) = len b2 .= len b1 .= len b1 .= len b1 .= len b1 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 + len b2 .= len b2 + len b2 .= len b2 + len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ implies f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[
assume that X is closed and a in X and a in X and a in X and y in a implies { { [ n , x ] } \/ { x } : x in X } \/ { x } in a ;
Z = dom ( ( ( id Z ) (#) ( arctan + arccot ) ) `| Z ) /\ dom ( ( ( id Z ) (#) ( arctan + arccot ) ) `| Z ) ;
func [: V , l :] -> Subset of V means : Def1 : for k st 1 <= k & k <= len l holds it . k in { 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 , c being Point of L st c is Point of N & c in N holds c <= N & N . ( c , d ) <= M . ( c , d )
for s being Element of NAT holds ( ( ( id NAT ) --> ( v + u ) ) . s + ( id NAT ) . s = ( ( id NAT ) --> ( v + u ) ) . s + ( id NAT ) . s
then z /. 1 = ( ( ( N-min L~ z ) .. z ) .. z ) .. z & ( ( ( N-min L~ z ) .. z ) .. z ) .. z < ( ( ( N-min L~ z ) .. z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and for x st x in Z holds f . x = x and f . x > 0 and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable commutative associative commutative associative commutative distributive non empty doubleLoopStr , I , J being Subset of R , I , J being Subset of R holds ( I + J ) *' I c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of B1 , y being Element of B2 holds f . x = F ( x , y ) and f . y = F ( y , x ) ;
dom ( x2 + y2 ) = Seg ( len x + ( x2 + y2 ) ) .= Seg ( len x + ( x2 + y2 ) ) .= Seg ( len ( x + ( x2 + y2 ) ) .= dom ( x + ( x2 + y2 ) ) .= Seg ( len x + ( x2 + y2 ) ) .= dom ( x + ( x2 + y2 ) ) .= Seg ( len x + ( x2 + y2 ) ) .= dom x + ( x2 + y2 ) .= dom x + ( x2 + y2 ) ;
for S being holds for C , B 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 f /\ ( dom f \/ f " { 0 } ) & b in dom f /\ ( dom f \/ f " { 0 } ) & f . a = f . ( f . a ) ;
a in Free ( ( H / ( x. 4 , x. 0 ) ) ) '&' ( ( H / ( x. 0 , x. 4 ) ) ) / ( x. 0 , x. 0 ) ;
for C1 , C2 being set , f , g being stable Function of C1 , C2 st ( for x being set st x in C1 holds f . x = g . x ) & ( for x being set st x in C1 holds f . x = g . x ) holds f = g
( W-min L~ go ) `1 = ( W-bound L~ go + E-bound L~ co ) / ( 2 * ( 1 + ( W-bound L~ pion1 ) ) / ( 2 * ( 1 + ( W-bound L~ co ) ) ) / ( 2 * ( 1 + ( W-bound L~ co ) ) ) ) .= ( W-bound L~ go + E-bound L~ co ) / ( 2 * ( 1 + ( W-bound L~ co ) ) / ( 2 * ( 1 + ( W-bound L~ co ) ) ) ) ;
consider u , y0 , z0 , z0 being Real such that f = <* x0 , y0 , z0 *> and f is partial & u in dom ( SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ) & SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = x & t . {} = x & t . {} = y ;
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 ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b <= a ;
func Class ( R , a ) -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & a in A & a in A & it c= A ;
defpred P [ Nat ] means ( ( ( \HM { the } \HM { vertices } \HM { of G ) . $1 ) `1 ) `1 c= G . ( ( \HM { the } \HM { vertices } \HM { of G ) . $1 ) `1 ) ;
assume that dim ( ( U1 ) ) = 0 and dim ( ( U1 ) ) = 0 and dim ( ( U1 ) ) = 0 and dim ( ( ( U1 ) ) . 0 ) = 0 and dim ( ( ( U1 ) ) . 1 ) = 0 and dim ( ( ( U1 ) ) . 2 ) = 0 ;
