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 ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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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 `2 is convergent
q in X ;
V ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCI-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , W be Walk of G ;
let G be _Graph , W be Walk of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom h ;
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 = 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 Point of X ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , i be Integer ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 ;
cluster uparrow x -> closed ;
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~ implies n >= 0. s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , x be set ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
p\pi c= PI ;
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 \not _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 + 1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
.= \emptyset ( C ) ;
x9 is increasing & x9 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
( G is non-decreasing ) ;
( G is non-decreasing ) ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is independent & finite S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT * ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , A be non-empty ManySortedSet of I ;
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 is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
an < n & a < b ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m be element ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g is_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 & P3 halts_on s ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be non empty Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable of f , A ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn ) ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> | | C ;
let R be non empty multMagma , x be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p `2 ;
assume f | X is lower ;
x in rng co & y in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .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 void maid ;
let N be non empty K be non empty K of M ;
let R be RelStr with finite finite finite finite finite for n being Nat ;
let n , k be Nat ;
let P , Q be be be be be be be be be be be be be be let RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I is not \leq lim [ a , b ] ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` & y in F ;
redefine func S --> T -> / S ;
assume that t1 <= t2 and t2 <= t1 ;
let i , j be even Integer ;
assume that F1 <> F2 and F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : x <> A2 & y <> A1 ;
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 | Z ) ;
assume [ x , y ] in R ;
set d = ( x - y ) / ( x - y ) ;
1 <= len g1 & g1 <= g2 ;
len s2 > 1 & len s2 > 0 ;
z in dom ( f1 (#) f2 ) ;
1 in dom ( D2 | 1 ) ;
( p `2 ) ^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 & X1 c= dom f ;
h . x in h . a ;
let G be in thesis of on ( on ) ;
cluster m * n -> invertible ;
let k9 be Nat , x be Element of X ;
i - 1 > m - 1 ;
R is transitive & R is transitive ;
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is thesis ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 (#) f2 ) ;
assume [ X , p ] in C ;
BX c= [ p3 , p4 ] ;
n2 <= ( 2 * n ) - 1 ;
A /\ ( cP ) c= A `
cluster x -valued for Function ;
let Q be Subset-Family of S , P be Subset of S ;
assume n in dom g2 & m in dom g2 ;
let a be Element of R ;
t `2 in dom e2 & t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , x be Element of S ;
i . y in rng i ;
REAL c= dom f & REAL c= dom f ;
f . x in rng f ;
mt <= ( r / 2 ) * 2 ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= NN . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [' S , T '] ;
let x be non positive ExtReal ;
let m be Element of M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , x be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom g c= dom y ;
n1 < n1 + 1 & n2 + 1 < n1 + 1 ;
n1 < n1 + 1 & n2 + 1 < n1 + 1 ;
cluster [: T , T :] -> \cdot ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 ) ;
b = upper_bound ( dom f ) & c = upper_bound ( dom f ) ;
x in Seg ( len q ) ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 & m in dom h2 ;
w + 1 = ( a - 1 ) + 1 ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 & k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support p ;
assume X is complete implies X is complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= n1 + 1 ;
let x be Element of REAL , y be Element of REAL ;
assume x in rng ( s2 - s1 ) ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( L (#) F ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & dom q = Seg n ;
j <= width ( M @ ) ;
let seq1 be real-valued subsequence of seq ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in ( a := . 0 ) ;
let i be set ;
n -' 1 = n-1 - 1 ;
len ( n-27 ) = n ;
\mathop { \rm \cal Z } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , x be Element of A ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg i ;
let s be Element of E -tuples_on omega ;
let B1 be Basis of x , B2 be Basis of y ;
L3 /\ L2 = {} implies L1 /\ L2 = {}
L1 /\ LSeg ( L2 , p2 ) = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c `2 ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let DD be non empty set , f be FinSequence of D ;
let K be right_zeroed non empty addLoopStr , M be Matrix of K ;
assume f `2 = f & h `2 = h ;
R1 - R2 is total & R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume that a , b are_maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster n[: -> ns[: for ;
not u in { ag } ;
the carrier of f c= B \/ A ;
reconsider z = x as VECTOR of V ;
cluster the
r (#) H is as as as as as as as as as as as as <> of X ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict universal MSAlgebra over S , x be Element of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in : ( y in : y in { x } ) ;
let x , y be Element of X ;
let A , I be contradiction of X ;
[ y , z ] in [: O1 , O2 :] ;
( not ( JumpPart i ) . 1 = 1 ) ;
rng Sgm ( A ) = A ;
q |- \! not 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 ;
( D . j ) `2 = {} & ( D . j ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: NAT , NAT :] ;
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 `2 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
M1 <= upper_bound M1 & M2 <= M1 implies M1 + M2 <= M1 + 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 H ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= h-14 ;
]. a , b .[ c= Z ;
X1 , X2 are_element & X2 , X1 are_element ;
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 empty addLoopStr , 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 ;
upper_bound B is upper & upper_bound B is upper ;
let L be non empty reflexive RelStr , x be Element of L ;
R is reflexive & X is transitive ;
E , g |= the_right_argument_of ( H ) ;
dom G `2 /. y = a ;
( 1 / 4 ) * ( 1 / 4 ) >= - r ;
G . p0 in rng G & G . p1 in rng G ;
let x be Element of [: F , G :] ;
D [ P-6 , 0 ] ;
z in dom id ( B ) & z in dom id ( C ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | X ) c= NAT & rng ( f | X ) c= REAL ;
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 & h . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = ( A +* {} ) +* ( A +* {} ) ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
Y = - ( a - b ) ;
assume the distance of V , Q is_w ;
let a be Element of ^ ( V ) ;
let s be Element of ( the carrier of P ) ;
let Py be non empty thesis RelStr , x be Element of Py ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= the carrier of f ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> 0 ) * ( j - 1 ) ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in ( \Omega T ) \/ ( { {} } ) ;
let h be Morphism of c , a ;
Y c= [: R , R :] & Y c= [: R , R :] ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L2 ) ;
