thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = Product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated ;
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 , c 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 ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in X ;
let x be element ;
let k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = that b = that b = that a = b ;
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 Element of REAL ;
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 in X ;
cluster uparrow x -> being being that L is being ;
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 ;
L~ L~ L~ L~ L~ L~ L~ L~ \subseteq h ;
G . y <> 0 ;
let X be RealNormSpace , A be Subset of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in - - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function of Y , BOOLEAN ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is upper-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= cos ;
1 <= i-15 ;
1 <= i-15 ;
LMP C in L ;
1 in dom f ;
let seq , n ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 & b2 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
z c= dom ( \HM { 0 } ) ;
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 , X be non empty set ;
assume P [ n ] ;
assume union S is finite independent finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , f be Function ;
b ` c= b9 & b ` c= b9 ;
assume not x in INT ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x & cosec . x > 0 ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I -> left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an1 < n & bn1 < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_continuous_in x0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , m = LifeSpan ( P3 , s3 ) ;
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 of f , A , B be Subset of f ;
let b be Element of X , c be Element of Y ;
R [ x , y ] ;
x ` = x & y ` = y ;
b \ x = 0. X ;
<* d *> in D * ;
P [ k + 1 ] ;
m in dom ( n |-> 0 ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> ] ] ;
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 ;
assume f | X is lower ;
x in rng co /\ rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be be be as as as as as mamaid ;
let N be non empty \HM K ;
let R be RelStr with finite finite -> finite 1 -element number ;
let n , k be Nat ;
let P , Q be let let Q ;
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 \HM { + } \cdot a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v + u ) ;
x <= c2 . x ;
x in F ` & y in F ` ;
redefine func S --> T -> o -is \in ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> {} ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> C ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = B ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 + f2 ) ;
x in dom ( sec | A ) ;
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 | X ) ;
1 in dom ( D2 | 1 ) ;
p `2 = 0 & p `2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom f ;
h . x in h . a ;
let G be .: \it let Z be non empty /. ;
cluster m * n -> square ;
let ( k + 1 ) ;
i - 1 > m - 1 ;
R is transitive & R is transitive implies R is transitive
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_halting_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 | X ) ;
assume [ X , p ] in C ;
BX c= ( X \/ Y ) ;
n2 <= ( 2 * n ) - 1 ;
A /\ cP c= A ` ;
cluster x -valued -> constant for Function ;
let Q be Subset-Family of S , P be Subset of Q ;
assume n in dom g2 & n + 1 in dom g2 ;
let a be Element of R ;
t `2 in dom ( e2 | ( dom e2 ) ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be K of X , M be Element of S ;
i . y in rng i ;
REAL c= dom f & f | A is bounded ;
f . x in rng f ;
mt <= r / 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S , T ]| ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng F1 & g in union rng 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 & dom g c= dom y ;
n1 < n1 + 1 & n2 + 1 <= n + 1 ;
n1 < n1 + 1 & n2 + 1 <= n + 1 ;
cluster [: T , T :] -> \overline ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 ) ;
b = sup dom f & b = sup dom f ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom ( h2 * h1 ) ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k2 ;
i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete thesis ;
assume f = g & p = q ;
n1 <= n1 + 1 & n2 + 1 <= n2 + 1 ;
let x be Element of REAL ;
assume x in rng ( s2 | n ) ;
x0 < x0 + 1 & x0 + 1 < r2 ;
len ( Carrier ( L ) ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued sequence of X ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in being not being not being not being Element of being Element of V ;
i be set ;
n -' 1 = n-1 - 1 ;
len ( n-27 ) = n ;
\cal ] c= F ;
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of [: A , B :] , C ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E * ;
let B1 be Basis of x , B2 be Basis of y ;
Carrier ( L3 /\ L2 ) = {} ;
L1 /\ LSeg ( L1 , j ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng f-129 & x in rng f-129 ;
set nnN = n + j ;
let D be non empty set , f be FinSequence of D ;
let K be right_zeroed non empty addLoopStr , p be Polynomial of K ;
assume f `1 = f & h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k implies k <= n
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & ( TOP-REAL 2 ) | K1 is open ;
assume a , b are_maximal 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 n[ -> nes] ;
not u in { ag } ;
the carrier of f c= B \/ C ;
reconsider z = x as VECTOR of V ;
cluster the bounded Str of L -> strict non empty ;
r (#) H is n " ;
s . intloc 0 = 1 & s . intloc 0 = 2 ;
assume that x in C and y in C ;
let U0 be strict universal universal MSAlgebra over S , a be Element of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ex y being Element of : x = y & y in { x } ;
let x , y be Element of X ;
let A , I be \HM { \vert A .| } ;
[ y , z ] in [: O , O :] ;
( \subseteq card Macro i ) = 1 & card Macro i = 1 ;
rng Sgm ( A ) = A ;
q |- \! \! \smallfrown All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , a , b ;
p . 2 = Z |^ Y ;
( DD2 ) `2 = {} & ( DD2 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: CQC-WFF ( Al ) , { x } :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f2 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative non empty for multMagma ;
x in support ( ( support t ) | ( support t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y | i ) ;
assume p divides b1 + b2 & p divides b2 + b2 ;
M . x0 <= upper_bound M1 & M . x0 <= upper_bound M2 ;
assume x in W-min ( X ) & y in W-min ( X ) ;
j in dom ( z | i ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uH = Vertices G ;
( seq " ) is non-zero & ( seq " ) is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 ( p ) ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 union X2 , X2 union X1 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non 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 Subset of B & sup B is upper Subset of B ;
let L be non empty reflexive antisymmetric RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= the_left_argument_of H implies E , g |= H
dom G `2 = a & cod G `2 = b ;
( 1 / 4 ) * ( - 1 ) >= - r ;
G . p0 in rng G & G . p0 in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom ( id B ) & z in dom ( id B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & g in the carrier of G ;
rng f\mathbb R c= [: NAT , NAT :] ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
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 . k2 in rng P ;
M = AM +* {} .= ( A \/ { x } ) ;
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 and Q is_v1 , v2 ;
let a be Element of op ( V ) ;
let s be Element of PL ;
let PA be non empty thesis , PA 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 = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS ( K , n ) ;
l <= ( -> -> id of L . j ;
assume x in downarrow [ s , t ] ;
x `2 in uparrow t & x `2 in uparrow t ;
x in ( JumpParts T ) \/ { {} } ;
let h be Morphism of c , a ;
Y c= R implies R .: ( the_rank_of Y ) c= R .: Y
A2 \/ A3 c= Carrier ( L1 ) \/ Carrier ( L2 ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> strict closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 , q1 in P ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = 1 / ( n + 1 ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b implies R ~ = R
let M be non empty Subset of V , a be Element of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( id the carrier of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W2 + W3 ;
assume that the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of ( Bool M ) . i ;
0 <= Arg a & Arg a < 2 * PI ;
o , a9 // o , y & o , a // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) * ;
assume D2 . k in rng D & D2 . k in rng D2 ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 & LIN c , a , e1 ;
cluster -> increasing for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= P ! ( l ) ;
redefine func J . i -> non empty TopStruct ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\HM { field W1 , field W2 } ;
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = Seg n ;
s4 misses s2 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an implies [ a , y ] in an
assume that I c= J and not I c= K and I c= J ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
sin * ( cos * ( sin * ( cos * sin ) ) ) is_differentiable_on Z ;
t3 . n = t3 . n .= s . n ;
dom ( ( dom } F ) | A ) c= dom ( F | A ) ;
W1 . x = W2 . x & W2 . x = W3 . x ;
y in W .vertices() \/ W .vertices() ;
( k + 1 ) <= len ( v | ( k + 1 ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
Gij = ( U /. 1 ) `1 .= G * ( i , 1 ) `1 ;
f . rr1 in rng f & rr2 in rng f ;
i + 1 + 1-1 <= len f - 1 ;
rng F = rng ( F | ( n + 1 ) ) ;
mode non empty multMagma is well unital associative associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of support ( m ) c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