mamam ( m . t ) = ( m . t ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 .= ( m . {} ) `1 ;
d11 = x9 ^ <* d *> .= f . ( y9 , d11 ) .= f . ( y9 , d11 ) .= f . ( y9 , d11 ) .= ( f . ( y9 , d11 ) ) `1 .= ( f . ( y9 , d11 ) ) `1 .= ( f . ( y9 , d11 ) ) `1 .= ( f . ( y9 , d11 ) ) `1 .= ( f . ( y9 , d11 ) ) `1 ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x iff g . x in f . x & f . x in f . x ;
x + 0. F_Complex = x + ( len x |-> 0. F_Complex ) .= ( x + len x |-> 0. F_Complex ) .= ( x + len x |-> 0. F_Complex ) .= ( x + len x |-> 0. F_Complex ) .= x + ( x + 1 ) .= x + ( x + 1 ) ;
( k -' ( k9 + 1 ) ) in dom ( f /. ( ( k -' 1 ) + 1 ) ) /\ ( ( k -' ( k + 1 ) ) + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p2 , p3 } and P1 = { p2 , p3 } and P2 = { p1 , p3 } and P1 = { p2 , p3 } and P2 = { p1 , p3 } and P2 = { p2 , p3 } and P1 = { p2 , p3 } and P2 = { p1 , p3 } and P2 = { p2 , p3 } and P2 = { p1 , p3 , p3 , p4 , p4 , p4 , p4 } and P2 = { p1 , p3 , p3 , p4 } and P2 = { p2 , p3 } and P2 = { p1 ,
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c1 = p , c2 = q , c2 = p , c1 = r , c2 = s , c2 = s , c1 = q , c2 = s , c2 = r , c1 = s , c2 = s , c2 = s , c2 = s , c1 = q , c2 = s , c2 = s , c2 = s , c2 = s , c1 = s , c2 = s , c1 = q , c2 = s , c2 = s , c1 = s , c1 = s , c2 = s , c2 = s , c2 = s , c1 = q , c2 = s , c2 = s , c1 = s , c2 = s , c1 = r , c2 = s , c2 =
reconsider thesis thesis thesis Gtb1f = G1 . ( t , b ) * F1 . f , FFf = G1 . ( t , b ) * F2 . f , FFf = G2 . ( t , b ) * F2 . f ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 + 1 ) ) \/ LSeg ( f /. ( i + 1 -' 1 + 1 ) , f /. ( i + 1 -' 1 + 1 + 1 ) ) ;
Integral ( P ` , P . m ) | dom ( P . n ) <= Integral ( M , P . m ) & Integral ( M , P . m ) <= Integral ( M , P . m ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( ( G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) ) `2 ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer st i in H holds a |^ i = b |^ i
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] and P [ x , B . x ] ;
reconsider K1 = { pp where pp is Point of TOP-REAL 2 : P [ pp ] & P [ p ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p where p is Point of TOP-REAL 2 : p in P & p `2 >= 0 } , K1 = { p where p is Point of TOP-REAL 2 : p in P & p `2 >= 0 } , K1 = { p : p `2 <= 0 } ;
( ( ( ( ( ( ( ( N ) ) - ( ( N ) ) / ( m + 1 ) ) ) ) / ( m + 1 ) ) ) / ( m + 1 ) ) <= ( ( ( ( ( N + 1 ) - ( N + 1 ) ) / ( m + 1 ) ) ) / ( m + 1 ) ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & Im ( F . n ) . x <= P . x & Im ( F . n ) . x <= P . x
len ( @ H ) = len ( @ H ^ <* 0 *> ) + len <* 2 *> .= len ( @ H ^ <* 1 *> ) + len <* 2 *> .= len ( ( @ H ) ^ <* 1 *> ) + len <* 2 *> .= len ( ( @ H ) ^ <* 1 *> ) + 1 ;
v / ( x. 3 , m1 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) ) = ( x. 4 , m3 ) / ( x. 0 , m1 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / (
consider r being Element of M such that M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 4 , m
func w1 \ w2 -> Element of Union ( G , R8 ) equals ( ( the Sorts of G ) * ( the ResultSort of G ) ) . ( w1 , w2 ) & ( ( the Sorts of G ) * ( the ResultSort of G ) ) . ( w1 , w2 ) = ( the Sorts of G ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s1 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - Partial_Sums ( |. seq .| ) . ( n + k ) + Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n + Partial_Sums ( |. seq .| ) . ( n + k )