assume LIN o , a , b & LIN o , a , b ;
b , c // d1 , e2 & b , c // d2 , e2 ;
x1 , x2 , x3 , x4 be set ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in [: X , X :] ;
for n be Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> > closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q2 in P ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] & y = [ y1 , y2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / ( n + 1 ) ) (#) ( 1 / ( n + 1 ) ) ;
rng g2 c= dom W & rng g2 c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( the InternalRel of R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W1 + W2 ;
assume func support L -> Subset of rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric RelStr ;
assume [ x , y ] in [: a9 , b9 :] ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool ( M ) ;
0 <= Arg a * PI ;
o , a9 // o , y & o , b9 // o , y ;
{ v } c= the carrier of l & { v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry g ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D2 . k in rng D ;
f " . p1 = 0 & f . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 & LIN c , e1 , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ S c= conv @ A & conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= P ! ( P ! l ) ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y2 , T ;
W1 , W2 are_well \in field W1 & W2 is well field W1 implies W1 + W2 is well \in field W1
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 s2 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in an ( f ) ;
assume that not Sgm I c= J and not Sgm J c= K ;
Im ( ( lim seq ) , ( lim seq ) ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z implies cos is_differentiable_on Z & for x st x in Z holds cos . x = sin . x / ( cos . x ) ^2
t3 . n = t3 . n & t3 . n = t3 . n ;
dom ( ( dom ( - F ) ) | ( dom F ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W1 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( ( k + 1 ) <= len ( v | k ) ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I & h . I = g2 . I ;
( G /. 1 ) `1 = ( G /. 1 ) `1 .= ( G /. 1 ) `1 ;
f . rr1 in rng f & f . rr2 in rng f ;
i + 1 + 1-1 <= len - 1 ;
rng F = rng ( F | n ) .= rng ( F | n ) ;
mode non empty multMagma over L is well unital associative non empty multMagma ;
[ x , y ] in [: A , { a } :] ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of m c= B & the carrier of m c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F | I ) = len I .= len ( F | I ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , x be Element of X ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be for non empty 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 Function equals J ;
K c= 2 -tuples_on the carrier of T & K is finite ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : ( a - 1 ) / ( a - 1 ) = 1 ;
assume that not 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 , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non trivial FinSequence of D ;
let FF2 be non empty element ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in [: F-8 , F-8 :] ;
reconsider pp = x , py2 = y as Subset of m ;
let A , B , C be Element of R ;
redefine mode non empty be strict be non empty be s3 -be union ;
rng c `2 misses rng ee `2 & rng c `2 misses rng ee `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 * cot ) & Z c= dom ( cot * cot ) ;
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 / ( g - r ) ) ;
pred f = u means : Def1 : a * f = a * u ;
for n holds P1 [ \mathop { \rm from n } ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = q2 ;
gcd ( n1 , n2 , k ) = 1 & gcd ( n1 , n2 , k ) = 1 ;
set oo = a * ( - 1 ) , oo = a * ( - 1 ) , oo = a * ( - 1 ) , oo = a * ( - 1 ) , oo = a
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= 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 , b ) `1 , ( a , b ) `2 ] in Indices G ;
assume that Y = { 1 } and s = <* 1 *> ;
IF1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W3 .last() = W3 . 2 ;
cluster trivial -> finite for non empty is finite connected _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ \bullet implies A , B are_that A , B are_that A , B are_that A , B are_that B , A are_that A , B are_that A , B are_that B , A are_that A
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 - cn ) ;
f1 is_<= <= <= <= <= g & g <= f implies f ^ g is non empty
( f `2 ) ^2 / ( |. q .| ) ^2 <= ( |. q .| ) ^2 / ( |. q .| ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b `2 ) ^2 / ( |. b .| ) ^2 <= ( |. b .| ) ^2 / ( |. b .| ) ^2 ;
let f , g be \cdot Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( ( max ( - 1 , 1 ) ) (#) f ) ;
p2 in [: N , { p } :] & p1 in [: N , { p } :] ;
len ( the_right_argument_of H ) < len ( H ) & len ( the_right_argument_of H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
pred 1 in C means : Def1 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 <> 0 ;
rng q1 c= rng C1 & rng q2 c= rng C2 ;
let A1 , L , A2 , A3 be non empty set ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in C ( p , Ss ) & c in C ( p , Ss ) ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T & Cl Int [#] T = [#] T ;
( f | A2 ) | A2 = f2 & ( f | A2 ) | A1 = f2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V & V c= Y \ Z ;
let X be Subset of [: S , T :] ;
consider H1 such that H = 'not' H1 and H1 in X ;
1_ ( 1 ) c= ( \mathop { t } * ( \mathop { t } ) ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set vFinSequence = v4 /. n , v5 = v5 /. n ;
r = 0. ( REAL-NS n ) * ( ||. x .|| ) ;
( f . p4 ) `1 >= 0 & ( f . p1 ) `2 >= 0 ;
len W = len ( W ) + len ( W ) .= len W ;
f /* ( s * G ) is divergent_to+infty to f /* ( s * G ) ;
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 - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 ] -> pair & [ x2 , x3 ] is pair ;
\HM { a /\ downarrow t where t is Ideal of T : t is Ideal of a } is Ideal of a
let X be \hbox { NAT , D } , F be non empty set ;
rng f = such \lbrace the carrier of S , X } ;
let p be Element of B , x be Element of the carrier' of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R1 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & p1 = 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 S ;
x in h " P /\ [#] T1 & x in h " P /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X-5 = X , Xelement = Y as non empty Subset of Tsuch
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & n2 <= len g2 & n2 <= len g2 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G1 ;
y = Re ( y ) + ( Im ( y ) * i ) ;
( ( - 1 ) * ( p / ( p ^2 ) ) ) ^2 = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng ( M | D2 ) c= rng ( M | D2 ) ;
for p being Real st p in Z holds p >= a
( ( X --> f ) | X ) = proj1 * f & ( X --> f ) | X = f ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p |-count M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( ( mod P ) * ( - 1 ) ) ;
reconsider i1 = i-1 , i2 = i-1 , j1 = 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being 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 & y in ( the Sorts of A ) . n ;
len } in Seg ( len ( f2 | Seg ( len f2 ) ) ) ;
pp1 c= the topology of T & pp2 c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , A be Subset of T2 ;
G * ( B * A ) = id o1 .= G * ( A * B ) ;
assume that p , u are_not zero and u , q , v be Element of V ;
[ z , z ] in union rng ( F | X ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , G = $1 .. S ;
LIN a1 , a3 , b1 & LIN a1 , a3 , b1 & LIN a1 , a3 , b1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom w2 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
IB * ( i , j ) = 0. K ;
|. f . ( s . m ) -g .| < g1 ;
( for x being element st x in rng ( q | n ) holds q . x = ( q | n ) . x ) ;
Carrier ( LLet ) misses Carrier ( Lq ) ` & Carrier ( Lq ) misses Carrier ( Lq ) ;
consider c being element such that [ a , c ] in G ;
assume that Nreal = o\HM and o8 = o\HM { o } ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( ( F |^ C ) * ( G |^ C ) ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . ( j + 1 ) .] ;