( s ^\ k1 ) is lower & ( s ^\ k1 ) is lower ;
len ( F | ( len F -' 1 ) ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , a be Element of REAL ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be empty Chain Chain Chain of T ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j2 ;
redefine func J => y -> total for I -valued Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def8 : ( a / a ) = 1 ;
assume that cf 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 & LIN o , b , c ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty thesis , F be non empty TopSpace ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp2 = x , pp2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine attr P is strict non empty as strict normal
rng c `1 misses rng ( e `1 ) \/ rng ( e `2 ) ;
z is Element of gr { x } & 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 / ( n + 1 ) ) ;
pred f = u means : Def8 : a * f = a * u ;
for n holds P1 [ ( being non zero Nat ) ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = p1 ;
( n1 gcd n2 ) = 1 & ( n1 gcd n2 ) = 1 ;
set oo = ( 2 * PI ) * ( 1 , j ) ;
seq . n < |. r1 .| & seq . n < 0 ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y2 , x2 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n to_power 1 : n in NAT } ;
k = a or k = b or k = c ;
a\leq , ag , bh , bh , bh , bh , bh , p ;
assume Y = { 1 } & s = <* 1 *> ;
Is1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W3 .last() = W3 . 2 ;
cluster trivial -> finite finite for R be finite is connected ;
reconsider u = u as Element of Bags X ;
A in B ^ -> Group implies A , B are_<* A , B *>
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) ;
f1 is_\HM \HM { the thesis of f2 : f2 in the carrier of P } ;
( f `2 ) ^2 / ( |. q .| ) ^2 <= ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
b `2 <= p `2 & p `2 <= ( p `2 + r ) / 2 ;
let f , g be s1 is Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( f , g ) ) ;
p2 in NL . p1 & p2 in NL . p2 ;
len ( the_left_argument_of H ) < len ( H ) & len ( H ) < len ( H ) ;
F [ A , F-14 F . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def8 : A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 & r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
A1 , L , A3 , A3 , A2 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in u implies { p , q } \/ { p , r } c= C
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
f12 | A2 = ( f2 | A2 ) | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & Y \ V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ 1 c= ( ( 1 - 1 ) * ( ( ( 1 - 1 ) * ( 1 - 1 ) ) ) ;
0 * a = 0. R .= a * 0 ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vp2 = ( v /. n ) `1 , vp2 = ( v /. n ) `1 ;
r = 0. ( REAL-NS n ) & r = 0. ( REAL-NS n ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `2 >= 0 ;
len W = len ( W -as Element of ( W -as non empty Subset of W ;
f /* ( s * G ) is divergent_to-infty & f /. ( s * G ) is divergent_to-infty ;
consider l being Nat such that m = F . l ;
t8 does not destroy b1 & not t8 does not destroy b1 ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i , j = j as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair for element ;
non empty downarrow a /\ downarrow t is Ideal of T ;
let X be non empty set , N be non empty set ;
rng f = being Element of implies S is \
let p be Element of B , s be Element of the connectives of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and Rx0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies q1 = q2
assume gR in the right of g & gR in the right of g ;
let A1 , A2 be Point of S , A be Subset of T ;
x in h " P /\ [#] ( T1 | P ) ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , Ximplies X' = Y as non empty Subset of Tsuch that X\cdot X = Y ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the in 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 & u in the carrier' of G2 ;
y = Re y + ( Im y ) * i ;
( as ( ( - 1 ) * p ) ) mod p = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng M5 c= rng D2 & len M5 = len D2 ;
for p being Real st p in Z holds p >= a
( for x being Element of X holds f . x = proj1 . x ) implies f is continuous
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A ++ ( B \ominus C ) = ( A ++ B ) \ominus C
h \equiv gg . ( mod P ) , T -ideal of P ;
reconsider i1 = i-1 , i2 = i2 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being strict Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , im2 = j as Element of NAT ;
dom f c= [: C ( ) , D ( ) :] ;
x in ( the Sorts of B ) . n ;
len } in Seg ( len ( f2 | i ) ) ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , a be Element of T2 ;
G * ( B * A ) = ( id o1 ) * A ;
assume that p , u , u is_collinear and u , v , q is_collinear ;
[ z , z ] in union rng ( F | ( X \ { x } ) ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , L = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , a3 , b1 & LIN a2 , a3 , b2 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom ( w2 * w1 ) = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 & ( ( g2 ) . I ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ip * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
q9 . x in rng ( q | ( n + 1 ) ) ;
Carrier Lxy misses Carrier ( Lxy ) \/ ( Carrier ( Lxy ) \/ { p } ) ;
consider c being element such that [ a , c ] in G ;
assume Na9 = oh & Na9 = oh & Na9 = Nh ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= F-12 |^ ( Cc ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 - q `2 <> 0. TOP-REAL 2 ;
redefine func \cal a] ( S , T ) -> non empty ;
let x be Element of [: S , T :] ;
the Arrows of F . ( a , b ) is one-to-one ;
|. i .| <= - ( 2 to_power n ) & |. i .| <= - ( 2 to_power n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = { 0 } ;
( n * ( n + 1 ) ) ! > 0 * ! ;
S c= ( A1 /\ A2 ) /\ A3 & S /\ A2 c= A2 /\ A3 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y = [ x , y ] ;
set v2 = ( v /. ( i + 1 ) ) ;
x = r . n .= r4 . n .= r . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: [: A2 , A2 :] , [: A1 , A2 :] :] ;
0 < ( p / ( ||. z .|| + 1 ) ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> \HM \HM { an } -valued for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , a be Element of U1 ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider px0 = p . x , px0 = p . y 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 - a is lower and - a is Element of - a ;
Int Cl A c= Cl Int Cl Int Cl A & Cl Int Cl Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 <= p `2 & p `2 <= p2 `2 or p2 `2 >= p `2 & p2 `2 <= p `2 ;
Cl Q ` = [#] ( ( TOP-REAL 2 ) | Q ) ;
set S = the carrier of T , T = the InternalRel of T ;
set I8 = for f be FinSequence of TOP-REAL n holds f is one-to-one iff f is one-to-one
len p -' n = len ( thesis ) - n .= len p - n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = nthat ( n + 1 ) = n7 ;
1 <= j + 1 & j + 1 <= len ( s | k ) ;
let q\subseteq , qT be Element of M ;
a in the carrier of S1 & b in the carrier of S1 & c in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c1 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) * S8 ) . x ;
consider x being element such that x in be N -_ set A ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n , h ) `1 , i1 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n , h ) `1 , i2 = ( n
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of ExtREAL ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 <= len p2 ;
let T1 , T2 be Scott Scott `1 of L , f be Function of T1 , T2 ;
then x <= y & ( x in : x in { y } ) ;
set M = n -tuples_on ( the carrier of K ) ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng the_arity_of a9 c= dom H & rng the_arity_of o c= dom H ;
z1 " = z9 " & z1 = z2 " & z1 = z2 " ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is that rng w /\ ( S \ L ) <> {} ;
set x-10 = x-9 ^ <* Z *> ^ xi2 ;
len w1 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of thesis , k be Element of thesis ;
x . n = ( |. a . n .| ) * ( A . n ) ;
p `1 <= Gik `1 & p `1 <= G * ( 1 , 1 ) `1 ;
rng ( g | ( L~ g ) ) c= L~ ( g | ( L~ g ) ) ;
reconsider k = i-1 * ( i + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x-10 = x-7 , x-10 = xas VECTOR of M ;
dom ( f | X ) = X /\ dom f .= X ;
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 = p . ag & m . ag = p . ag ;
a / ( s . m - 1 ) / ( s . n - 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and C2 \/ C2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= ( X * ) . i ;
r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ;
that \mathclose { 0. R } = a and b-0 = b ;
FF is_closed_on t2 , Q2 & FF is_halting_on t2 , Q2 implies thesis
set T = the non empty non empty TopSpace , X = the topology of T ;
Int Cl Int Cl R c= Int Cl R & Int Cl R c= Cl Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( Flen F ) = { Flen F } ;
G-23 \ ( { c } ) c= B \/ S ;
fbeing Relation of [: X , Y :] , X ;
set RQ = the \HM { Point of P , p = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k1 be Element of NAT ;
reconsider pp = u , pq = v as Element of ( TOP-REAL n ) | ( i + 1 ) ;
g . x in dom f & x in dom g implies g . x = f . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of carr ( G ) ;
len Pt <= len P-35 & len Pt <= len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple function ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL ;
rng f = the carrier of ( ( Carrier A ) * ( i , j ) ) ;
assume s1 = sqrt ( 2 - p ) - ( p |^ 2 ) ;
pred a > 1 & b > 0 & a to_power b > 1 ;
let A , B , C be Subset of Ias Subset of Ias Subset of Ias ;
reconsider X0 = X , Y0 = Y , Y0 = Z as Real ;
let f be PartFunc of REAL , REAL , r be Real ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , tt2 be Relation of n -tuples_on BOOLEAN ;
Q [ e-14 \/ { vN } , va1 ] ;
g \circlearrowleft W-min L~ z = z implies ( g /. 1 ) .. z < ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = v\vert x - y ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z + x