set F = S \! \mathop { 0 } ;
( Partial_Sums ( seq ) ) . K + Partial_Sums ( seq ) . K + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . K + Partial_Sums ( seq ) . K + 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- ( x - x0 ) ) + R . ( x- ( x - x0 ) ) ;
func the closed of \HM { a , b , c , d } = ( the distance of \HM { a , b , c } ) ` \/ ( the such that d = ( the such that d = a and c = b and d = d ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 >= 6 * a * b * c ;
v / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) ;
consider consider consider consider \mathop { \rm such that Q ^ <* x *> , M *> , M ^ <* x *> ) = ( ( \mathop { \it true } ( K , 0 ) ) +* ( ( \mathop { \it true } ( K , 0 ) ) --> TRUE ) ) . ( ( \mathop { \it true } ( K , 0 ) ) --> TRUE ) ;
Sum ( FM ) = ( r |^ n1 ) * Sum ( CM ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) .= C . ( n1 + 1 ) ;
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 .= ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means Partial_Sums ( s ) . $1 = ( a * ( $1 + 1 ) ) * ( 1 / ( $1 + 1 ) ) + b / ( $1 + 1 ) * ( 1 / ( $1 + 1 ) ) ;
( the_arity_of g ) . g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the ResultSort of S ) . g ] ) `1 .= ( ( the Arity of S ) . g ) . g .= ( the Arity of S ) . g ;
( X ~ ) ^ Z tolerates X ~ , Y ^2 & card ( ( X ~ ) * Z ) = card ( X ~ ) * Z ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = G . ( n + 1 ) holds b = N . ( s . n ) \ G . s ;
E , f |= All ( x. 2 , ( x. 0 ) ) '&' ( x. 2 , ( x. 1 ) ) '&' ( x. 0 , ( x. 1 ) ) '&' ( x. 1 , ( x. 2 ) ) '&' ( x. 2 , ( x. 1 ) ) '&' ( x. 1 , ( x. 1 ) ) ) ;
ex R2 be 1-sorted st R2 = ( p | ( n + 1 ) ) . i & ( the carrier of p ) = the carrier of R1 & ( the carrier of p ) c= the carrier of R2 ;
[. a , b + ( 1 / ( k + 1 ) ) .[ is Element of the \rbrace & ( ( the partial of f ) . k ) . ( ( the partial F of f ) . ( k + 1 ) ) is Element of the \overline of ( the , the carrier of a ) . ( k + 1 ) ) ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( ( a , s ) := ( 2 , 1 ) ) .= Exec ( ( a , s ) := ( 2 , 3 ) ) ;
card ( h1 ) . k = power ( F_Complex , n ) . ( ( - 1_ F_Complex ) * k ) .= ( ( - 1_ F_Complex ) * k ) * Sum u .= ( ( - 1_ F_Complex ) * ( ( - 1_ F_Complex ) * k ) ) * u .= ( ( - 1_ F_Complex ) * ( k * k ) ) * u .= ( ( - 1_ F_Complex ) * ( k * k ) ) * u ;
( f / g ) /. c = f /. c * ( g /. c ) " .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) ;
len ( ( C /. len ( C /. 1 ) ) - ( ( ( C /. 1 ) ) - ( ( C /. 1 ) ) ) = len ( ( C /. 1 ) ) - ( ( C /. 1 ) ) .= len ( ( C /. 1 ) ) - ( ( C /. 1 ) ) .= len ( ( C /. 1 ) ) - ( ( C /. 1 ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) /\ X .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X ) .= dom ( f | X )
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) ) * Fib ( n ) ;
consider f being Function of INT , INT such that f = f ` and f is onto and f is onto and n < k and f is onto and for n st n < k holds f " { f . n } = { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( A , B ) and ( for A , B st A in S holds E . A = Prob . ( A \/ B , C ) ) and ( E . A ) = Prob . ( B , C ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x , y ] } , y ) and P [ y , x ] ;