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 for S being aa\circ ( S , T ) ;
let x be Element of [: S , T :] ;
the ObjectMap of F ( a , b ) is one-to-one ;
|. i .| <= - ( - 2 to_power n ) / ( n + 1 ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
! * ( n + 1 ) ! > 0 * ! ;
S c= ( A1 /\ A2 ) /\ ( A1 \/ A2 ) & S c= A2 ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & A is non empty ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y Joins X , Y ;
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = ( v /. ( i + 1 ) ) `2 ;
x = r . n .= ( r . n ) / ( r . n ) ;
f . s in the carrier of S2 & g . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A1 :] & dom d2 = [: A2 , A2 :] ;
0 < ( p / ||. z .|| + 1 ) / ( ||. z .|| + 1 ) ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X /\ Y
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> for operation of X ;
let U1 , U2 be non-empty MSAlgebra over S , f be Function of U1 , U2 ;
Proj ( i , n ) * g is_differentiable_on X & Proj ( i , n ) * g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , py0 = 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 - b is lower ;
Int Cl A c= Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
p2 `2 / |. p2 .| <= p2 `2 / |. p2 .| ;
Cl ( Q ` ) = [#] ( ( T | A ) | A ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = for f |^ n , I8 = f |^ n ;
len p - n = len ( thesis - n ) + n ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n0. L , n6 = n6 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( s | j ) ;
let q\lbrace q , p , q \rbrace be State of M , s be State of M ;
( a in the carrier of S1 ) & ( b in the carrier of S2 ) ;
c1 /. n1 = c1 . n1 & c2 /. n1 = c2 . n1 & c1 /. n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , x be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) * ( S * x ) ) . y ;
consider x being element such that x in Assume an in Assume G ;
assume r in ( ( dist ( o ) ) .: P ) ;
set i2 = ( \mathopen { - } h : h in L~ h } ) ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 1 ) / 2 as Element of REAL ;
let U1 , U2 be strict Subspace of U0 , A be Subset of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 < len p1 + 1 ;
let T1 , T2 be complete Scott thesis of L , x be Element of T1 ;
then x <= y & : x in : x in : y in { x } ;
set M = n -tuples_on ( Seg m ) ;
reconsider i = x1 , j = x2 , k = x3 as Nat ;
rng the_arity_of ( a9 ) c= dom H & rng the_arity_of ( a9 ) c= dom H ;
z1 " = ( z " ) * ( z * z1 ) .= ( z * z1 ) * ( z * z2 ) ;
x0 - r / 2 in L /\ dom f & x0 - r / 2 in dom f ;
then w is non empty implies rng w /\ L <> {} & L is non empty ;
set x9 = ( x ^ <* Z *> ) | Z , y9 = ( x ^ <* Z *> ) | Z ;
len w1 in Seg ( len w1 + len w2 ) & len w2 = len w2 + len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( x . n ) ;
p `1 <= G * ( len G , 1 ) `1 & p `1 <= G * ( len G , 1 ) `1 ;
rng ( g | 1 ) c= L~ ( g | 1 ) \/ rng ( g | 1 ) ;
reconsider k = i-1 * ( i-1 + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 , z9 = z9 as Element of M ;
dom ( f | X ) = X /\ dom f & dom ( f | X ) = 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 ) .= p . ( bg ) ;
a to_power ( s . m - 1 ) / ( n + 1 ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and B2 = C2 \/ C1 ;
X . i = { x1 , x2 } . i .= ( x1 | dom x1 ) . i ;
r2 in dom ( h1 + h2 ) & r2 in dom ( h1 + h2 ) ;
- ( 0. R ) = a & b-0 = b ;
F8 is_closed_on t2 , Q2 & F8 is_halting_on t1 , Q1 ;
set T = -> in and { X , x0 } is non empty ;
Int Cl ( Int Cl R ) c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F[: f , g :] ) = { F[: f , g :] } ;
( G-23 ( { c } ) ) \/ S c= B \/ S ;
fbeing Relation of [: X , Y :] , X ;
set RQ = the Element of P , RQ = the Element of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , i be Element of NAT ;
reconsider p9 = u , q9 = v , p9 = w as Element of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n
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 G / ( N , 1 ) ;
len ( ( P | k ) ^ ( P | k ) ) <= len ( P | k ) ;
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 ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of ( is non empty Subset of ( A + B ) * ;
assume s1 = sqrt ( 2 * ( p / 2 ) - 1 ) ;
pred a > 1 & b > 0 & a to_power b > 1 ;
let A , B , C be Subset of [: I , J :] ;
reconsider X0 = X , Y0 = Y as RealNormSpace , Y = Z as RealNormSpace ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of 2 -tuples_on BOOLEAN , t-3 be Relation ;
Q [ e-14 \/ { v-5 } , f ] & f . v-5 = [ e-5 , f . v-5 ] ;
g \circlearrowleft W-min L~ z = z & g /. len g = z implies g /. len g = z
|. |[ x , v ]| - |[ x , y ]| .| = v\vert x \vert ;
- f . w = - ( L * w ) .= - ( L * w ) ;
z - y <= x iff z <= x + y & y <= z ;
( 7 * p1 ) to_power ( 1 / e ) > 0 ;
assume X is BCK-algebra of 0 , 0 & 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v1 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( tan (#) sec ) `| Z ) . x in dom ( sec (#) sec ) ;
i2 = ( f /. len f ) & i2 = ( f /. len f ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 implies 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 = the carrier of I[01] & g2 is one-to-one ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & f2 is continuous ;
( proj2 | X ) .: X = proj2 .: X & proj2 .: X = proj2 .: X ;
f . ( x , y ) = h1 . ( x `2 , y `2 ) ;
x0 - r < a1 . n & x0 < a1 . n ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A & len Line ( B , i ) = width B ;
SFinSequence ^ S = ( S . g ) ^ ( S . g ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized p & intloc 0 in dom Initialized p ;
i1 := i2 := i3 does not destroy ( b := ( a , b ) ) & not i1 does not destroy ( b ) ;
arccos r + arccos r = ( PI / 2 + 0 ) / ( PI / 2 + 0 ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = ( q - x ) / ( 1 - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j ;
assume f in the carrier of [: X \to Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( ( true T ) at ( C , u ) ) = TRUE implies ( ( {} T ) at ( C , u ) ) = TRUE
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom ( f | [. 0 , 1 .] ) ;
p2 `1 - x1 > - g / 2 & p2 `2 - g / 2 > - g / 2 ;
|. r1 - `2 .| = |. a1 .| * |. ` .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .( n ) = D0W .( n ) + 1 ;
i1 = ( a + n ) & i2 = ( K + n ) & i1 = ( K + n ) ;
f . a [= f . ( f . O1 "\/" a ) ;
pred f = v & g = u , h = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( [: T1 , T2 :] , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k2 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R4 * R4 ) & L~ M1 meets L~ ( R4 * R4 * ( len M1 ) ) ;
set h = the continuous Function of X , R , x be Point of X ;
set A = { L . ( ( k + 1 ) + 1 ) where k is Element of NAT : k in dom L } ;
for H st H is atomic holds P7 [ H ] ;
set b\HM = S5 ^\ ( i + 1 ) , Selement = S5 ^\ ( i + 1 ) , bREAL = S5 ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 / ( n + 1 ) ) < ( 1 / ( n + 1 ) ) * ( 1 / ( n + 1 ) ) ;
( l `1 = [ dom l , cod l ] ) `1 .= ( [ dom l , cod l ] ) `2 ;
y +* ( i , y /. i ) in dom g & y +* ( i , y ) in dom g ;
let p be Element of CQC-WFF ( Al ( ) ) , x be Element of D ( ) ;
X /\ X1 c= dom ( f1 - f2 ) & X c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) & p1 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) & indx ( D2 , D1 , j1 ) + 1 <= len D1 ;
assume x in K1 /\ K0 \/ K1 /\ K0 \/ K1 /\ K0 \/ K1 /\ K0 ;
- 1 <= ( ( f2 ) . O ) `2 & ( ( f2 ) . O ) `2 <= 1 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c be Real ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 + 1 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= dom f ;
g2 in ]. x0 , x0 + r .[ & g2 in ]. x0 , x0 + r .[ ;
sgn ( p `2 , K ) = - ( 1_ K ) * ( - 1_ K ) ;
consider u being Nat such that b = p |^ ( y * u ) ;
ex A being as as as as \mathopen normal ' of T st a = Sum A ;
Cl ( union ( H ) ) = union ( ( union ( H ) ) \/ ( union ( H ) ) ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 ;
v-29 = v + w |-- v + A8 .= v + ( A + B ) ;
v ( ) <> DataLoc ( t0 . GBP , 3 ) & v ( ) <> DataLoc ( t0 . GBP , 3 ) ;
g . s = upper_bound ( d " { s } ) & g . s = upper_bound ( d " { s } ) ;
( \dot { y } ) . s = s . ( \dot { y } . s ) ;