( 7 / p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan | A ) * ( f | A ) ) . x in dom ( sec | A ) ;
i2 = ( f /. len f ) & i2 = ( f /. len f ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & a * b = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & rng g2 c= the carrier of I[01] ;
dom f2 = the carrier of I[01] & rng f2 c= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: ( X ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & a1 . n < x0 + r ;
|. ( f /* s ) . k - GM .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
Sthesis / op = ( S . g ) / op .= ( S . g ) / op ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 := i2 := i3 does not destroy b1 & i2 does not destroy b2 implies not ( i1 , i2 ) := i2 does not destroy b2
arccos r + arccos r = PI / 2 + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= len ( 0 qua Nat ) ;
assume f in the carrier of [' X , Omega Y '] ;
F . a = H / ( x , y ) . a ;
( ( TRUE T ) at ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 2 * PI = 1 ;
p2 `1 - x1 > - g / 2 & p2 `1 - x1 < p2 `1 - g / 2 ;
|. r1 - thesis .| = |. a1 .| * |. thesis .| ;
reconsider S-14 L = 8 as Element of ( len S ) -tuples_on ( the carrier of K ) ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .3 = D0W .3 + 1 ;
i1 = ma + n & i2 = 0. ( K , n ) implies i1 = i2
f . a [= f . ( f . O1 "\/" f . a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , ( len T1 ) ) . s = 1 & chi ( T2 , ( len T2 ) ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ R4 implies ( M1 \/ M2 ) * ( i , j ) = ( M1 + M2 ) * ( i , j )
set h = the continuous Function of X , R , g = the Function of X , R ;
set A = { L . ( ( k . n ) `1 ) : not contradiction } ;
for H st H is atomic holds P7 [ H ] ;
set b' = S5 ^\ ( i + 1 ) , bseq = S5 ^\ ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
1 / ( n + 1 ) < ( 1 / s ) " ;
l `1 = [ dom l , cod l ] & l `2 = [ cod l , cod l ] ;
y +* ( i , y /. i ) in dom g & y in dom g ;
let p be Element of CQC-WFF ( Al ) , P be Subset of CQC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) & X /\ X1 c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( i + 1 ) ) & p2 in rng ( f /^ i ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) + 1 <= len D2 ;
assume x in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( K , K ) ) ) ) | K0 ) ) | K0 ) ) | K0 ) ) | K0 ) ) ) ) ) ;
- 1 <= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( q q ) ) ) ) ) ) ) ) ) . O ) ) ) `2 ) ) ) ) ) ^2 ;
let f , g be Function of I[01] , ( TOP-REAL 2 ) | P , p1 , p2 be Point of TOP-REAL 2 ;
k1 -' k2 = k1 - k2 & k1 -' k2 = k2 - k2 ;
rng ( seq ^\ k ) c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & g2 in ]. x0 - r , x0 .[ ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ y * u ;
ex a being thesis or a is thesis or for A being Ordinal st a = Sum A holds a in A
Cl ( union Ha ) = union ( ( union rng ( H | a ) ) ) ;
len t = len t1 + len t2 & len t = len t1 + len t2 ;
v-29 = v + w |-- ( A , w ) + A8 ;
cv <> DataLoc ( t0 . GBP , 3 ) & cv <> DataLoc ( s . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( \dot { y } ) ;
{ s : s < t } in REAL & t = {} implies t = {}
s ` \ s = s ` \ 0. X .= s ` \ ( s ` \ s ) ;
defpred P [ Nat ] means B + $1 in A & B + $1 in B ;
( 339 + 1 ) ! = 333335 ! * ( 339 + 1 ) ;
1. ( A , succ A ) = 1. ( A , A ) .= 1. ( A , A ) ;
reconsider y = y , z = z as Element of COMPLEX * ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) \mathop \mathop { \rm \hbox { - } F } ;
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 , p1 , p2 be Point of TOP-REAL n ;
( ( SAT M ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 0. ( V ) , r = 0. ( A ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + |. s .| + a ;
consider y being Element of S such that z <= y and y in X ;
a being being being being being Element of 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c )
||. xy0 - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - ( g - p ) ) ) ) ) ) ) ) ) ) ) )
b9 , a9 // b9 , c9 & b9 , c9 // c9 , a9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k1 + 1 = k2 & k2 + 1 = k2 + 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in cell ( RR , 1 , 1 ) & E-max L~ Cage ( C , 1 ) in rng R ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p , a // a , b or p , a // b , a ;
g . n = a * Sum fx1 .= f . n ;
consider f being Subset of X such that e = f and f is strict ;
F | ( N2 , S ) = CircleMap * ( F | ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , r ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re seq is summable and Im seq is summable and Im seq is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = 0 as 0 string of S2 , F = the not ( X , D ) --> 1 ;
reconsider xd = seq . n , xe = seq . n as sequence of REAL ;
assume that C meets L~ pion1 and E-max C in L~ pion1 and E-max C in L~ pion1 and E-max C in L~ pion1 ;
- ( Partial_Sums ( 1 / ( n + 1 ) ) . x ) < F . n - ( Partial_Sums ( 1 / ( n + 1 ) ) . x ) ;
set d1 = thesis , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d1 = dist ( x1 , z2 ) ;
2 |^ ( 2 -' 1 ) = 2 |^ ( 2 |^ 100 ) - 1 ;
dom ( v | ( len ( v | k ) ) ) = Seg ( len ( v | k ) ) ;
set x1 = - k2 + |. k2 .| , x2 = - k2 + |. k2 .| + |. k1 .| ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( L1 + L2 ) ) c= I2 & the carrier of ( Carrier ( L1 + L2 ) ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal \hbox of {} ;
Z c= dom ( ( sin * f1 ) `| Z ) & Z c= dom ( ( sin * f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `1 ) ^2 .| < r / 2 - ( q `1 ) ^2 ;
non empty not not not not not not not not not not not thesis & not ( A , t ) in ConsecutiveSet2 ( A , t ) ;
E = dom L8 & L8 is_measurable_on E implies ( for x st x in E holds L8 . x = 0 )
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC ss2 = P . IC ss2 .= ( card I + 2 ) .= ( card I + 2 ) ;
pred x > 0 means : Def8 : 1 / x = x to_power ( - 1 ) ;
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 ) are_connected ;
assume that f = id the carrier of OO and g = id the carrier of OO ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { 0. V } ) ;
reconsider g = f " as Function of U2 , U1 , f " ;
A1 in the points of G_ ( k , X ) & A2 in the \cap the carrier of G ;
|. - x .| = - ( - x ) .= - x .= - x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * contradiction ;
vM /. ( k + 1 ) = vM . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = n & width M1 = n ;
Line ( St , j ) = St . j .= St . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| ) * ( card b * h ) is nonnegative ;
assume that x = ( a1 ^ <* x1 *> ) ^ b1 and y = ( a1 ^ <* x1 *> ) ^ b1 ;
MI is_closed_on IExec ( I , P , s ) , P & MI is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y implies |. x .| = - y
LIN c , q , b & LIN c , q , c & LIN c , q , b & LIN c , q , c ;
f\rbrace . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flim . a = flim . a & v in InputVertices S & not v in InputVertices S ;
p `1 <= ( E-max C ) `1 & ( E-max C ) `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E7 = Cage ( C , n ) ;
p `1 >= ( E-max C ) `1 & p `1 >= ( E-max C ) `1 or p `1 >= ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p /. i and p in L~ f ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 , x0 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f /. len f <> f /. 1
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) c= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * t ) ) ;
dom B = 2 -tuples_on the carrier of V & rng B c= the carrier of V ;
consider r such that r _|_ a and r \not _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 in [. 1 / 2 , 1 .] ;
for L being complete LATTICE holds \mathbb <* \HM { A } , L *> , L are_isomorphic ;
[ gi , gj ] in [: Ii \ Ij , Ij \ Ij :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 ex g st r < g & g < x0 & g < x0 & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) / ( a ` ) , z = ( a ` ) / ( a ` ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - 1 ) ) * f ) . c <= h . c ;
set G3 = the \HM { of G , v } , G2 = the Vertex of G , e = the Element of G ;
reconsider g = f as PartFunc of REAL n , REAL-NS n , r be Real ;
|. s1 . m / p .| / |. p .| < d / p / p / p ;
for x being element st x in ( ( for u being element st u in ( ( x \ y ) \ { x } ) holds x in ( x \ y ) )
P = the carrier of ( TOP-REAL n ) | ( P ` ) .= ( TOP-REAL n ) | ( P ` ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , \square ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 + ( 2 * c * d ) ;
let f , g , h be Point of the carrier of X , Y be non empty set , g be Function of X , Y ;
set h = Hom ( a , g ) , g = Hom ( b , f ) ;
then idseq ( n ) | Seg m = idseq ( m ) ^ <* n *> & m <= n ;
H * ( g " * a ) in the right * ( g " * a ) ;
x in dom ( ( cos * sin ) `| Z ) & ( ( cos * sin ) `| Z ) . x = f . x ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 implies LE q2 , p , P , p1 , p2
attr B is an component of A means : Def8 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( ( p | n ) ^ ( p | n ) ) ;
attr a <> 0. K means : Def8 : the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom /\ I and I = len k + j ;
consider x1 such that z in x1 and x1 in ( P \ { x1 } ) and x2 in ( P \ { 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 ) , CP2 = P2 ;
set cv = 3 -tuples_on { a , b , c } , cw = 3 -tuples_on { a , b , c } ;
conv @ W c= union ( F .: ( E " W ) ) & conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * arccot ) & 1 in [. - 1 , 1 .] ;
r3 <= s0 + ( r0 / |. v2 - v1 .| - ( r / 2 ) ) / ( 2 * ( 1 + 1 ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gq = gq as Point of ( TOP-REAL n ) | P ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom being `1 & ( Frege Frege A ) . o = ( Frege Frege A ) . o ;
for I being non degenerated integral of I holds the carrier of I is commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* I ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( lpp . i ) = ( v *' lpp ) . i .= ( v *' lpp ) . i ;