assume that A c= Z and f = Z and f = ( ( - 1 ) (#) ( ( id Z ) ^ ) ) (#) ( ( id Z ) ^ ) and Z c= dom f and f = ( - 1 ) (#) ( ( id Z ) ^ ) and f = ( - 1 ) (#) ( ( id Z ) ^ ) and Z c= dom f and f = ( id Z ) ^ ) and Z c= dom f ;
( 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 ;
dom Shift ( Seq q2 , len Seq q1 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 = ( j + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 +
consider G1 , G2 , G3 being Element of V such that G1 <= G2 & G2 <= G2 and f . 1 = G1 & g . len f = G2 & g . len f = G2 & g . len g = G2 & f . len g = G2 & f . len f = G2 & g . len g = G2 & f . len f = G2 & g . 1 = G2 ;
func - f -> PartFunc of C , V means : Def1 : 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 & it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a and for v st v <> a holds ( for v st v in a holds v in union L holds L . ( v , a ) = L . ( v , a ) iff L . ( a , v ) = H . ( v , a ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 + 1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 + 1 , j1 + 1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i / n ) * ( n / ( n + 1 ) ) and for n1 , n2 being Integer st 1 <= i & i <= n & n2 <= n holds ( i / ( n + 1 ) ) * ( n / ( n + 1 ) ) = ( i / ( n + 1 ) ) * ( n / ( n + 1 ) ) ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) ) (#) ( ( 1 / 2 ) ) (#) ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 )
cell ( G1 , i1 -' 1 , ( 2 |^ ( m -' 1 ) ) * ( ( m -' 1 ) + ( 2 |^ ( m -' 1 ) ) * ( ( m -' 1 ) + ( 2 |^ ( m -' 1 ) ) * ( ( m -' 1 ) + ( 2 |^ ( m -' 1 ) ) ) * ( ( m -' 1 ) + ( 2 |^ ( m -' 1 ) ) * ( ( m -' 1 ) + ( m -' 1 ) ) ) ) ) \ ( ( m -' 1 ) + ( m -' 1 ) ) ) \ ( ( m -' 1 ) ) ) \ ( ( m -' 1 ) + ( m -' 1 ) ) c= BDD L~ f ) \ ( ( m -' 1 ) + ( m -' 1 ) + ( m -' 1 ) ) \ ( ( ( m
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st [: Q1 , F8 :] c= F & F8 is open & F8 is open & F8 is open & F8 is open & F8 is open & F8 is open & F8 is open ;
gcd ( A , ( ( 1 , 1 ) * ( r , s1 ) ) , ( ( 1 , 1 ) * ( r , s2 ) ) ) = 1 & gcd ( A , ( ( 1 , 1 ) * ( r , s1 ) ) , ( ( 1 , 1 ) * ( r , s2 ) ) ) = 1 ;
R8 = ( ( ( h . ( ( + 1 ) + 1 ) ) + 1 ) ) . ( m2 + 1 ) .= ( ( ( h . ( ( + 1 ) + 1 ) ) + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( ( h . ( ( + 1 ) + 1 ) ) + 1 ] ;
CurInstr ( P-6 , Comput ( P-6 , s , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 + 1 ) ) .= halt SCMPDS .= ( CurInstr ( P3 , s3 ) ) ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) = { p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p2 ) \/ LSeg ( p1 , p2 ) \/ LSeg
func -> Subset of the Sorts of A means : Def1 : a in it iff ex p being FinSequence of the carrier of A st p in dom f & a = f . i & p . 1 = f . i & for i st i in dom f holds f . i = p . i & f . ( i + 1 ) = q . i ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. b .| >= 1 holds f . a * ( f . b ) = 1 & f . b * ( f . b ) * ( f . b ) * ( f . b )