{ s : s < t } in INT implies t = {} & s = {}
s ` \ s = s ` \ 0. X .= ( s ` \ ( s ` ) ` ) \ ( s ` \ ( s ` ) ) ;
defpred P [ Nat ] means B + $1 in A & A c= B ;
( 339 + 1 ) ! = 3329 ! * ( 339 + 1 ) ;
IC ( ( Carrier A ) * ( ( Carrier A ) * ( o , i ) ) ) = T ;
reconsider y = y , z = z as Element of ( len y ) -tuples_on COMPLEX ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT , i be Nat ;
set f = ( S , U ) \mathop { z } , g = ( S , U ) \mathop { z } , h = ( S , U ) \mathop { z } , f = ( S , U ) \mathop { z } , F = ( S , U ) \mathop
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , g be Function of I[01] , TOP-REAL n ;
( ( SAT M ) . [ n + i , 'not' A ] ) <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n -tuples_on REAL ) , x be Element of REAL ;
reconsider l = 0. ( V ) , v = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. n .| + |. w .| + a ;
consider y being Element of S such that z <= y and y in X ;
a be being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being being Element of Y holds ( a 'or' b ) 'or' c = 'not' ( a 'or' b )
||. x9 - g .|| < r2 & ||. x9 - g .|| < r ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // c9 , a9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k2 & k2 + 1 = k2 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
E-max C in cell ( RG , 1 , 1 ) & ( E-max C ) `2 = ( GoB f ) * ( 1 , j ) `2 ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `2 , a // b `1 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is |^ ;
F | ( N2 , S ) = CircleMap * ( F-4 , S ) .= ( CircleMap * ( F | N2 ) ) | ( N2 , S ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( cos | [. - 1 , 1 .] ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . n ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = I as string of S2 , t2 = I as string of S2 ;
reconsider x9 = seq , y9 = seq2 as sequence of REAL-NS n , x be Element of REAL n ;
assume that r meets ( L~ go \/ L~ pion1 ) and p in L~ go /\ L~ pion1 and p in L~ pion1 ;
- ( ( - 1 ) / ( n + 1 ) ) < F . n - ( - 1 ) / ( n + 1 ) ;
set d1 = being thesis , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( x2 , y2 ) , d2 = dist ( y2 , z2 ) , d2 = dist ( y2 , z2 ) , M = dist
2 |^ ( 2 |^ |. 00 .| - 1 ) = 2 |^ ( |. 100 .| - 1 ) ;
dom ( v | Seg ( len d6 ) ) = Seg ( len d6 ) .= dom ( v | ( dom v6 ) ) ;
set x1 = - k2 + |. k2 .| + 4 , x2 = - ( k2 + 1 ) + 4 ;
assume for n being Element of X holds 0. X <= 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 ( LT ) + L2 ) c= I2 & the carrier of ( Carrier ( LT ) + L2 ) c= I2 ;
'not' Ex ( x , p ) => All ( x , p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal be normal be Subgroup of {} ;
Z c= dom ( ( ( - 1 ) (#) ( ( sin * f1 ) ^ ) ) `| Z ) ;
|. 0. TOP-REAL 2 - q .| < r / 2 + r / 2 ;
not not not not not not not not not ( not ( A , succ d ) ) c= not ( A , L )
E = dom ( L (#) F ) & L (#) G is_measurable_on E & ( L (#) F ) | E is_measurable_on E & ( L (#) F ) | E is_measurable_on E implies ( L (#) F ) | E is_measurable_on E
C / ( A + B ) = C / B * C / A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC s2 = P . IC s2 .= ( I . IC s2 ) ;
pred x > 0 means : Def1 : ( 1 / x ) ^2 = x ^2 / ( 1 - x ^2 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c are_connected & - C , - C + - D + ( - D , 1 ) are_connected ;
assume f = id ( the carrier of O1 ) & g = id ( the carrier of O1 ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( {} ( the carrier of V ) ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of G_ ( k , X ) ;
|. - x .| = - - x .= - x + - x .= - x + - x .= - x ;
set S = ) ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) - 1 ) >= 4 * 8 ;
v3 /. ( k + 1 ) = v3 . ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S\mathopen { - 1 , j } , i ) = S\mathopen { - 1 , j } ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y2 , y1 ] ;
|. f .| - Re ( |. f .| * h ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & 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 ;
LIN c , q , b & LIN c , q , c & LIN c , q , c ;
f| ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
flet . a = flim a & flim a in InputVertices S & flim a in InputVertices S ;
p `1 <= ( E-max C ) `1 & p `1 <= ( E-max C ) `1 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , R7 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) ^2 + ( p `2 ) ^2 ;
consider p such that p = p-20 and s1 < p & p <= s2 ;
|. ( f /* ( s * F ) ) . l - G .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N .= width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = ( f1 /* s1 ) /* s1 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 implies f ^ <* f /. 1 *> is | ( len f + 1 )
dom ( Proj ( i , n ) * s ) = REAL m & rng ( Proj ( i , n ) * s ) c= REAL ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on ( the carrier of V ) \ { {} } .= the carrier of V ;
consider r such that r \not _|_ 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 / 2 in dom f ;
for L being complete LATTICE holds <* <* \mathclose ( L ) , L *> , <* 3 , 1 *> *> are_isomorphic ;
[ gi , gj ] in [: I , I :] \ [: I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c , d ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st x0 < r ex g st g < r & g in dom ( f2 * f1 ) ;
reconsider y = ( a ` ) / ( F . ( a ` ) ) , z = ( a ` ) / ( F . ( a ` ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) ) . c <= h . c ;
set G3 = the as Vertex of G , v be Vertex of G , x be set ;
reconsider g = f as PartFunc of REAL , REAL-NS n , x be Element of REAL ;
|. s1 . m - p .| / |. p .| < d / ( p |^ m ) ;
for x being element st x in ( ( for u being element st u in ( ( for t being element st t in ( holds u in holds t in A ) ) holds u in B ) ) holds x in B
P = the carrier of ( TOP-REAL n ) | ( P ` ) & Q = the carrier of ( TOP-REAL n ) | P ;
assume p01 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) to_power ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 + ( 2 * c * d ) ;
let f , g , h be Point of the complex normed space of X , Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) & m <= n implies m <= n ;
H * ( g " * a ) in the right of H & I * ( g " * a ) in the right of H ;
x in dom ( ( cos * sin ) `| Z ) & ( ( cos * sin ) `| Z ) . x = sin . x / ( cos . x ) ^2 ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q2 , p4 , P , p1 , p2 ;
attr B is BDD closed means : Def1 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p + - n ) + ( - n ) ;
attr a <> 0. K means : Def1 : the_rank_of M = the_rank_of ( a * M ) & the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom /\ /\ dom /\ I and I = len of of I + j ;
consider x1 such that z in x1 and x1 in ( P * f ) . x1 and x2 in ( P * f ) . x2 ;
for n ex r being Element of REAL st X [ n , r ] ;
set CP1 = Comput ( P2 , s2 , i + 1 ) , CP2 = Comput ( P2 , s2 , i + 1 ) , CP2 = P2 ;
set cv = 3 / ( 3 * ( a , b , c ) ) , cw = 3 / ( 3 * ( a , b , c ) ) , cw = 4 * ( a , b , c ) , cw = 5 / ( 3 * ( a , b , c ) ) ,
conv ( @ W ) c= union ( ( F .: ( E " ( W ) ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( arccot * ( f1 + f2 ) ) ;
r3 <= s0 + ( r0 - ( |. v2 - v1 .| ) / ( 2 * ( 1 - r ) ) ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 & dom ( f (#) f4 ) = dom ( f (#) f4 ) /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) F ) /\ Seg k ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp , gq = gq , gr = gr as Point of TOP-REAL 2 ;
( T * h . s ) . x = T . ( h . s . x ) ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom [ *> <* the *> of ( commute ( ( Frege ( A . o ) ) ) , ( commute ( ( Frege ( A . o ) ) ) ) *> ] ;
for I being non degenerated commutative commutative Ring holds the carrier of I is commutative commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) , P1 = 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 . ( lw . i ) = ( v *' lw ) . i .= ( v *' lw ) . i ;
consider n being element such that n in NAT and x = ( cn M ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F2 . x <> F2 . x ;
Funcs ( X , 0 , x1 , x2 ) = { E } & card ( X \ { x1 , x2 } ) = k ;