consider n being element such that n in NAT and x = ( sn succ n ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and x in F1 . x ;
card Funcs ( X , 0 , x1 , x2 ) = { EC } & card Funcs ( X , 0 ) = k ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 2 ) ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n2 >= len crossover ( p2 , p1 , n2 ) ;
( for HT ( ( g2 ) , T ) st HT ( ( g2 ) , T ) <> 0. L holds HT ( ( g2 ) , T ) = HT ( ( g2 ) , T )
then H1 , H2 are_<* H1 , H2 *> & card H1 , card H2 are_<> 0 & H1 , H2 are_<* H2 , H1 *> ;
( ( N-min L~ ( f ) ) .. ( f ) ) .. ( ( f ) ) > 1 ;
]. s , 1 .] = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( ( TOP-REAL 2 ) | L~ g ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , f be PartFunc of REAL , the carrier of S ;
DigA ( t-23 , z ) is Element of k -tuples_on ( the carrier of k ) ;
I is d2\rm `1 & I is k2 & I is k2 -2NAT implies I is k2
[: [: { u } , { u } :] = { [ a , u9 ] } ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u2 in W3 ;
for y st y in rng F ex n st y = a |^ n & a |^ n in F
dom ( ( g * ( ( g * ( f , C ) ) | K ) ) | K ) = K ;
ex x being element st x in ( ( ( U0 ) \/ A ) \/ B ) . s ;
ex x being element st x in ( \HM { OO } \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x = |[ r , r ]| ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p10 } & L1 /\ LSeg ( p00 , p2 ) c= { p10 } ;
( b + bs ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the [: of ( the carrier of X ) , the carrier of Y :] ) . ( x , y ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q `2 = 0 ;
f | EK1 ` = g | EK1 ` .= g | EK1 ` .= g | EK1 ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 as Element of NAT ;
( a * A * B ) @ = ( a * ( A * B ) ) @ ;
assume ex n0 being Element of NAT st f to_power n0 is min ;
Seg len ( ( ( len ( f2 ) ) | ( len ( f2 ) ) ) ) = dom ( ( ( len ( f2 ) ) | ( len ( f2 ) ) ) ) ;
( Complement ( A ) ) . m c= ( ( Complement ( A ) ) . n ) . m ;
f1 . p = p9 & g1 . p = d & g1 . p = b & g2 . p = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| to_power n ) / ( ( r |^ n ) * ( n + 1 ) ) <= ( r2 |^ n ) / ( ( n + 1 ) * ( n + 1 ) ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( G ) c= 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 W3 ;
||. t-15 . x .|| = lim ( ||. xbeing .|| ) .= ||. ( ||. x .|| ) . x .|| ;
assume that i in dom D and f | A is lower bounded and g | A is lower bounded ;
( ( p `2 ) ^2 - 1 ) * ( ( q `2 ) ^2 ) <= ( ( - ( q `2 ) ) * ( ( q `2 ) ) ^2 ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) .= id ( Sphere ( p , r ) ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , j ) ;
pred a <> 0 means : Def8 : ( A \+\ B ) Let a = ( A \+\ a ) \+\ ( B be set ) ;
then f is_\mathbin { \frac 2 } 2 , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c <> 1 and b > 0 and c > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s-7 = p2 . IC s-7 .= ( p2 . IC s-7 ) + ( p2 . IC sU ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind B - ind ( T-10 | b ) ;
[ a , A ] in the \HM { of - ( 2 * ( A + B ) ) : not contradiction } ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( a , CompF ( PA , G ) ) . z ) = FALSE & ( a , CompF ( PA , G ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 22 , phi = phi /. 22 as Element of ( S , D ) * ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 * ( len s2 - 1 ) ;
delta ( D ) * ( f . ( upper_bound A ) - f . ( lower_bound A ) ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and z in { x } ;
[#] V1 = { 0. V1 } .= the carrier of (0). V1 .= { 0. V1 } .= { 0. V1 } ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p *> ^ <* p *> ^ <* q *> ^ <* p *> ^ <* q *> ^ <* p *> ^ <* q *> ^ <* p *> ^ <* q *> ^ ( len q ) *> ^ ( len q ) ;
c /. |[ b , c ]| = c .= c /. |[ a , c ]| .= |[ a , c ]| ;
reconsider t1 = p1 , t2 = p2 , t1 = p3 as Term of C , V ;
1 / 2 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 in the carrier of [. 1 / 2 , 1 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= C * ( p1 `2 ) + D ;
R . b - b + a = 2 * PI * b .= 2 * b - b .= b ;
consider \cdot ] such that B = ( - 1 ) * x + ( - 1 ) * A and 0 <= \vert 1 .| ;
dom g = dom ( ( the Sorts of A ) * a9 ) & dom ( ( the Sorts of A ) * b9 ) = dom a9 ;
[ P . ( l6 ) , P . ( l6 ) ] in => ( ( P => ( P => ( Q => ( P => ( P => ( Q => ( P => ( P => Q ) ) ) ) ) ) ) ) ;
set s2 = Initialize s , P2 = P +* I ;
reconsider M = mid ( z , i2 , i1 ) , N = len z - 1 , M = z - 1 as Element of REAL ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 = 0 & 1 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g & y in the left of g ;
consider M being strict Subspace of Ai2 such that a = M and T is Subspace of M and M is Subgroup of M ;
for x st x in Z holds ( ( ( #Z n ) + f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + 1 ;
reconsider h1 = ( vseq . n ) - ( t-16 . n ) as Lipschitzian LinearOperator of X , Y ;
( ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is_\emptyset and F in the { of s2 : s2 in the rng s2 } ;
( ( ( ( ( ( ( ( ( ( ( ( ( x , y ) , 3 ) ) , 3 ) ) , 1 ) ) , 1 ) ) mod ( n + 1 ) ) ) mod ( n + 1 ) = 1 ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m
for B being Subset of u-5 st B in E holds A = B or A misses B or B misses C
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W1 = p ^ W2 ;
x in { X where X is Ideal of L : X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W2 ;
( for a , b being Element of L holds a * ( a + b ) = ( a + b ) * ( a + b )
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( i - 2 ) |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) & ( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b1 . r = c1 . r ;
ex P st a1 on P & a2 on P & b on P & c on P & c on P & a , b // P ;
reconsider gf = g * f `1 , hg = h * g `2 , hf = h * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in V and v1 in W ;
n in { i where i is Nat : i < n0 + 1 + 1 & i < n0 + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn * |. p .| >= sn & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ( ConsecutiveSet ( A , O1 ) ) * ( A * O1 ) ) * ( A * O1 ) ;
set Ii1 = in dom let AddTo ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = SubFrom ( a , intloc 0 ) , Ii2 = goto 2 , Ii2 = goto 3 , Ii2 = goto 3 , Ii2 = goto 4 , I5 = goto 0 , I5 = goto 0 , I5 = goto 3
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & ( the carrier of L1 ) \/ ( the carrier of L2 ) c= the carrier of L1 ;
consider x-40 be Element of GF ( p ) such that x-40 |^ 2 = a and x-40 |^ 2 = b ;
reconsider e3 = e4 , f4 = f-5 , f5 = f-5 , f6 = f5 as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . n in U2 ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) . x ) . x0 ;
defpred P [ Nat ] means A + succ $1 = succ A + $1 & A in dom ( A + ) ;
the left \geq the left of - g & the left dom of g = the left of g implies g = f
reconsider pM = x , pM = y , pM = z , pM = w as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= g3 and g3 <= x0 and for x st x in dom g2 holds g2 . x = x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of set & y = ( the set of positive ) | ( n + 1 ) ;
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p11 ) = {} & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func thesis of thesis , X -> set equals [: [: the carrier of X , the carrier of Y :] , { id X } :] ;
len ( ( CR ) /. ( len ( Cg ) ) ) <= len ( Cg ) & len ( ( CR ) /. ( len ( Cg ) ) ) <= len ( Cg ) ;
attr K is with_a , b means : Def8 : a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t `2 . {} = [ o , the carrier of S ] and o in rng t ;
for x st x in X ex y st x c= y & y in X & y is \rm - 1 } implies f . x = f . y
IC Comput ( P-6 , sseq , k ) in dom ( PJ +* I ) & IC Comput ( PJ , sJ , k ) in dom I ;
pred q < s means : Def8 : r < s & ]. r , s .[ \not c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in { 3 , 4 , 5 } ;
func the ResultSort of S2 -> id the carrier' of S2 means : Def8 : for x being Element of the carrier' of S2 holds it . x = ( the ResultSort of S2 ) . x ;
set yxy = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & ri2 in cell ( GoB f , i , width GoB f ) \ L~ f ;
q `2 >= ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ) / ( ( Cage ( C , n ) /. ( i + 1 ) ) `2 ) ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + len f1 -' len f & len f + 1 <= len f -' len f + 1 ;
for n ex x st x in N & x in N1 & h . n = x- ( x0 - r )
set sx0 = ( \mathop { a , I , p , s ) . i , sx0 = ( \mathop { a , I , p , s ) . i , sx0 = ( \mathop { a , I , p , s ) . i , sx0 = ( \mathop { a , I , p , s ) . i , sx0 = ( \mathop { a , I , p , s )
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 & p ( k ) . 0 = 1 or p ( k ) . 0 = 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider xset being set such that x in xset and xset in Vset and x8 = [ x, x8 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - len p ) ;
g + h = gg + hh & A1 + h = g + h & A2 + h = g + h ;
L1 is distributive & L2 is distributive implies [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive & [: L1 , L2 :] is distributive
pred x in rng f & y in rng ( f | x ) implies f \leftarrow x = f \mathclose { y } ;
assume that 1 < p and 1 / p + 1 / q = 1 and 0 <= a and 0 <= b and a <= b ;
F* ( f , M ) = rpoly ( 1 , M ) *' + t *' .= ( 0. F_Complex ) *' + t *' ;
for X being set , A being Subset of X holds A ` = {} implies A = X & A = {} or A = {}
( ( N-min X ) `1 ) ^2 + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 1 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 <= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 1 1 )
for c being Element of the Sorts of A , a being Element of the free of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= 0 ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies b >= 0 & a >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = y ` \ x ;
mode BCK-algebra of i , j , m , n , m , n be BCK-algebra of i , j , m , n , m be Element of NAT ;