defpred P [ Nat ] means ( 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & G * ( i , j ) = G * ( i , j ) & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 be <> {} and for f being State of C1 , g being State of C2 , s being State of C2 , n being Nat st n = len f & n = len g holds f . n is stable iff for n being Nat st n in dom f holds f . n is stable & g . n is stable & f . n = n ;
( ||. f .|| ) . c = ||. f .|| . c .= ||. f .|| /. c .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of TT st F is open & {} in F & not {} in F & for A , B being Subset of TT st A in F & B in F & A is open holds card A = card B & card B = card A implies card B c= card F
assume that len F >= 1 and len F = k + 1 and len F = k and for i st i in dom F holds F . i = G . i & for k st k in dom F holds F . k = g . ( k + 1 ) and for k st k in dom F & k in dom F holds F . k = g . ( k + 1 ) ;
i |^ ( ( \mathop { \rm Let n ) - ( i |^ k ) ) |^ s = i |^ ( s + k ) - ( i |^ k ) .= i |^ ( s + k ) - ( i |^ k ) .= i |^ ( s + k ) - ( i |^ k ) .= i |^ ( s + k ) - ( i |^ k ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and q . ( q . 1 ) = v1 and ( for q st q in rng ( p ^ q ) holds q . ( q . 1 ) = v2 . ( p . 1 ) ) and rng q c= rng ( p ^ q ) ;
defpred P [ Element of NAT ] means $1 <= len ( I , Z , I ) implies ( ( ( ( f , Z , I ) ^ <* 0 *> ) ^ ( I , I ) ) . $1 = ( ( ( f , Z , I ) ^ ( I , I ) ) . ( ( I , I , J ) . ( $1 + 1 ) ) ) . ( ( I , I , J ) . ( $1 + 1 ) ) ;
for A , B being Matrix of n , REAL for B , C 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 B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s holds ex a , b being Element of R st s . i = a * b & a * s . i = b * a & b * s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x , Re y ) , ( Im x ) * ( Im y ) + ( Im y ) * ( Im x ) + ( Im y ) * ( Im y ) + ( Im x ) * ( Im y ) + ( Im y ) * ( Im y ) + ( Im y ) * ( Im y ) + ( Im y ) * ( Im y ) + ( Im y ) * ( Im y ) ;
consider g2 being FinSequence of Fy such that g2 is continuous and g2 is continuous and rng g2 c= A and g2 . 1 = x1 and g2 . len g2 = x2 and g2 . len g2 = y1 and g2 . len g2 = y2 . 1 and g2 . len g2 = y1 and g2 . len g2 = y2 and g2 . len g2 = y1 and g2 . len g2 = y2 and g2 . len g2 = y2 and g2 . len g2 = y2 and g2 . 1 = y2 and g2 . len g2 = x2 and g2 . len g2 = x2 and g2 . len g2 = y2 and g2 . len g2 = x2 and g2 . len g2 = y2 and g2 . len g2 = x2 and g2 . len g2 = y2 and g2 . len g2 = y2 and g2 . len g2 = y2 and g2 . len g2 = x2 and g2 . 1 = x2 and g2 . 1 = x2 and g2 . len g2
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , 8 , n4 , 8 , n3 , n4 , n4 , n4 , 8 , n3 , n4 , 8 , n4 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 7 ,
( q `1 ) ^2 * a <= ( q `1 ) ^2 * a & ( q `1 ) ^2 * a <= ( q `1 ) ^2 * a & ( q `1 ) ^2 * a <= ( q `1 ) ^2 * a & ( q `1 ) ^2 * a <= ( q `1 ) ^2 * a ;
Fv . ( p9 . ( len p9 ) ) = Fv . ( p . ( len p9 + 1 ) ) .= ( v . ( p . ( len p9 + 1 ) ) ) .= ( v . ( p . ( len p9 + 1 ) ) ) .= ( v . ( p . ( len p9 + 1 ) ) ) .= ( v . ( p . ( len p9 + 1 ) ) ) .= v . ( p9 . ( len p9 + 1 ) ) ;
consider k1 being Nat such that k1 + k = 1 and a = ( <* a *> ^ ( k + 1 ) ) and a = ( <* a *> ^ ( k + 1 ) ) ^ ( k + 1 ) and a = ( k + 1 ) + ( k + 1 ) ;