j + ( 2 * ( k + 1 ) ) + m1 > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on A3 & { s , t } on B3 & { s , t } on B3 ;
n1 > len crossover ( p2 , p1 , n1 , n2 , n3 ) & n2 > len crossover ( p2 , p1 , n2 , n3 ) ;
mg . HT ( mg , T ) = 0. L & mg . HT ( mg , T ) = 0. L ;
then that H1 , H2 are_that card H1 , ( card H2 ) / ( 2 , 1 ) are_relative_prime and ( H1 , H2 ) / ( 2 , 1 ) / ( 2 , 1 ) " ;
( N-min L~ f ) .. ( ( f | ( len f -' 1 ) ) .. ( f | ( len f -' 1 ) ) ) + 1 > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( k -tuples_on ( k + 1 ) ) ;
I is k2 & I is k2 & I is k2 & I is k2 is k2 implies I is k2
[: u , { u9 } :] = { [ a , u9 ] } & [: u , { u9 } :] = [: { a , b } , { a , b } :] ;
( w | p ) | ( p | ( w | w ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W3 ;
for y st y in rng F ex n st y = a |^ n & F . y = a |^ n
dom ( ( g * ( f . ( V \dot \to C ) ) ) | K ) = K ;
ex x being element st x in ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( the Sorts of U0 ) ) ) \/ A ) ) ) . s ) ) ) ) ) ) . s ) ;
ex x being element st x in ( ( and ( T \/ A ) ) . s ) & x in ( T \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x = ( - 1 ) * ( 1 - r ) ;
( the carrier of X1 union X2 ) /\ ( ( the carrier of X1 ) \/ ( the carrier of X2 ) ) <> {} ;
L1 /\ LSeg ( p01 , p2 ) c= { p01 } & L2 /\ LSeg ( p01 , p2 ) c= { p01 } ;
( b + ( bs ) / 2 ) in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A ;
( the sequence of ( ( the carrier of X ) ) | ( the carrier of X ) ) . ( id the carrier of X ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 + 1 .= len w + 1 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | E-4 ` = g | E-4 ` & g | Ed = h | Ed ` ;
reconsider i1 = x1 , i2 = x2 , j1 = x3 , j2 = x4 , i1 = x4 , i2 = x4 , j1 = x4 , j2 = x4 as Element of NAT ;
( a * A * B ) ` = ( a * ( A * B ) ) ` ;
assume ex n0 being Element of NAT st f to_power n0 & f to_power n0 < r & f to_power n0 < r ;
Seg len ( ( ( ( f1 ^ f2 ) | Seg len f1 ) ) ^ ( ( f2 | Seg len f1 ) ) ^ ( ( f2 | dom f1 ) | dom f1 ) ) = dom ( ( f2 | dom f1 ) ^ ( ( f2 | dom f1 ) | dom f1 ) ) ;
( Complement ( A ) ) . m c= ( ( Complement ( A ) ) . n ) . m ;
f1 . p = p9 & g1 . p = d & g1 . ( p , q ) = c & g2 . ( p , q ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| |^ n ) / ( n + 1 ) <= ( r2 |^ n ) / ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & rng ( F ) 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 W3 is Subspace of W3 and W3 is Subspace of W3 ;
||. ( t-15 . x ) .|| = lim ||. ( x - y ) .|| .= ||. ( x - y ) .|| .= ||. x .|| ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 ) ^2 - 1 ) / ( 1 + { p `2 } ) <= ( - 1 ) / ( 1 + { p `2 } ) ;
g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) & g | Sphere ( p , r ) = id ( Sphere ( p , r ) ) ;
set N8 = N-min L~ Cage ( C , n ) , N8 = N-bound L~ Cage ( C , n ) , N8 = S-bound L~ Cage ( C , n ) , N8 = S-bound L~ Cage ( C , n ) , N8 = S-bound L~ Cage ( C , n ) , N8 = N-bound L~ 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 ) .= B * ( i , j ) ;
pred a <> 0 means : Def1 : ( A \+\ B ) \ a = ( A Let a ) \+\ ( B Y. a ) ;
then f is_is_\cal 2 2 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and a > 0 and b <> 1 and c > 0 and d > 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w2 , w2 } ;
p2 /. IC s = p2 . IC s .= p2 . IC s .= ( p2 . IC s ) .= ( p2 . IC s ) ;
ind ( T-10 | b ) = ind b .= ind B .= ind ( b | b ) ;
[ a , A ] in the \leq of ( the \cdot ( 2 * ( A ) ) ) & [ a , A ] in the carrier of ( 2 * ( A ) ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & ( the Arrows of C ) . ( o1 , o2 ) = ( the Arrows of C ) . ( o2 , o2 ) ;
( ( ( a , CompF ( PA , G ) ) 'imp' ( b , CompF ( PA , G ) ) ) ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi , phi = phi as Element of ^2 ;
len s1 - 1 * ( len s2 - 1 ) + 1 > 0 + 1 * ( len s2 - 1 ) ;
delta ( D ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in [: the carrier' of A , the carrier' of B :] ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and g2 . z = g2 . z ;
[#] V1 = { 0. V1 } .= the carrier of ( (0). V1 ) /\ the carrier of ( (0). V1 ) ;
consider P2 be FinSequence such that rng P2 = M and P2 is one-to-one and P2 is one-to-one and P2 is one-to-one ;
assume that x1 in dom ( f | X ) and ||. x1 - x0 .|| < s and s . x1 - f /. x0 .|| < r ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ^ <* p *> .= h ^ <* p *> ;
c /. ( |[ b , c ]| ) = c .= |[ a , c ]| `2 .= |[ a , c ]| `2 .= |[ a , c ]| `2 ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 , t1 = p4 as Term of C , V ;
( 1 / 2 ) * ( 1 / 2 ) in the carrier of [. 1 / 2 , 1 .] & ( 1 / 2 ) * ( 1 / 2 ) in the carrier of I[01] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= ( h . p2 ) `2 + D ;
R . ( b - a ) = 2 * - b .= 2 * - b .= - b ;
consider \hbox such that B = - 1 * strict * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( ( the Sorts of A ) * ( ( the Sorts of B ) * ( the Arity of S ) ) * ( the Arity of S ) ) ) ;
[ P . ( l ) , P . ( l ) ] in => ( ( P => Q ) => ( 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 NAT ;
y in product ( ( Carrier J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 0 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g & y in the right of g & x = g & y = g ;
consider M be strict Subgroup of Abe such that a = M and T is Subgroup of M and T is Subgroup of M ;
for x st x in Z holds ( ( ( #Z 2 ) * f ) + ( #Z 2 ) * f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m + len W3 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian LinearOperator of X , Y ;
( - ( i mod len ( p + q ) ) + 1 ) in dom ( p + q ) ;
assume that s2 is_>= s1 and F in the not empty & F in the not empty & G in the not empty ;
( ( ( ( ( ( ( ( ( ( ( x , y ) ) ) * ( 1 , 3 ) ) ) + 1 ) ) * ( ( ( x , y ) * ( 1 , z ) ) * ( 1 , y ) ) ) ) / ( 2 * ( ( x , y ) * ( 1 / 2 ) ) ) ) ) / (
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B be Subset of u-5 st B in E holds A = B or A misses B or B = C ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ W1 , W3 = the carrier of W2 ;
x in { X where X is Ideal of L : X is non empty & X is non empty } ;
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 ;
( not 1 / ( a + b ) ) * ( id a + b ) = ( 1 / ( a + b ) ) * ( id a + b ) ;
( ( X --> f ) . x ) = ( X --> dom f ) . x .= ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) , y = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => q => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( being C - 2 ) |^ ( n -' m ) + 1 ) + 2 ;
( 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 ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & d on P & c on P & d on P ;
reconsider gf = g `2 * f `2 , hg = h `2 * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v2 ) ` ;
n in { i where i is Nat : i < n0 + 1 & i <= n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn & p `2 >= 0 & p <> 0. TOP-REAL 2 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( ( ConsecutiveSet ( A , O1 ) ) . succ O1 ) ;
set IF1 = in dom ( Macro ( a , intloc 0 ) ) , IF2 = SubFrom ( a , intloc 0 ) , IF2 = SubFrom ( a , intloc 0 ) , IF2 = SubFrom ( a , intloc 0 ) , IF2 = Initialize ( s ) , IF2 = s ;
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. ( len z ) <> z /. ( len z )
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 3 = b ;
reconsider ee = ee , fe = fe , fe = fe , fe = f , ee = g as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( \overline { O } ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and n <= m ;
f * g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + succ $1 = succ A & A = ( succ $1 ) + ( succ $1 ) ;
the left of - g = the left of g & the left of - g = the left of g implies g = f & g = g
reconsider pp = x , p-> Point of TOP-REAL 2 , p = y , q = z as Point of TOP-REAL 2 ;
consider ex g2 such that g2 = y and x <= g2 & g2 <= x0 and g2 <= x0 and x0 < g2 & g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 & len ( x2 ^ y2 ) = len x2 + len y2 + len y2 ;
for x being element st x in X holds x in the set of ( the set of 0 ) * ( ( m + 1 ) / 2 ) & ( m + 1 ) * ( ( n + 1 ) / 2 ) = ( m + 1 ) / 2