set x2 = |( Re ( y - Im y ) , Im ( y - Im y ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & upper_bound divset ( D , k ) = upper_bound A ;
0 <= delta ( S2 ) . n & |. delta ( S2 ) . n .| < ( e / 2 ) * ( ( n + 1 ) * ( n + 1 ) ) ;
( - ( q `1 / |. q .| - cn ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = 2 / b-a ;
for x , y being set st x in R" holds x , y are_\hbox { $ \subseteq $ } -let x , y
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in ( -> element ) iff s in ( -> Element of \rm means : Def8 : s in ( { f } \/ { g } ) ) ;
for S being non empty non void non void holds S is H -holds S is thesis iff S is connected
max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 & max ( degree ( z `1 ) , degree ( z `2 ) ) >= 0 ;
consider n1 be Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B /\ A ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . x qua Element of BOOLEAN ) , n-15 = M . ( n + 1 ) ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D & f .: V in D ;
rng ( ( a is :] ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being WWR being Wof G1 such that y `1 = y and dom y `1 = WWWWR ;
dom ( 1 / ( f . x0 ) ) /\ ]. -infty , x0 .[ c= ]. -infty , x0 .[ & ( 1 / ( f . x0 ) ) * ( f . x0 ) < 0 ;
as Morphism of i , j , n , r be Element of non zero Element of NAT ;
v ^ ( n-3 |-> 0 ) in Lin ( ( B | c1 ) ^ ( B | c2 ) ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i = c := k2 & i = c := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ ( i1 + 1 ) .= succ ( i1 + 1 ) .= ( i1 + 1 ) + 1 ;
assume that F is bbSubset-Family and rng p = F and dom p = Seg ( n + 1 ) and rng p c= Seg ( n + 1 ) ;
( not b , b9 // b , a & not LIN a , a9 , c & not LIN a , a9 , b & not LIN b , a9 , c ) & not LIN a , a9 , c
( L1 := ( L2 , O ) ) \& O c= ( L1 \& O ) => ( L2 \& O )
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) and for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( - w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) & Sum ( |. $1 .| ) <= Sum ( |. $1 .| ) ;
u = cos / sin . ( x , y ) * x + ( cos / cos . ( x , y ) * y ) .= cos . ( x , y ) * y ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ <* p *> , {} ] implies 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 ininininand X is inin;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= g } ;
vol ( ( G . n ) vol ) <= ( ( G . n ) vol ) * vol ( ( G . n ) vol ) ;
f . y = x .= x * ( 1_ L ) .= x * ( power L ) . ( y , 0 ) .= x * ( 1. L ) ;
NIC ( <% i1 , i2 , j2 %> , k ) = { i1 , succ ( i1 , i2 ) } .= { i1 , succ ( i1 , i2 ) } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } & LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
Product ( ( the Sorts of I-15 ) +* ( i , { 1 } ) ) in ( the Sorts of I-15 ) . ( i + 1 ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s1 , n ) ;
W-bound ( QA2 ) <= q1 `1 & q1 `1 <= E-bound ( QA2 ) & ( for i st i in dom ( QA2 ) holds ( ( Q ) . i ) `1 <= ( ( Q ) . i ) `1
f /. i2 <> f /. ( len f + 1 -' 1 ) & f /. ( len f + 1 -' 1 ) <> f /. ( len f + 1 -' 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) |= H / ( x. 4 , a ) ;
len ( ( P ^ ( P ^ ) ) | ( len P + 1 ) ) in dom ( ( P ^ ( P ^ ) ) | ( len P + 1 ) ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , l ) c= A |^ ( k , l ) implies A = B
REAL n \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X \not c= Z ;
CurInstr ( P3 , Comput ( P3 , 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 |. ( ( id the carrier of l1 ) | v ) .|
for phi holds phi in X implies ( phi in X & phi in X & phi in X ) & ( phi in X implies phi in X )
rng ( ( Sgm dom f-6 ) | ( dom f-9 ) ) c= dom ( ( Sgm dom fE ) | ( dom f-9 ) ) ;
ex c being FinSequence of D ( ) st len c = k & ( P [ c ] ) & ( P [ c ] implies P [ c ] ) ;
the_arity_of ( \in Hom ( a , b , c ) ) = <* Hom ( b , c ) , Hom ( a , b ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 or a1 = b3 & a3 = b3 & a2 = b2 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) .= D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. |[ r .|| , r ]| .|| /. 1 .= <* r *> . 1 .= <* r *> . 1 .= x ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . n = C-25 . m and C-25 . n = C-25 . n ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative associative means : Def8 : for b being Element of X holds F -Sum { b } = f . b ;
p = - ( - p0 + 0. TOP-REAL 2 ) + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= ( 1 - p ) * p0 + 0. TOP-REAL 2 .= ( 1 - p ) * p0 + 0. TOP-REAL 2 .= ( 1 - p ) * p0 + p * p0 ;
consider z1 such that b , x3 , x3 is_collinear and o , x1 , x1 is_collinear and o , x2 , x3 is_collinear and o , x1 , x2 is_collinear ;
consider i such that Arg ( Rotate ( s ) ) . q = s + Arg q + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and rng g = { x } ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} ;
attr F is associative means : Def8 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F .: ( g , h ) ) ;
ex x being Element of NAT st m = x & x `1 in z & x `2 < i or m in { i } & m in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in P-2 . k2 and ( P`1 . k2 ) `1 = ( P`1 . k2 ) `1 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) , ( id a ) ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & x in D2 } ;
consider z being element such that z in dom ( ( dom F ) | ( dom F ) ) and ( ( dom F ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 }
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , ' of n , ' ) ) . ( thesis , thesis ) .= ( Mx2Tran J ) . ( thesis , thesis ) ;
- 1 / ( n + 1 ) = mm (#) D | n .= mm (#) D .= ( - m ) (#) ( - m ) .= ( - m ) (#) ( - m ) .= ( - m ) (#) ( - m ) ;
pred for x being set st x in dom f /\ dom g holds g . x <= f . x & - ( f | A ) is nonnegative ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( All ( 'not' a , A , G ) , B , G ) '<' Ex ( 'not' All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k0 + 1 ) & LSeg ( E , k + 1 ) c= RightComp Cage ( C , k0 + 1 ) ;
x \ a |^ m = x \ ( ( a |^ k ) * a ) .= ( x \ ( a |^ k ) ) \ a ;
k -inth ininininininininC = ( commute ( I ) ) . k .= ( commute ( I ) ) . k .= ( ( commute I ) . k ) . i ;
for s being State of A^ ( s , n ) 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 ( thesis ) \/ support ( ( support ( s ) ) \ support ( m ) ) c= support ( ( support ( s ) ) \ support ( m ) ) \/ support ( ( support ( s ) ) \ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of S , ( the carrier' of B ) * the Arity of S ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( b . a ) = f . ( g . a ) & phi . ( a . a ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 } \/ { x2 }
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 "\/" U2 c= the Sorts of U2 & the Sorts of U1 c= the Sorts of U2 ;
( - ( 2 * a * ( b / 2 ) ) + b / 2 ) ^2 - delta ( a , b , c ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in [: N , N :] & P [ z ] and not ( ex z being element st z in N & z in N & z <> 0. N ) ;
assume ( the Arity of S ) . o = <* a *> & ( the ResultSort of S ) . o = r & ( the ResultSort of S ) . o = <* r *> ;
Z = dom ( ( exp_R (#) ( arccot + arccot ) ) `| Z ) & f1 + ( ( arccot (#) ( arccot + arccot ) ) `| Z ) = dom ( ( exp_R (#) ( arccot + arccot ) ) `| Z ) ;
integral ( f , SS1 ) is convergent & lim ( \HM { 0 } , SS2 ) = integral ( f , SS1 ) ;
( X ( ) ) => ( ( f => g ) => ( x_ ( X ( ) ) ) ) in is ) & ( X ( ) => ( x_ ( X ( ) ) ) in
len ( M2 * M3 ) = n & width ( M3 ~ * M2 ) = n & width ( M2 * M3 ) = n & width ( M2 * M3 ) = n ;
attr X1 union X2 is open SubSpace of X means : Def8 : X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X2 are_separated ;
for L being upper-bounded antisymmetric RelStr for X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) , f-129 = F2 . ( b `2 ) ;
consider w being FinSequence of I such that the InitS of M is_\HM { w } \HM { \frac s } { s } ^ w \HM { 1 } } and w in q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= 1_ H ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in rng f ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 ;
reconsider oY = o `1 , oY = o `2 , oY = o `1 , oY = o `2 `1 , oY = o `2 `1 , oY = o `2 `1 , oY = o `2 `1 , oY = o `2 `2 ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace ( 0 , \dots , 0 ) *> .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x3 ) .= x1 + ( 0 * x2 ) ;
EL " . 1 = ( EL qua Function ) " . 1 .= ( ( 1 - 2 ) * ( ( 1 - 2 ) * ( 1 - 2 ) ) ) " . 1 .= ( 1 - 2 ) * ( ( 1 - 2 ) * ( 1 - 2 ) ) " .= ( 1 - 2 ) * ( 1 - 2 ) ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , u2 = the carrier of U0 , u1 = the carrier of U0 as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . l1 ) .| < ( 1 / |. M .| + 1 ) ;
LSeg ( ( Lower_Seq ( C , n ) ) /. ( i + 1 ) , ( Lower_Seq ( C , n ) ) /. ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( - ( g . c ) * f . c ) + f . c <= h . c * ( ( - ( g . c ) ) * f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ( ColVec2Mx f ) in the set of K and ( ColVec2Mx f ) . ( len ( ColVec2Mx b ) ) = width A and ( ColVec2Mx f ) . ( len ( ColVec2Mx b ) ) = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( TOP-REAL n ) | ( the carrier of thesis ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1 implies pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 1
pred a <> 0 & b <> 0 & Arg a = Arg b & Arg ( - a ) = Arg ( - b ) & Arg ( - a ) = Arg ( - b ) ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the non empty set of set , the topology of T ) & not c in Intersection ( the topology of T , the topology of T )
assume that V1 is linearly-independent and V2 is linearly-independent and V2 = { v + u : v in V1 & u in V2 } and V1 is open and V2 is open and V1 is open and V2 is open and p in V1 ;
z * x1 + ( 1 - z ) * x2 in M & z * y1 + ( 1 - z ) * y2 in N implies z * y1 + ( 1 - z ) * y2 in M
rng ( ( PS1 qua Function ) " * SS2 ) = Seg ( card ( ( card ( dom ( PS2 ) " ) ) ) ) .= Seg ( card ( dom ( PS2 ) " ) ) ;
consider s2 being rational number such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b and s2 . n <= b ;