consider B9 be Subset of [: B1 , B2 :] , y1 , y2 be set such that [: B1 , y2 :] is finite and y1 is finite and for x , y being Element of B1 , z being Element of [: B1 , B2 :] st [ x , y ] in D & z = [ x , y ] holds [ z , y ] in \mathop { \rm T , { 0 } , { 0 } , { 0 } :] ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry ( F2 , g ) ) . b2 ) * ( ( curry ( F2 , g ) ) . b2 ) .= ( curry ( F2 , g ) ) . b2 .= ( ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) . b2 ) ) * ( ( curry ( F2 , g ) ) . b2 ) .= ( ( curry ( F2 , g ) ) . b2 ) * ( ( curry ( F2 , g ) ) . b2 ) ;
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st dbeing Real st dbeing > 0 & for h be Real st h <> 0 & |. h .| < ( 1 / 2 ) * ||. ( L . h ) .|| + ( 1 / 2 ) * ||. ( L . h ) .|| < e / 2 ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { G * ( len G , 1 ) } \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } ;
( ( - x ) .|. y ) = - ( ( 1 - x ) * ( x .|. y ) ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( p `1 ) ^2 + ( p `2 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 / sqrt ( 1 + ( p `2 ) ^2 ) ;
( ( ( U * ( W * ( W , m ) ) ) * ( ( W * ( W , m ) ) ) ) . ( x , y ) = ( ( ( U * ( W * ( W , m ) ) ) * ( W * ( m , m ) ) ) * ( ( W * ( W , m ) ) ) ) . ( x , y ) ) .= ( ( U * ( W * ( W , m ) ) ) * ( ( W * ( W * ( m , m ) ) ) ) * ( ( W * ( m , m ) ) ) * ( ( m , m ) ) ) * ( ( W * ( m , m ) ) ) * ( ( W * ( m , m ) ) ) * ( m , m ) ) .= ( ( W * ( m , y ) ) * ( ( ( x , y ) ) * ( ( W * ( x , y ) ) .= (
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = REAL & for x be Element of REAL , h be PartFunc of REAL , REAL st x in dom it holds it . x = ( - h ) * f . x & for x be Element of REAL st x in dom it holds it . x = ( - h ) * f . x + ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
( not y in Free H implies x in Free H & not x in Free H or x in Free H & not y in Free H or x in Free H & y in Free H or x = y ) & not x in Free H & y in Free H implies x = y
defpred P11 [ Element of NAT , Element of NAT ] means ( p |-count ( p |-count ( $1 -' 1 ) ) ) / ( $1 -' 1 ) < ( p |-count ( $1 -' 1 ) ) / ( $1 -' 1 ) ) & ( $1 -' 1 ) <= ( p |-count ( $1 -' 1 ) ) / ( $1 -' 1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A , B being Subset of X st A c= it holds A c= C & B c= C implies for A , B being Subset of X st A c= B & B c= C holds C . A c= C . B & C . B c= C . ( A \/ B ) ;
[#] ( ( dist ( ( ( ( ( ( ( P ) ) ) ) ) .: Q ) ) ) ) = ( ( ( dist ( ( ( ( P ) ) ) ) ) ) .: Q ) ) .: Q & ( ( ( ( ( ( ( ( ( ( ( ( ( P ) ) ) ) ) ) ) ) ) .: Q ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( P ) ) ) ) ) ) ) ) ) ) .: Q ) ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 1 , 2 } or rng ( F | ( [: S , T :] ) ) = { 2 , 3 } ;
( f " ( rng f ) ) . i = f . i " . i .= ( f . i ) " . i .= ( f . i ) " . i .= ( f . i ) " . i .= ( f . i ) " . i .= ( f . i ) . i .= ( f . i ) . i .= ( f . i ) . 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 C = { p1 , p2 } and C = { p2 } and C = { p1 , p2 } and C = { p2 , p3 } and C = { p1 , p2 } and C = { p2 , p3 } and C = { p2 , p3 } and C = { p1 , p3 } and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and C is closed and
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 / ( 1 + ( p2 `1 / p2 `2 ) ^2 ) , ( p2 `2 ) ^2 / ( 1 + ( p2 `1 / p2 `2 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 / ( 1 + ( p2 `1 / p2 `2 ) ^2 ) , ( p2 `2 ) ^2 ]| ;