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = {} & LSeg ( p1 , p01 ) /\ LSeg ( p01 , p2 ) = {} ;
func of ) -> set equals ( [: [: X , X :] , [: X , Y :] ) --> ( id X ) ;
len ( ( the carrier of ( ( C | ( len C -' 1 ) ) ) | ( len C -' 1 ) ) ) <= len ( C | ( len C -' 1 ) ) ;
pred K is has a , v means : Def1 : 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 p ;
for x st x in X ex y st x c= y & y in X & y is - f . x = f . y
IC Comput ( P-6 , scmeans : Def1 : IC Comput ( PJ , scJ , k ) in dom ( sJ +* I ) & IC Comput ( PJ , scJ , k ) in dom I ;
pred q < s means : Def1 : r < s & ]. r , s .[ \not c= ]. p , q .[ & ]. p , q .[ c= ]. p , q .[ ;
consider c being Element of Class ( f , 3 ) such that Y = ( F . c ) `1 and c in Class ( f , 3 ) ;
func the ResultSort of S2 -> Function of the carrier' of S2 , the carrier' of S2 means : Def1 : for x being set st x in the carrier' of S2 holds it . x = [ x , x ] ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] , z9 = [ <* x , y *> , f3 ] ;
assume x in dom ( ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) ) & ( ( ( #Z 2 ) * ( arccot ) ) `| Z ) . x = f . x ;
r-7 in Int cell ( GoB f , i , ( GoB f ) * ( i , j ) + ( GoB f ) * ( i + 1 , j ) ) & ri2 in Int cell ( GoB f , i + 1 , j ) ;
q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 & q `2 >= ( Cage ( C , n ) /. ( i + 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i - len f <= len f + len f1 - len f & i - len f <= len f + len f - len f ;
for n ex x st x in N & x in N1 & h . n = - ( x0 - h . n )
set s0 = ( ( a , I , p , s ) +* I ) . i , s1 = ( ( a , I , p ) +* I ) . i , s2 = ( a , I , J ) . i , s3 = ( ( a , I , J ) +* J ) . i , s5 = ( ( a , I , J ) +* J ) . i
p ( k ) . 0 = 1 or p ( k ) . 0 = - 1 or p ( k ) . 0 = 1 & p ( k ) . 1 = - 1 ;
u + Sum L-18 in ( U \ { u } ) \/ { u + Sum L-18 } ;
consider x9 being set such that x in x9 and x9 in V1 and y = [ x9 , the carrier of L ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) + ( len p ) ) ;
g + h = gg + h1 & h = g + h & h = h + h & g = h + 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 ) & f | y in rng ( f | y ) implies f | x = f | y ;
assume that 1 < p and ( 1 - p ) / ( 1 - q ) = 1 and 0 <= a and a <= b ;
F* ( f , t ) = rpoly ( 1 , the carrier of F_Complex ) *' t + 0. F_Complex .= 0. F_Complex + 0. F_Complex .= 0. F_Complex ;
for X being set , A being Subset of X , B being Subset of X holds A ` = {} implies A = B & B = {}
( ( N-min X ) `1 ) ^2 <= ( ( ( N-min X ) `1 ) ^2 + ( ( E-max X ) `2 ) ^2 ) ;
for c being Element of the \vert of A , a being Element of the \subseteq the \subseteq of a & c <> a holds c = a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= s2 . GBP .= Exec ( i2 , s2 ) . GBP .= s2 . GBP .= s2 . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 ) & b >= 0 implies a * b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = ( x \ y ) ` ;
mode BCK-algebra of i , j , m , n , m , n be Element of NAT , i , j , m be Element of NAT ;
set x2 = |( Re ( y - y ) , Im ( y - y ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y & [ y , x ] in dom g ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & [. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .] c= A ;
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) * ( 2 * ( n + 1 ) ) ;
( - ( ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ) ) ^2 <= ( - ( q `1 / |. q .| - cn ) ) ^2 ;
set A = ( 2 / b-a ) / ( 2 * b-a ) ;
for x , y being set st x in R" holds x , y are_let x , y implies x , y are_let x , y
deffunc FF2 ( Nat ) = b . $1 * ( M * G ) . $1 & ( M * G ) . $1 = ( M * G ) . $1 ;
for s being element holds s in -> ( f 'or' g ) iff s in ( f \/ g ) \/ ( f \/ g )
for S being non empty non void non void non empty non void non empty non void holds S is connected ;
max ( degree ( ( z `1 ) ) , degree ( ( z `2 ) ) ) >= 0 & degree ( ( z `2 ) ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s and seq . ( n1 + k ) < r ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = nz '&' ( M . ( x qua Element of BOOLEAN ) ) , ny = ( M . ( x qua Element of BOOLEAN ) ) * ( M . ( x qua Element of BOOLEAN ) ) ;
f " V in ' ( X ) & f " V in D & f " V in D & f " V in D & f " V in D & f .: V = D & f .: V = D ;
rng ( ( a sequence ( c ) ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being as | of G1 such that y `1 = y and dom y `2 = WWG and y `2 = WWG ;
dom ( ( 1 / f ) (#) ( ]. - 1 , 0 .[ ) ) c= ]. - 1 , 1 .[ & dom ( ( 1 / f ) (#) ( g `| ]. - 1 , 0 .[ ) ) c= dom ( ( 1 / f ) (#) ( g `| ]. - 1 , 0 .[ ) ) ;
as of as Matrix of n , j , n , r , - r , ( n + 1 ) * ( - r ) ;
v ^ ( ( n |-> 0 ) ^ ( ( B | ( n -' 1 ) ) ) in Lin ( ( B | ( n -' 1 ) ) ) ;
ex a , k1 , k2 st i = a /. k1 & j = b /. k2 & k2 = c /. k2 & k2 = d /. k1 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ i1 .= succ i1 .= succ ( 5 + 1 ) .= NAT ;
assume that F is bbfamily and rng p = F and dom p = Seg ( n + 1 ) and for i being Nat st i in Seg ( n + 1 ) holds p . i = F . i ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN c , b9 , a & LIN c , c9 , b
( L1 Let L2 ) \& O c= ( L1 Let O ) Let ( L2 Let O ) Let ( L2 Let O ) ;
consider F be ManySortedSet of E such that for d be Element of E holds F . d = F ( d ) and for d be Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( 0. V ) = b * ( -w ) and 0 < a & a < b ;
defpred P [ FinSequence of D ] means |. Sum $1 .| <= Sum |. $1 .| & |. Sum $1 .| <= Sum |. $1 .| ;
u = cos / ( x , y ) . v * x + ( cos / ( x , y ) . v * y ) .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| (#) |. p .| ] , id ( the Sorts of A ) . [ 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 inininand X is ininand X is non empty ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & h <= g } & g <= f & f <= g & g <= h implies g <= h
vol ( ( G . n ) vol ) <= ( Partial_Sums ( ( G . n ) vol ) ) * vol ( ( G . n ) vol ) ;
f . y = x .= x * 1_ L .= x * ( power L ) . ( y , 0 ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( <% i1 , i2 %> , n ) = { i1 , succ i2 } & NIC ( a , b ) = { i1 , succ i2 } ;
LSeg ( p01 , p2 ) /\ LSeg ( p1 , p01 ) = { p1 } & LSeg ( p1 , p01 ) = { p1 } ;
product ( ( ( Carrier ( I ) ) +* ( i , { 1 } ) ) ) in ( Z . i ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) .= Following ( s2 , n ) ;
W-bound ( Qs2 ) <= ( q1 `1 ) / ( 1 + ( q1 `1 ) ^2 ) & ( q1 `1 ) / ( 1 + ( q2 `1 ) ^2 ) <= ( q2 `1 ) / ( 1 + ( q2 `2 ) ^2 ) ;
f /. i2 <> f /. ( ( len f + ( len g -' 1 ) -' 1 ) ) & f /. ( len f + ( len g -' 1 ) ) = f /. ( len f + ( len f -' 1 ) -' 1 ) ;
M , f / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) |= H / ( x. 4 , m ) ;
len ( ( P ^ ) ^ ( ( P ^ ) ) ^ ( ( P ^ ) ) ) in dom ( ( P ^ ) ^ ( P ^ ) ) ;
A |^ ( mn ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , l ) ;
( R |^ n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a }
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and p1 . ( n1 + 1 ) = p2 . ( n1 + 1 ) ;
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 |. ( seq_id v ) .| & |. ( |. v .| ) * ( |. v .| ) = |. ( |. v .| ) * ( |. v .| )
for phi holds phi in X implies phi in X & phi in X & phi in X & phi in X & phi in X
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom f ) ) ) ) ) ) | dom ( ( f | dom ( f | dom ( f | dom ( f | dom f ) ) ) ) ) ) c= dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom f ) ) ) ) ;
ex c being FinSequence of D ( ) st len c = k & a = c & c = c & a = c & b = d ;
the_arity_of ( a , b , c ) = <* o , \mathop { \rm , c } , \mathop { \rm and } ( a , b , c ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and for x be Element of X st x in the carrier of X holds f1 . x = x ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 & a3 = b3 or a1 = b3 & a2 = b2 & a3 = b3 & a3 = b3 & a4 = b3 & a4 = 6 ;
D2 . indx ( D2 , D1 , n1 + 1 ) = D1 . ( n1 + 1 ) & D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. ( r (#) f ) .|| /. 1 .= <* r * ( f /. 1 ) .|| .= r * ( f /. 1 ) .= r * ( f /. 1 ) .= r * ( f /. 1 ) .= r * ( f /. 1 ) ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . n ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & d <= b ;
||. L /. h .|| - ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative means : Def1 : for b being Element of X holds F \hbox { b } = f . b & F . b = f . b ;