h2 " . n = h2 . n " & 0 < - 1 & 0 < - 1 & 1 < ( - 1 ) / ( n + 1 ) & ( - 1 ) < 1 ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= ||. ( seq1 . m ) - ( seq2 . m ) .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = ( - 1_ Gv ) * v & - w = ( - 1_ Gv ) * w & ( - w ) * v = ( - 1_ Gv ) * w & ( - w ) * v = ( - 1_ Gv ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= k . ( sup D ) .= sup ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , l ) ) = ( A |^ ( k , l ) ) ^^ ( A |^ ( k , l ) ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) gcd ( a * b ) = ( ( a * b ) gcd ( a * b ) )
consider A5 being countable ( Al ) such that r is Element of CQC-WFF ( Al ) & A5 is ( Al ) `1 & A5 is ( Al ) `1 & A5 is ( Al ) `1 & not contradiction ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st y in M holds x + y in x + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= [: { x1 , y1 } , { y2 } :] & { x1 , x2 } in [: { x1 } , { x2 } :] ;
h . ( f . O ) = |[ A * ( ( f . O ) `1 ) + B , C * ( ( f . O ) `2 ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) & ( Gauge ( C , n ) * ( k , i ) ) `2 = ( Gauge ( C , n ) * ( k , i ) ) `2 ;
cluster m , n are_relative_prime means : Def8 : for p being prime Nat holds it is ( m mod p ) & ( p divides n implies p divides n ) & ( p divides n & p divides n implies p divides n ) ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c & a \+\ c <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b and b in { x } ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 7 , G ;
( H H H H H H H H H H ) . ( 2 * n ) . x = ( H H H H ) . ( 2 * n ) . x + ( H . n ) . x ;
j + 1 = ( len h11 + 2 ) - 1 + 1 .= i + 1 - 1 + 2 .= i + 1 - 1 + 1 .= i + 1 - 1 + 1 .= i + 1 - 1 ;
*' ( S *' ) . f = S *' . ( opp f ) .= S . ( opp f ) .= S . ( opp f ) .= S . ( opp f ) .= S . ( opp f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L1 ) and Sum ( L1 ) = Sum ( L2 ) ;
attr R is b2 means : Def8 : p in R & p <> q implies ex P st P is special arc p & P c= R & P c= R & P c= R ;
dom product ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( ( X --> dom f ) . 0 ) .= meet ( ( X --> dom f ) . 0 ) .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Upper_Arc C ) ) <= upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) & upper_bound ( proj2 .: ( C /\ Vertical_Line w ) ) <= upper_bound ( proj2 .: ( C /\ [ w , e ] ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - pp .| < r
i * fN - fN = i * fN - ( i * ( - y ) ) .= i * ( fN - ( i * ( - y ) ) ) .= i * ( fN - ( i * ( - 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 ) and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in union C and g2 in C and g2 in D ;
func d |-count n -> Nat means : Def7 : d |^ n divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n & d divides n ;
f\in . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = M . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( n + 1 ) ;
( q `1 ) ^2 / ( |. q .| ) ^2 <= ( ( q `1 ) ^2 / ( |. q .| ) ^2 & ( q `2 ) ^2 / ( |. q .| ) ^2 <= ( ( q `1 ) ^2 / ( |. q .| ) ^2 ) ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) .= h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o in { the carrier' of S } ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & a >= b & b >= a
||. h1 .|| . n = ||. h1 . n .|| .= |. h .| . n .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n ;
( ( ( - 1 ) (#) ( #Z n ) ) `| Z ) . x = f . x - ( #Z n ) . x .= ( ( - 1 ) (#) ( #Z n ) ) . x .= ( ( - 1 ) (#) ( #Z n ) ) . x ;
pred r = F .: ( p , q ) means : Def8 : len r = min ( len p , len q ) ;
( rmin / 2 ) ^2 + ( rmax / 2 ) ^2 <= ( r ^2 + ( r ^2 ) ) ^2 + ( r ^2 + ( r ^2 ) ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( thesis ) * ( i , j ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p . ( j -' 1 ) * r3 ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 & ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 = L . ( h . n ) ;
assume that the carrier of H1 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H1 ) and the InternalRel of H1 = f .: ( the carrier of H2 ) and the InternalRel of H1 = f .: ( the carrier of H2 ) and the InternalRel of H1 = f .: ( the carrier of H2 ) ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 & H2 = |. 2 to_power ( n + 1 ) + h .| .= n + 1 & H1 = n + 1 & H2 = n + 1 implies H1 = H2
( O = 0 & O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 or O = 1 ) implies O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O = 1 or O
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 1 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( a - b ) / ( b - c ) = ( - ( e / d ) ) / ( b - b\bf ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= D ;
for i be set st i in dom g ex u , v be Element of L , a be Element of B st g /. i = u * a * v & u in A & v in B
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) * g `2 .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( ex i st i in dom s1 & i <> s1 & i <> s1 ) & ( not i in dom s1 & s1 . i <> s1 . s1 ) ;
h5 | ]. a , b .[ = ( g | Z ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected & [ s2 , t2 ] , [ t2 , t2 ] are_connected ;
then H is negative & H is not negative & H is not conjunctive & H is not negative implies H is not g-gOne & H is not non empty ;
attr f1 is total means : Def8 : 1 / ( f1 - f2 ) is total & ( f1 - f2 ) . c = f1 . c * f2 . c " & ( f1 - f2 ) . c = f1 . c * f2 . c " ;
z1 in W2 -Seg ( z2 ) or z1 = z2 & not z1 in W2 & not z1 in W2 & not z2 in W2 & not z1 in W2 & not z2 in W1 & not z2 in W2 & not z1 in W2 & not z2 in W1 & not z1 in W2 & not z2 in W1 & not z2 in W2 & not z1 in W1 & not z2 in W2 & not z1 in W2 & not z1 in W2
p = 1 * p .= a " * a * p .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) ;
for seq1 be Real_Sequence for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng seq1 <= upper_bound ( seq ) & upper_bound rng seq1 <= upper_bound ( seq )
x0 in C or C meets L~ go \/ L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C meets L~ pion1 or C c= L~ pion1 or C c= L~ pion1 \/ L~ pion1 \/ L~ pion1 ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( B .--> B ' +* ( C .--> D ' ) +* ( E .--> F ) +* ( F .--> J ) +* ( J .--> M ' ) ) +* ( A .--> N ) +* ( A .--> N ) +* ( A .--> M ' ) +* ( A .--> N ) ;
|. ( ( ( ex H be || A ) || A ) . k - ( ( ( ( ( ( k - 1 ) * ( k - 1 ) ) ) | A ) . n ) ) .| <= e * ( ( ( k - 1 ) * ( k - 1 ) ) ) ;
( (
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } \/ { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 }
assume that A = [. 0 , 2 * PI .] and integral ( ( #Z n ) * sin , A ) = 0 and integral ( ( #Z n ) * sin , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 " = ( ( Sgm Y ) " ) * ( p " ) & p `2 = ( ( Sgm Y ) " ) * ( p " ) ;
for x , y st x in A & y in A holds |. 1 / ( f . x - 1 ) .| <= 1 * |. f . x - 1 .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) - sn * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 ) .= |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is_continuous_on dom f & f is_continuous_on dom f & f is_continuous_on dom f holds rng f is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 and for k be Nat st k in n1 holds Q [ k , F3 . k ] and Q [ k , F3 . k ] ;
ex u , u1 st u <> u1 & u , u1 / ( u , v1 ) / ( v , v1 ) / ( u , u1 ) / ( v , v1 ) / ( u , u1 ) = ( u , u1 ) / ( v , v1 ) / ( v , v1 ) ;
for G being Group , A , B being non empty Subset of G , N being normal Subgroup of G holds ( N ` A ) * ( N ` B ) = N ` A * N
for s be Real st s in dom F holds F . s = integral ( R to_power 0 ) - integral ( ( R to_power k ) (#) ( e to_power k ) , ( f to_power k ) (#) ( e to_power k ) )
width AutMt ( f1 , b1 , b2 ) = len b2 .= width ( ( a * b1 ) * ( b1 * b2 ) ) .= width ( ( a * b1 ) * ( b1 * b2 ) ) .= width ( ( a * b1 ) * ( b1 * b2 ) ) ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f " ]. - PI / 2 , PI / 2 .[ = ]. - 1 , 1 .[ implies f | ]. PI / 2 , PI / 2 .[ is continuous
assume that X is closed w.r.t. being set and a in X and a c= X and y in { { [ n , x ] } \/ { y : x in X } \/ { x } in a } ;
Z = dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) /\ dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) `| Z ) & Z = dom ( ( #Z 2 ) * ( arctan + arccot ) ) ;
func <* V ( ) *> -> Subset of V means : Def1 : 1 <= k & k <= len l & l . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of N , c being Point of L st c is Point of N & M is net of N holds c is inf of M
for s being Element of NAT holds ( ( ( ( ( ( ( ( ( ( ( C\mathop ( C\mathop ( C\mathop ( C\mathop ( Ct ) ) ) ) ) + 1 ) ) | ( C\mathop { 0 } ) ) ) ) | ( C\mathop { 0 } ) ) . s = ( ( ( ( ( C\mathop { 0 } ) ) | ( C\mathop { 0 } ) ) ) | ( C\mathop { 0 } ) ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( ( N-min L~ z ) .. z ) .. z < ( ( N-min L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) *> .= len p + 1 .= 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 f . x > 0 ;
for R being right_zeroed right_zeroed right right_zeroed right_complementable non empty doubleLoopStr , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B12 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f is Function of B1 , B2 ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 + z2 ) .= dom ( x (#) ( y | ( i -' 1 ) ) ) .= dom ( x (#) ( y | ( i -' 1 ) ) ) .= Seg ( len ( x | ( i -' 1 ) ) ) ;
for S being Functor of C , B for c being Object of C holds S . ( id c ) = id ( ( Obj S ) . c ) & S . ( id c ) = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & for a st a in f6 /\ f6 holds holds holds holds \cal ( f6 , a ) = or { a , a } in \rm \rm ^2
a in Free ( H2 / ( x. 4 , x. 0 ) ) '&' H2 / ( x. 4 , x. 0 ) & a in Free H2 & a in Free H2 implies ( ( ( x. 4 , x. 0 ) '&' ( x. 4 , x. 0 ) ) '&' ( ( x. 4 , x. 0 ) '&' ( x. 0 , x. 0 ) ) ) ;