( ( ( a , X ) ) * ( ( a , X ) " ) ) . x = ( ( ( a , X ) " ) * ( ( a , X ) " ) ) . x .= ( ( a , X ) " ) * ( ( a , X ) " ) . x .= ( ( a , X ) " ) * ( a , X ) ) . x .= ( ( a , X ) " ) * ( a , X ) .= ( a , b ) * ( a , b ) * ( b , b ) ;
for T being non empty normal TopSpace , A , B being closed Subset of T , p being Point of T st A <> {} & A misses B & B misses B holds A is open & B is open implies for p being Point of T st p in A & p in B holds p in Cl ( A /\ B ) & p in B implies p in Cl ( A /\ B )
for i st i in dom F for i , 1 being strict normal Subgroup of G1 , G1 , G2 being strict normal Subgroup of G2 st G1 = F . i & G2 = F . i & G1 is strict Subgroup of G2 & G2 is strict Subgroup of G1 holds G1 * ( i , 1 ) = G1 * ( i , 1 ) & G2 * ( i , 1 ) = G2 * ( i , 1 )
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) `| Z ) . x = ( ( 1 / 2 ) * ( f1 + #Z 2 ) ) / ( 1 + x ^2 )
synonym f is right & f /. x0 = lim ( f /* a ) & f /. x0 = ( f /. x0 ) implies for a st a in dom f & a in dom f & ( for x st x in dom f holds f /. x = ( f /. x ) - ( f /. x0 ) ) & for x st x in dom f holds f /. x = ( f /. x ) - ( f /. x0 ) ) & ( for x st x in dom f holds f /. x = ( f /. x ) - ( f /. x ) & ( f /. x ) / ( f /. x ) & ( for x st x in dom f ) implies f /. x = ( f /. x ) / ( f /. x ) / ( f /. x )
then X1 , X2 are_separated or X1 , Y2 are_separated or ex Y1 , Y2 being SubSpace of X st Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y2 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated implies X1 , Y1 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated & Y1 , Y2 are_separated ;
ex N be 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- ( 1 , f , u ) ) + R . ( SVF1 ( 1 , f , u ) )
( ( p2 `1 ) ^2 + ( p2 `1 ) ^2 ) / ( 1 + ( p2 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) / ( 1 + ( p2 `2 ) ^2 ) ;
( ( 1 / ( t1 * t2 ) ) |^ n ) / ( ( t1 * t2 ) |^ m ) = ( ( 1 / ( t1 * t2 ) ) * ( t2 * t1 ) ) / ( ( t2 * t1 ) |^ m ) / ( t2 * t1 ) ) / ( t2 * t1 ) |^ m .= ( ( 1 / ( t1 * t2 ) ) |^ m ) / ( t2 * t1 ) ) / ( t2 * t1 ) ;
( for x holds f . x = ( ( - 1 / ( x + h ) ) (#) ( sin * ( x + h ) ) ) & for x st x in dom ( ( - 1 / ( x + h ) ) (#) ( sin * ( x + h ) ) ) holds ( h h ) . x = ( ( - 1 / ( x + h ) ) (#) ( sin * ( x + h ) ) ) . x
consider X-23 being Subset of Y , Y1 being open Subset of X such that t = [: Xf1 , Y1 :] and Y1 is open and Y1 is open and ex Y1 being Subset of X st Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) + b * d + b * d + b * d + c * d + d * b + b * d + c * d + d * b + c * d + d * b + d * a + b * d + c * d + d * b + d * c + d * b + c * d + d * b + d * b + c * d + d * b + d * b * d + d * a + d * a + d * a + d * a + d * a + d * a + d * a + d * a + d * a + d * a + b * a + d * a + c * a + c * a + c + c + 1 / a + c + b * a + c + c + c + c + c + c + c + b * a + c * a + c * a + c * a + c + b * a + c * a + c * a + c + c + b * a +
( ( ( W-bound D ) - ( W-bound D ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( i + 1 ) ) ) ) ) ) / ( 2 * ( ( i + 1 ) + ( i + 1 ) ) / ( 2 * ( ( i + 1 ) + 1 ) ) ) ) = ( ( i + 1 ) + ( ( i + 1 ) + ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) ^2 ) ) * ( 2 * ( ( i + 1 ) ) ) ^2 ) ) ^2 ) = ( ( i + 1 ) ) ^2 ) * ( ( i + 1 ) ) * ( ( i + 1 ) ) / ( 2 * ( ( i + 1 ) ) ) / ( 2 * ( ( i +