p = - ( - ( - ( p0 + 0. TOP-REAL 2 ) ) ) .= 1 * p0 + 0. TOP-REAL 2 .= ( 1 - ( p0 + 0. TOP-REAL 2 ) ) * p0 .= ( 1 - ( p0 + 0. TOP-REAL 2 ) ) * p0 + 0. TOP-REAL 2 .= ( 1 - ( p0 + 0. TOP-REAL 2 ) ) * p0 ;
consider z1 such that b , x3 , z1 is_collinear and o , x1 , z1 is_collinear and o , x1 , z1 is_collinear and o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 ;
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 g is one-to-one and g is one-to-one ;
assume that A = P2 \/ Q2 and P2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f .: ( g , h ) , h ) ;
ex x being Element of NAT st m = x `2 & x in z & x in { 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 x = P-2 . k2 ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n & seq is convergent & rng seq = dom ( r (#) seq )
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 } & x in D1 "\/" D2 & y in D2 & x in D2 "\/" D1 & y in D2 "\/" D1 & x in D2 ;
consider z being element such that z in dom ( ( dom ( ( dom F ) | ( dom F ) ) ) ) and ( ( ( F | ( 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
G * ( i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 & G * ( i + 1 , 1 ) `2 <= s } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( ( Mx2Tran ( J , BT , i ) ) * ( t /. j ) ) ) * ( ( ( Mx2Tran ( J , i ) ) * ( t /. j ) ) ) ;
- 1 / ( ( - 1 ) (#) D ) = ( m (#) D ) | n .= ( ( m (#) D ) (#) ( - 1 ) ) | n .= ( Det M ) (#) ( ( - 1 ) (#) ( - 1 ) ) .= ( Det M ) (#) ( ( - 1 ) (#) ( - 1 ) ) ;
attr x being set means : Def1 : x in dom f /\ dom g & g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len f2 /. j .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 | j ) ;
All ( All ( 'not' a , A , G ) , B , G ) |= Ex ( All ( 'not' a , B , G ) , A , G ) ;
LSeg ( E . k0 , F . k0 ) c= Cl RightComp Cage ( C , k + 1 ) & LSeg ( E , k + 1 ) c= RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( ( a |^ k ) * a ) .= ( x \ ( a |^ k ) ) \ a ;
k = ( ( commute th ) | ( I . k ) ) . i .= ( ( commute IB ) | ( I . k ) ) . i .= ( ( commute IB ) | ( I . k ) ) . i ;
for s being State of A[: s , n :] holds Following ( s , n + 2 * n + 1 ) is stable & Following ( s , n + 2 * n + 1 ) is stable ;
for x st x in Z holds f1 . x = a / ( 2 * x ) & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 <> 0
support ( support ( n ) \/ support ( support ( m ) ) ) c= support ( max ( support ( m ) , support ( n ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of S , ( the carrier' of C ) * the Arity of S ;
- ( a * sqrt ( 1 + ( b ^2 ) ) / ( 1 + ( a ^2 ) ) ) <= - ( b * sqrt ( 1 + ( a ^2 ) ) / ( 1 + ( b ^2 ) ) ) ;
phi / ( succ b1 ) . a = g . a & phi / ( b , a ) . a = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) . i ) and i = len ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 } = { x1 } \/ { x2 , x3 , x4 } \/ { x4 } .= { x1 } \/ { x2 , x3 , x4 } \/ { x4 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U2 c= the Sorts of U2 implies ( the Sorts of U1 ) "\/" ( the Sorts of U2 ) c= the Sorts of U2
( - ( 2 * a * ( b * ( 2 * a ) + b ) ) / ( 2 * a ) ) ^2 - ( 2 * a * ( 2 * a ) + c ) / ( 2 * a ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] & Q [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r and ( the Arity of S ) . o = <* r *> ;
Z = dom ( ( exp_R (#) ( arccot ) - ( arccot (#) ( f1 + f2 ) ) ) (#) ( ( exp_R (#) ( f1 + f2 ) ) ) ) ;
sum ( f , SS1 ) is convergent & lim ( \HM { the carrier of S1 , the carrier of S2 } ) = integral ( f , SS2 ) & lim ( f , SS2 ) = integral ( f , SS2 ) ;
( as set ) => ( ( a => f ) => ( x9 => x9 ) ) in -> Element of \rm of \rm WFF ( Al ) & ( a => f ) => ( x9 => x9 ) in
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M1 * M2 ) = n & len ( M1 * M3 ) = n & width ( M1 * M2 ) = n ;
attr X1 union X2 is an open SubSpace of X means : Def1 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated implies X1 , X2 are_separated ;
for L being upper-bounded antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-129 = F1 . ( ( b - a ) / ( b - a ) ) , f29 = F2 . ( ( b - a ) / ( b - a ) ) as Function of M , M ;
consider w being FinSequence of I such that the InitS of M is_\HM { w } ^ w ^ w ^ w ^ w = q ^ w ^ w ^ q ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= ( g . ( a |^ 0 ) ) |^ ( a * ( a |^ 0 ) ) .= ( g . ( a |^ 0 ) ) |^ ( a * ( a |^ 0 ) ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds L /\ K <> {} & K is closed ;
( 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 C2 ;
reconsider oY = o `2 , oY = p `2 , oY = p `2 , oY = p `2 , o = p `2 , o = p `2 , o = p `2 , o = p `2 , Y = p `2 , h = p `2 , G = G ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) .= x1 + ( 0 * x2 ) ;
Es " . 1 = ( Es qua Function ) " . 1 .= ( ( ( 1 - s ) * ( 1 - s ) ) * ( 1 - s ) ) " .= ( ( 1 - s ) * ( 1 - s ) ) " ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . l1 + 1 ) - f . ( s1 . l1 ) .| < ( 1 - |. M .| ) * ( M . l1 + 1 ) ;
LSeg ( ( Upper_Seq ( C , n ) ) /. ( i + 1 ) , ( Upper_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 + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the set of the carrier of A and ColVec2Mx b in the carrier of A and len f = width A and width ( ColVec2Mx f ) = width A and width ( ColVec2Mx f ) = width A and width ( ColVec2Mx f ) = width A ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & len ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of ( ( the InternalRel of ( n + 1 ) ) * ( i + 1 ) )
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 ;
pred a <> 0 & b <> 0 & Arg a = Arg b & 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 } , ( the ) , ( the set ) , ( the InternalRel of a ) , ( the InternalRel of b ) )
assume that V1 is linearly-independent and V2 is closed and V2 is closed and V1 = { v + u : v in V1 & u in V1 & u in V1 } ;
z * x1 + ( 1 / ( 1 - a ) ) * x2 in M & z * ( y1 + ( 1 - a ) * y2 ) in N implies z * ( x1 + ( 1 - a ) * x2 ) in N
rng ( ( ( ( ( ( ( ( ( ( ( ( P ) qua Function ) ) * ) ) " ) ) * ( ( P * ( P * ( Q * ( P * ( Q * ( P * ( Q * ( P * ( Q * ( P * Q ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = Seg card ( dom ( Q * ( P * ( Q * ( Q * ( P * ( Q * ( Q * ( Q * ( Q * ( Q * ( Q
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b & s2 . n <= b ;
h2 " . n = h2 . n " & 0 < h2 . n & 0 < h2 . n & h2 . n < 1 implies h2 " . n = h2 . n & h2 . n = h2 . n
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. seq1 .|| . m .= ( ||. seq1 .|| ) . m .= ( ||. seq1 .|| ) . m .= ( ||. seq .|| ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = ( - 1_ ( G ) ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v & - w = ( - 1_ G ) * v ;
upper_bound ( ( k .: D ) .: D ) = upper_bound ( ( k .: D ) .: D ) .= k . ( upper_bound D ) .= upper_bound ( ( k .: D ) .: D ) .= upper_bound ( ( k .: D ) .: D ) ;
A |^ ( k , l ) ^^ ( A |^ ( n , .. A ) ) = ( A |^ ( k , .. A ) ) ^^ ( A |^ ( k , .. A ) ) ;
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 / p `1 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for n being Nat holds support ( a * b ) = support ( a ) + support ( b ) ) & ( for n being Nat holds a . n = ( support ( a ) ) + ( support ( b ) )
consider A9 being countable set such that r is countable & A9 is Element of CQC-WFF ( Al ) & ( for n being Nat holds A9 . n = A ( n ) ) & ( not ( ex x being Element of NAT st x in A & x in A ) & ( x in A ) & ( x in A ) & ( x in A ) & ( x in A ) & ( x in A ) & ( x in A ) & ( A is countable ) ) ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] , [ y1 , y2 ] } \/ { [ y1 , y2 ] } ;
h . ( f . O ) = |[ A * ( f . O ) + B , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) & ( Gauge ( C , n ) * ( k , i ) ) `1 = ( Gauge ( C , n ) * ( k , i ) ) `1 ;
cluster m , n are_relative_prime means : Def1 : for p being prime Nat holds p divides m & p divides n & p divides n & p divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ a <= c holds a \+\ b <= c
consider b being element such that b in dom ( H / ( ( x , y ) / ( x , y ) ) ) and z = ( H / ( x , y ) ) / ( x , y ) ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( ( F * g ) . y ) * ( g . x ) ;