for C1 , C2 being v1 , C2 being Function of C1 , C2 st `1 = ( i , len g ) `1 & ( for f being Function of C1 , C2 st f in C2 holds f = g ) holds f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ go \/ ( L~ pion1 \/ L~ pion1 ) .= W-bound L~ pion1 \/ ( L~ pion1 \/ L~ co ) .= W-bound L~ pion1 \/ ( L~ pion1 \/ L~ co ) .= W-bound L~ pion1 \/ ( L~ pion1 \/ L~ co ) .= W-bound L~ pion1 \/ ( L~ pion1 \/ L~ co ) ;
assume that u = <* x0 , y0 , z0 *> and f is_is_is_is_is_is_is_is_or SVF1 for u , 3 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . z0 = z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( x in Vars & x in Vars ) & ( x in Vars & x in Vars & x = [ x , s ] ) ;
Valid ( p '&' q , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b >= 0 ;
func Class R -> Subset-Family of R means : Def6 : for A being Subset of R holds A in it iff ex a being Element of R st A = Class a & a in Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( G ) ) ) ) ) ) ) ) ) ) . $1 ) ) ) . $1 ) ) ) ) . $1 ) `1 c= G
assume that dim W1 = 0 and dim W1 = 0 and dim W2 = 0 and ( for i st i in dom W1 holds W1 . i = 0. ( U1 ) ) and ( for i st i in dom W1 holds W1 . i = 0. ( U1 ) ) & ( i in dom W1 implies W1 . i = 0. ( U1 ) ) ;
mamas ( m ) . t = ( m . t ) `1 .= ( [ m . t , the carrier of C ] ) `1 .= [ m . t , the carrier of C ] `1 .= m . t ;
d11 = x11 ^ d22 .= f . ( y22 , d22 ) .= f . ( y22 , d22 ) .= ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) .= ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) .= ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) ;
consider g such that x = g and dom g = dom fx0 and for x being element st x in dom fx0 holds g . x in fx0 and g . x in fx0 ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= ( x + len x ) |-> 0. F_Complex .= x *' + ( x *' ) .= x *' ;
( k -' ( k + 1 ) ) + 1 in dom ( f | ( k -' ( k + 1 ) ) | ( k -' ( k + 1 ) ) ) & ( f | ( k + 1 ) ) /. ( k + 1 ) = ( f | ( k + 1 ) ) . ( k + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P1 \/ P2 and P1 = P1 \/ P2 and P2 = P2 \/ P2 and P1 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 \/ P2 and P2 = P1 \/ P2 and P2 = P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2 \/ P2 and P2 = P2
reconsider a1 = a , b1 = b , c1 = c , c1 = p , c1 = p , c2 = p , c2 = q , c1 = r , c2 = s , c2 = r , c1 = s , c2 = r , c2 = s , c1 = s , c2 = r , c2 = s , c1 = s , c2 = r , c2 = s , c2 = s , c1 = s , c2 = r , c2 = s , c1 = s , c2 = s , c1 = s , c2 = r , c2 = s , c2 = s , c2 = s , c1 = s , c1 = s , c2 = r , c2 = r , c2 = r , c1 = r = r , c2 = s , c1 = s , c2 = s ,
reconsider set set _ ttb1f = G1 . ( t * b ) , F1 = G1 . ( t * b ) , F2 = G2 . ( t * b ) , F2 = G2 . ( t * b ) , F2 = G2 . ( t * b ) , F2 = G2 . ( t * b ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + i1 -' 1 ) ) .= LSeg ( f , i + i1 -' 1 ) ;
Integral ( M , P . m ) | dom ( P . n -| ( dom P . m ) ) <= Integral ( M , P . n ) | dom ( P . m -| ( dom P . m ) ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ y , x ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( - G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) ) `1 ) ;
for G being Group , H being Subgroup of G , a being Element of G st a = b holds for i being Integer st i in H holds a |^ i = b |^ i & a |^ i = b |^ i
consider B be Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } , K1 = { p : p `2 >= 0 & p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p <> 0. TOP-REAL 2 } , K1 = { p : p
( ( N-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-bound C ) / 2 ) <= ( ( N-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-bound C ) / 2 ) * ( ( S-bound C - S-bound C ) / 2 ) ;
for x be Element of X , n be Nat st x in E holds |. Re ( F . n ) .| . x <= P . x & |. Im ( F . n ) .| . x <= P . x
len @ ( @ ( @ p ^ @ q ) ) = len ( @ p ^ @ q ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ @ q ) + len ( @ p ^ @ q ) .= len ( @ p ^ @ q ) + len ( @ p ^ @ q ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) = m3 / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) ;
consider r be Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , n ) / ( x. 0 , m ) / ( x. 4 , n ) / ( x. 4 , n ) |= H2 iff ( ( ( x. 4 , n ) / ( x. 4 , n ) ) / ( x. 0 , m ) ) / ( x. 4 , n ) / ( x. 4 , n ) ) / ( x. 4 , n ) |= r ;
func w1 \ w2 -> Element of Union ( G , R6 ) means : DefREAL : for w1 , w2 being Element of Union ( G , R6 ) holds it . ( w1 , w2 ) = ( ( ( ( ( ( ( ( ( ( G . i ) \ R ) | ( i -' 1 ) ) | ( i -' 1 ) ) ) | ( i -' 1 ) ) ) . w1 ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= s . b2 .= s . b2 .= s . b2 ;
for n , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . ( n + k ) - ( Partial_Sums ( |. seq .| ) ) . n + ( Partial_Sums ( |. seq .| ) ) . ( n + k )
set F = S \! \mathop { {} } ;
( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . ( K + 1 ) + Partial_Sums ( seq ) . ( K + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- ( x - x0 ) ) + R . ( x- ( x - x0 ) ) ;
func the closed \HM { closed L of \HM { a , b , c , d } -> Subset of \HM { the } \HM { closed } , P , Q : P [ a , b , c ] } , R = the closed } ;
a * b ^2 + ( a * c ^2 + b * c ^2 ) + ( b * c ^2 + c * a ^2 ) >= 6 * a * a * b * c + ( c * a ^2 + c * a ^2 ) ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x1 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) ;
N = ( Q ^ <* x *> , M ) +* ( ( Q ^ <* x *> , M ^ <* y *> ) +* ( ( ( Q ^ <* x *> ^ <* y *> ) ^ <* TRUE *> ) ) ) .= ( Q ^ <* x *> ^ <* y *> ^ <* TRUE *> ) ;
Sum ( F ) = r |^ ( n1 + 1 ) * Sum ( Cv ) .= C ( n1 ) * ( C . n1 ) .= Cv . n1 * ( C . n1 ) .= ( C ^\ n1 ) . n1 * ( C ^\ n1 ) .= ( C ^\ n1 ) . n1 * ( C ^\ n1 ) . n1 ;
( GoB f ) * ( len GoB f , 2 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * ( $1 + 1 ) ) * ( ( $1 + 1 ) * ( $1 + 1 ) ) + b * ( ( $1 + 1 ) * ( $1 + 1 ) ) ;
the_arity_of g = ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g .= ( the Arity of S ) . g ;
( X , Y ) |^ Z tolerates X |^ Z & card ( ( X , Y ) |^ Z ) = card ( X |^ Z ) & card ( ( X , Y ) |^ 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 + 1 ) holds b = N . ( s . ( n + 1 ) \ G . s ) ;
E , f |= All ( x. 2 , ( x. 2 ) .--> ( x. 0 , ( x. 2 ) .--> ( x. 1 , ( x. 2 ) --> ( x. 2 , x. 1 ) ) ) '&' ( x. 2 , ( x. 2 ) --> ( x. 0 , ( x. 2 ) --> ( x. 1 , ( x. 2 ) --> ( x. 2 , x. 1 ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n-11 ) ) . i & ( ( p | ( n-11 ) ) . i = ( p | ( n-11 ) ) . i ) & ( p | ( n-11 ) ) . i = ( p | ( n-11 ) ) . i ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the non empty set & ( the partial of L~ f ) . k is Element of the non empty set & ( the partial of L~ f ) . k is Element of the carrier of G & ( the non empty set of f ) . k is Element of the carrier of G ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := a2 , Comput ( P , s , 2 ) ) ;
card h1 = power ( F_Complex ) . k .= ( power ( F_Complex ) ) . ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) * Sum ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) * Sum ( ( - 1_ F_Complex ) * u ) .= ( ( - 1_ F_Complex ) * u ) * ( ( - 1_ F_Complex ) * u ) ;
( f / g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( ( 1 - g ) * ( 1 - g ) ) ) .= ( f / g ) /. c ;
len Cv - len ( ( the { of ( ( the carrier of ( ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of C ) ) ) ) ) ) ) ) ) ) ) ) ) = len Cv - len ( ( the { the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of ( the carrier of
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f `1 and f is onto and ( n < k implies f " { f . n } = { n } ) & ( n < k implies f " { n } = { n } ) ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( for A st A in S holds E . A = Prob . A ) & ( for A st A in S holds E . A = Prob . A ) & ( for A st A in S holds E . A = Prob . A ) & E . A = Prob . A ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] ;
assume that A c= Z and f = ( ( #Z 2 ) * ( sin + cos ) ) / ( sin + cos ) / ( sin + cos ) and Z c= dom f and f | A is continuous ;
( f /. i ) `2 = ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 .= ( ( GoB f ) * ( 1 , j2 + 1 ) ) `2 ;
dom Shift ( Seq q2 , len Seq q1 ) = { j + len Seq q1 where j is Nat : j in dom Seq q1 } .= { len Seq q1 where j is Nat : len Seq q1 + 1 <= len Seq q1 } .= { len Seq q1 where q1 is FinSequence of NAT : len Seq q1 = len Seq q1 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G2 , G3 and g is Morphism of G2 , G3 and f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 ;
func - f -> PartFunc of C , V means : Def5 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a for v st v in ( union L ) | [. v , v .] holds L . a |= ( union L ) | [. v , v .] ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for i1 , i2 being Nat st n1 <> 0 & i2 <> 0 & n < len p & n < len p holds sqrt ( n + 1 ) <= ( i - n ) / ( n + 1 ) ;
assume that not 0 in Z and Z c= dom ( ( arccot * ( 1 / 2 ) ) (#) ( ( arccot * ( 1 / 2 ) ) (#) ( ( #Z 2 ) * ( 1 / 2 ) ) ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( 1 / 2 ) ) (#) ( ( #Z 2 ) * ( 1 / 2 ) ) `| Z ) . x = - 1 ;
cell ( G1 , i1 -' 1 , 2 |^ ( m -' 1 ) ) \ ( ( Y -' 1 ) * ( ( Y -' 1 ) , ( Y -' 1 ) + ( Y -' 1 ) ) \ ( ( Y -' 1 ) * ( ( Y -' 1 ) + ( Y -' 1 ) ) ) c= BDD L~ f1 ;
ex Q1 being open Subset of [: X , Y :] st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & F8 is finite & ( for x being Point of Y , Q1 being Subset of Y st x in F1 & Q = F2 holds ( x in Q1 & Q = union F8 ) & ( x in Q1 & Q = union Q1 ) & ( x in Q1 ) & ( x in Q1 ) & ( x in Q1 implies x in Q1 ) & ( x in Q1 implies x in Q1 ) & ( x in Q1 implies x in Q1 implies x in Q1 ) & ( x in Q1 ) & ( x in Q1 implies x in Q1 ) & ( x in Q1 implies ex y in Q1 ) & ( y in Q1 ) & ( y in Q1 implies ex Q1 being Subset of [: Q1 in
gcd ( A-27 , ( ( the carrier of R ) , s1 , s2 ) , ( the InternalRel of R ) , s2 , s1 ) = 1 & gcd ( ( the InternalRel of R ) , ( ( the InternalRel of R ) , s1 , s2 ) , s2 ) = 1 ;
R8 = ( ( j , ( ( j + 1 ) ) ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( j + 1 ) + ( ( j + 1 ) + 1 ) ) . ( m2 + 1 ) .= [ 3 , ( ( j + 1 ) + 1 ) . ( m2 + 1 ) ] .= [ 3 , ( ( j + 1 ) + 1 ) . ( m2 + 1 ) ] ;
CurInstr ( P-6 , Comput ( Pmeans , m1 + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt ( SCMPDS , s3 ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) ) /\ ( LSeg ( p1 , p2 ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) .= { p1 } \/ ( { p2 } \/ { p2 } ) /\ ( LSeg ( p1 , p2 ) \/ { p2 } ) ;
func not the still of f -> Subset of the Sorts of Al means : Def5 : a in it iff ex i , p st i in dom f & p = f . i & a in dom f & p in dom f & for i st i in dom f & i <> p holds it . i = f . i ;
for a , b being Element of F_Complex for f being Polynomial of F_Complex st |. a .| > 1 & f is non zero holds f is non or a * ( - b ) is non or f is non or f is non ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 + 1 ) & 1 <= j & j + 1 <= width G & j + 1 <= width G holds LSeg ( G * ( i , j ) , G * ( i , j ) ) = LSeg ( G * ( i , j ) , G * ( i , j ) ) ;
assume that C1 , C2 are_`2 and for f being State of C1 , g being State of C2 , s1 , s2 being State of C1 st s1 = s2 holds s1 is stable iff for f being State of C1 , s2 being State of C2 st s2 = s2 holds s1 is stable & s2 is stable & s1 is stable & s2 is stable & s1 is stable & s2 is stable ;
( ||. f .|| | X ) . c = ||. f .|| /. c .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of [: T , T :] st F is open & not {} in F & for A , B being Subset of [: T , T :] st A in F & B in F & A <> B & B <> {} & B <> {} & B <> {} & B <> {} & B <> {} holds card F = card B & card F = card B & card F = card B & card F = card A & card F = card B = card B
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . ( F . k , G . k ) and for k st k in dom F & k <> 1 & k <> 1 holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( \mathop { \rm mod n ) |^ n ) - i |^ s = i |^ ( ( s + k ) - i |^ s ) .= i |^ ( ( s + k ) - i |^ s ) .= i |^ ( ( s + k ) - i |^ s ) .= i |^ ( ( s + k ) - i |^ s ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( F . ( q . 1 ) = v1 ) & ( for i st i in dom F holds F . ( q . len q ) = v2 ) & ( for i st i in dom q holds q . i = ( p . i ) `1 ) & ( for i st i in dom p holds p . i = ( p . i ) `1 ) ;
defpred P [ Element of NAT ] means $1 <= len I implies ( ( g = g , Z ^ I ) . ( len I ) ) . ( $1 + 1 ) = ( ( ( g , Z ^ I ) . ( len I ) ) . ( len I + 1 ) ) . ( len I + 1 ) ;
for A , B being square Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a ;
func |( x , y )| -> Element of COMPLEX equals |( Re ( x , y ) , ( Re ( x ) ) * ( ( Re ( x ) ) * ( ( Re ( x ) ) * ( ( Re ( x ) ) * ( ( Re ( x ) ) * ( ( Im x ) * ( ( Im x ) * ( ( Im x ) * ( ( Im x ) * ( Im x ) ) ) ) ) ) ;
consider g0 be FinSequence of FH such that h0 is continuous and rng h0 c= A & for i be Nat st i in dom h0 & i in dom g2 & g2 . 1 = x1 & g2 . i = x2 holds g2 . ( len g2 ) = x1 & g2 . ( len g2 ) = x2 & len g2 = len g2 ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , p1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 ,
q `1 * a <= q `1 & - q `1 * a <= q `1 & - q `1 * a <= q `1 * a or q `1 * a >= q `1 & q `1 * a >= q `1 & q `1 * a >= q `1 & q `1 * a >= q `1 * a & q `1 * a >= q `1 * a & q `1 * a >= q `1 * a & q `1 * a >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 >= q `1 * a & q `1 >= q `1 >= q `1 >= q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q `1 >= q `1 * a & q
Fv . ( len pp ) = Fv . ( p . ( len pp ) ) .= vv . ( len p + 1 ) .= vv . ( len p + 1 ) .= vv . ( len p + 1 ) .= vv . ( len p + 1 ) .= vv . ( len p + 1 ) .= vv . ( len p + 1 ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( k1 --> SubFrom ( a , intloc 0 ) ) ) ^ <* a := intloc 0 *> ^ ( k1 --> SubFrom ( a , intloc 0 ) ) ^ <* a := intloc 0 *> ^ <* a *> ^ <* a *> ^ ( k1 --> intloc 0 ) *> ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , B2 such that B8 is finite and D8 = the carrier of ( A1 \/ B1 ) and D8 = the InternalRel of ( A1 \/ B1 ) and D8 = the InternalRel of ( A1 \/ B1 ) and B8 = the InternalRel of ( A1 \/ B1 ) and B8 = the InternalRel of ( A1 \/ B1 ) ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry Map ( F2 , g ) ) . b2 ) . ( ( ( curry Map ( F2 , g ) ) . b2 ) ) .= ( ( curry Map ( F2 , g ) ) . b2 ) . ( ( ( curry Map ( F2 , g ) ) . b2 ) . b2 ) .= ( ( curry Map ( F2 , g ) ) . b2 ) . b2 .= ( ( curry Map ( F2 , g ) ) . b2 .= ( ( ( F2 , g ) * ( ( ( F2 id B ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( F2 id B ) . b2 ) . b2 ) . b2 .= ( ( ( F2 , g ) * ( ( ( ( id B ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2
dom IExec ( I-35 , P , Initialize s ) = the carrier of SCMPDS .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) .= dom IExec ( I , P , Initialize s ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 & h <> 0 holds |. h .| " * ||. ( R2 + R1 ) /. h .|| < e / ( ( R2 + R1 ) /. h ) * ||. ( R2 + R1 ) /. h .|| ;
LSeg ( G * ( len G , 1 ) + |[ 1 , 0 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 0 ) \/ { G * ( len G , 1 ) }
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) .= LSeg ( h /. ( i + i1 -' 1 ) , h /. ( i + i1 -' 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 & LE p1 , p2 , P , p1 , p2 & LE p2 , p1 , P , p1 , p2 } ;
( ( - x ) .|. y ) = ( - 1 ) * ( ( - 1 ) * ( x .|. y ) ) .= ( - 1 ) * ( ( - 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 `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) .= ( p `2 ) ^2 / sqrt ( 1 + ( p `1 / p `2 ) ^2 ) ;
( ( U + W ) * ( W7 ) ) * ( ( W7 ) * ( p ) ) = ( ( ( U + W ) * ( p ) ) * ( ( p ) ) ) * ( ( p ) ) .= ( ( ( U + W ) * ( p ) ) * ( p ) ) * ( ( p ) ) .= ( ( U + W ) * ( p ) ) * ( p ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def8 : dom it = [: { x } , dom h :] & for x st x in [: { x } , dom h :] holds it . x = ( f + h ) . x + ( h + h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and x in Free H and ( not x in Free H ) and ( x in Free H ) \/ { y } and ( x in Free H ) \/ { y } and ( x in Free H ) \/ { y } and ( x in Free H ) \/ { y } ;
defpred P11 [ Element of NAT , Element of NAT ] means ( P [ $1 , $2 ] ) & ( for k being Element of NAT st k in dom p holds $2 = p |^ k ) & ( $1 = p |^ k implies $2 = p |^ k ) & ( not $1 = p |^ k implies $2 = p |^ ( k -' 1 ) ) & ( not $1 = p |^ ( k -' 1 ) ) & ( not $1 = p |^ ( k -' 1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def8 : for A being Subset of X holds A in it iff for W being Subset of X st W c= A & W c= A & W c= A & W c= A holds C . W <= C . ( W \/ A ) ;
[#] ( ( dist ( P ) ) .: Q ) = ( dist ( P ) ) .: Q & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( Q ) ) .: Q ) ;
rng ( F | ( S S ) ) = {} or rng ( F | ( S S ) ) = { 1 } or rng ( F | ( S S ) ) = { 1 } or rng ( F | ( S S ) ) = { 2 } or rng ( F | ( S S ) ) = { 1 } ;
( f " ( rng f ) ) . i = f . i " ( ( rng f ) . i ) .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P2 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p2 , p1 } and P2 = { p1 , p2 } ;
f . p2 = |[ ( p2 `1 ) ^2 + sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 / ( p2 `2 ) ^2 + ( p2 `2 ) ^2 / ( p2 `1 ) ^2 ) ]| ;
( ( AffineMap ( a , X ) ) " ) . x = ( ( ( AffineMap ( a , X ) qua Function ) " ) . x ) " .= ( ( ( AffineMap ( a , X ) ) " ) . x ) " .= ( ( ( TOP-REAL n ) | ( a , X ) ) . x ) " .= ( ( TOP-REAL n ) | ( a , X ) ) . x .= ( ( TOP-REAL n ) | a ) . x .= ( ( ( TOP-REAL n ) | a ) . x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A misses B for p being Point of T , r being Element of ( dom G ) , r being Element of ( dom G ) , p being Point of ( dom G ) , r being Element of ( dom G ) , r being Element of ( dom G ) st r in dom ( ( Q ) . r ) & p in ( ( Q ) . r ) & r in ( ( ( dom ( Q ) ) . r ex s being Point of ( dom ( ( Q ) . s ) . s & p in ( ( dom ( ( Q ) . s & ( ( ( ( Q ) ) . s & ( ( Q ) ) . s & ( ( ( ( ( ( Q ) ) . s ) . s ) . s ) . s ) . s ) . s in ( ( ( len G ) . s being Point of ( ( ( dom G ) . s ) . s st s in ( ( dom G ) . s ) . s & ( for r being Point of ( ( ( dom G ) . s
for i , j st i + 1 in dom F for G1 , G2 being strict Subgroup of G st G1 = F . i & G2 = F . ( i + 1 ) & G2 is strict Subgroup of G1 & G1 is strict Subgroup of G2 & G2 is Subgroup of G1 & G2 is Subgroup of G1 & G2 is Subgroup of G2 holds G1 is strict Subgroup of G2
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x = ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x / ( ( ( #Z 2 ) * ( arctan - arccot ) ) . x )
synonym f is_right continuous means : Def8 : x0 in dom ( f /* a ) & for x0 st x0 in dom f & x0 in dom f & x0 in dom f & x0 in dom f & f . x0 = lim ( f , x0 ) holds ( f /* a ) . x0 = lim ( ( f /* a ) /* a ) ;
then X1 , X2 are_separated & ( X1 union X2 ) is SubSpace of X & ( X1 meet X2 ) is SubSpace of X1 & ( X1 union X2 ) is SubSpace of X2 & ( X1 union X2 ) is SubSpace of X1 & ( X1 union X2 ) is SubSpace of X2 & ( X1 union X2 ) is SubSpace of X2 implies X1 union X2 is SubSpace of X2 & X2 union X1 is SubSpace of X1 & X1 union X2 is SubSpace of X2 ;
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 ) . x ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `1 / p3 `2 ) ^2 ) ) ^2 * ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) ;
( ( 1 / t1 ) (#) ( ||. f1 .|| ) ) . x = ( ( 1 / t1 ) (#) ( ||. g1 .|| ) ) . x & ( ( 1 / t ) (#) ( ||. g1 .|| ) ) . x = ( ( 1 / t ) (#) ( ||. g1 .|| ) ) . x ;
assume that for x holds f . x = ( ( sin + cos ) (#) ( cos + sin ) ) . x and x in dom ( ( sin + cos ) (#) ( sin + cos ) ) and for x st x in dom ( ( sin + cos ) (#) ( sin + cos ) ) holds ( ( sin + cos ) (#) ( sin + cos ) ) . x = ( ( sin + cos ) (#) ( sin + cos ) ) . x ;
consider X-23 be Subset of Y , X-22 be Subset of X such that X-22 is open and X-22 is open and for Y1 being Subset of Y st Y1 = X-23 & 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 where d is Element of GF ( p ) : [ d , Y ] in Indices GF ( p ) & [ d , Y ] in Indices GF ( p ) & [ d , Y ] in Indices GF ( p ) } .= { 1 , 2 , 3 } ;
( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / ( m + 2 ) ) = ( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / ( m + 2 ) ) .= ( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / ( m + 2 ) ) .= ( W-bound D - W-bound D ) / ( m + 2 ) ;