assume ex e being element st e Joins W . 1 , W . 5 , G & e Joins W . 3 , G & W . 7 = G . 6 ;
( h (#) ' ( h ) ) . ( 2 * n ) . x = ( h (#) ( h . n ) ) . ( 2 * n + ( h . n ) ) . x ;
j + 1 = ( - len h + 2 ) + 1 .= i + 1 - len h + 2 - 1 .= i + 1 - 1 + 2 - 1 .= i + 1 - 1 + 1 .= i + 1 - 1 ;
( ^ ( S , f ) ) . f = S *' . ( ( S *' ) . f ) .= S *' . ( ( S *' ) . f ) .= S *' . ( ( S *' ) . f ) .= S *' . ( ( S *' ) . f ) .= S *' . ( ( S *' ) . f ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L2 * H ) = Sum ( L2 ) ;
attr R is >= + means : Def1 : for p , q st p in R & q in R ex P st P is special & p in P & q in P & P c= R ;
dom product ( X --> f ) = meet ( ( X --> f ) . ( X --> f ) ) .= meet ( ( X --> f ) . ( X --> f ) ) .= meet ( ( X --> f ) . ( X --> f ) ) .= ( ( X --> f ) . ( X --> f ) ) . ( X --> f ) .= ( ( X --> f ) . ( X --> f ) ) . ( X --> f ) ;
upper_bound ( proj2 .: ( Upper_Arc ( C ) /\ Lower_Arc ( C ) /\ \hbox { w } ) ) <= upper_bound ( proj2 .: ( C /\ \hbox { w } ) /\ \hbox { w } ) & upper_bound ( proj2 .: ( C /\ \hbox { w } ) <= ( proj2 .: ( C /\ \hbox { w } ) ) /\ ( 2 * w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - x0 .| < r
i * ( f - fcccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
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 ) & f | Y is one-to-one ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in C and g2 in D and g2 in D ;
func d |-count n -> Nat means : Def1 : d divides d |^ n & ( d |^ n ) divides ( d |^ n ) & ( d |^ n ) divides ( d |^ n ) ;
f{ [ 0 , t ] } = f . [ 0 , t ] .= ( - P ) . [ 2 * ( x `1 ) , t `2 ] .= 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 = h . M or t = h . N ;
consider m1 being Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) ^2 ) ;
( ( q `1 ) / |. q .| ) ^2 <= ( ( q `2 ) / |. q .| ) ^2 + ( ( q `2 ) / |. q .| ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 - len h11 + 2 ) .= h21 . ( i + 1 + 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and [ o , x2 ] in the carrier' of S ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a >= b & a >= b & b >= a & b >= a & b >= a
||. h1 .|| . n = ||. h1 . n .|| .= |. h . n .| .= ||. h . n .|| .= ||. h .|| . n .= ||. h .|| . n .= ||. h .|| . n ;
( ( - ( exp_R * f ) ) `| Z ) . x = f . x - exp_R . x .= ( ( exp_R * f ) `| Z ) . x .= ( ( exp_R * f ) `| Z ) . x .= ( ( exp_R * f ) `| Z ) . x ;
pred r = F .: ( p , q ) means : Def1 : len r = min ( len p , len q ) & for i being Nat st i in dom r holds r . i = F . ( p . i ) ;
( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) + ( ( r / 2 ) ^2 + ( r / 2 ) ^2 ) <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det M = Sum ( ( Det M ) * ( i , j ) ) & Det M = ( Det M ) * ( i , j )
then a <> 0. R & a " * ( a * v ) = 1 / ( a * v ) & a " * ( a * v ) = 1 / ( a * v ) & a * v = 1 / ( a * v ) ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 -' j ) = Sum ( p . ( j -' 1 ) * r3 ) .= Sum ( p ) * r3 ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) " ) . $1 & ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) . $1 = ( ( h ^\ n ) /* ( h ^\ n ) ) . $1 ;
assume that the carrier of H2 = f .: ( the carrier of H1 ) and the carrier of H2 = f .: ( the carrier of H2 ) and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H2 and the carrier of H1 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o & ( the Arity of S ) . o = ( the Arity of S ) . o ;
H1 = n + 1 - ( 2 to_power ( ( 2 to_power ( n + 1 ) + h ) ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) .= n + 1 - ( 2 to_power ( n + 1 ) ) ;
( O1 = 0 & O1 = 0 & O2 = 1 & O2 = 1 & O1 = 1 & O2 = 1 & O2 = 1 & O2 = 1 & O2 = 1 & O2 = 2 & O2 = 3 & O2 = 3 & O2 = 1 & O2 = 2 & O2 = 3 & O2 = 3 implies O1 = 3 & O2 = 4 & O2 = 5
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
pred b <> 0 & d <> 0 & d <> 0 & b <> d & ( a - b ) / ( d - c ) = ( - ( e / 2 ) ) / ( d - c ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D .= ( dom f \/ dom g ) /\ D ;
for i being set st i in dom g ex u , v being Element of L st g /. i = u * a * v & u in B & v in C & v in C & u in C
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 if i = s2 & not ( f . ( i + 1 ) <> s1 & f . ( i + 1 ) <> s1 & s1 = s2 ) & s1 = s2 ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= h | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] , [ s2 , t2 ] are_connected & [ s2 , t2 ] in the InternalRel of G & [ s2 , t2 ] in the InternalRel of G & [ s2 , t2 ] in the InternalRel of G & [ s2 , t2 ] in the InternalRel of G & [ s2 , t2 ] in the InternalRel of G ;
then H is negative & H is non negative & H is non negative & H is not conjunctive -gex F being non empty FinSequence of NAT st F is not non -gex H being non empty elements of NAT st H is not negative -gex F being non [ F ]
attr f1 is total means : Def1 : ( f1 (#) f2 ) is total & ( f1 (#) f2 ) . c = ( f1 . c ) (#) ( f2 . c ) (#) ( f2 . c ) " & ( f1 (#) f2 ) . c = ( f1 . c ) (#) ( f2 . c ) " ;
z1 in W2 " ( W2 " ( W1 ) ) or z1 = z2 & not z1 in W2 " ( W2 " ( W1 ) ) & ( z1 in W2 & z2 in W2 implies z1 = z2 ) & ( z1 = z2 implies z1 = z2 )
p = 1 * p .= a " * a * p * a .= a " * ( b * q ) .= a " * ( b * q ) .= a " * ( b * q ) .= ( a " * ( b * q ) ) * ( b * q ) ;
for seq1 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 . n ) <= ( seq . n ) * ( seq1 . n )
x0 meets ( L~ go \/ L~ pion1 ) \/ ( L~ pion1 ) or ( C /\ L~ pion1 ) meets ( L~ pion1 ) \/ ( L~ pion1 ) or ( C /\ L~ pion1 ) meets ( L~ pion1 ) \/ ( L~ pion1 ) or ( C /\ L~ pion1 ) meets ( L~ pion1 ) \/ ( L~ pion1 ) ;
||. f . ( g . ( k + 1 ) ) - g . ( g . k ) .|| <= ||. g . 1 - g . 0 .|| * ( K to_power k ) ;
assume h = ( ( ( B .--> B ' ) +* ( C .--> D ) +* ( E .--> F ) ) +* ( F .--> J ) +* ( J .--> M ) +* ( N .--> N ) +* ( F .--> N ) +* ( M .--> N ) +* ( M .--> N ) +* ( N .--> A ) +* ( N .--> A ) +* ( N .--> A ) +* ( N .--> A ) +* ( N .--> N ) +* ( M .--> A ) +* ( N .--> A ) +* ( N .--> A ) +* ( N .--> A ) +* ( N .--> A ) +* (
|. ( ( ( ( ( ( ( ( ( ( ( H . n ) || A ) ) ) | A ) ) - ( ( ( ( H . n ) || A ) ) | A ) ) ) ) | A ) . k - ( ( ( ( ( H . n ) || A ) ) | A ) . k ) .| <= e * ( 2 * ( ( H . n ) || A ) ) ;
( ( ( ( ( ( the Sorts of A ) . i ) ) | ( the carrier of I ) ) ) . e ) = [ ( ( the Sorts of A ) at ( the carrier' of I ) ) . e , ( ( the Sorts of A ) * ( the ResultSort of I ) ) . e ] ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 } = { x1 } \/ { x1 } .= { x1 } ;
assume that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) cos ) , A ) = ( ( exp_R (#) cos ) | A ) . ( upper_bound A ) - ( ( exp_R (#) cos ) | A ) . ( upper_bound A ) ;
p `2 is Permutation of dom f1 & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p & p `2 = ( Sgm Y ) " * p ;
for x , y st x in A holds |. ( 1 - ( f . x ) ) .| <= 1 * |. ( f . x ) .| - ( f . y ) .|
p2 `2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) / ( 1 + sn ) - sn ) / ( 1 + sn ) * ( 1 + sn ) / ( 1 + sn ) ;
for f be PartFunc of the carrier of CNS , REAL , x be Element of REAL , r be Real st f is compact & f is continuous & x in dom f & f | X is compact holds rng ( f | X ) is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider F3 such that dom F3 = n1 and for k be Nat st k in n1 holds Q [ k , F3 . k ] & ( for k be Nat st k in n1 holds F [ k , F3 . k ] ) & ( for k be Nat st k in NAT holds F [ k , F3 . k ] ) implies F ( k ) = F ( k ) ;
ex u , u1 st u <> u1 & u , u1 , u1 , v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1 ^ u1 ^ v1 ^ u1 ^ u1 ^ v1 ^ v1 ^ u1 ^ v1
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N : A ) * ( N { N } ) = N ` A * B
for s be Real st s in dom F holds F . s = integral ( ( R to_power 0 ) (#) ( f + g ) ) + Integral ( M , ( f + g ) (#) ( f - g ) ) ;
width AutMt ( f1 , b1 , b2 ) = len b2 .= len b1 .= len ( ( f | ( len b1 ) ) + len b2 ) .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 .= len b1 + len b2 + len b2 .= len b1 + len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & dom f = ]. - PI / 2 , PI / 2 .[ & rng f = ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ is continuous ;
assume that X is closed and a in X and a c= X and y in a ^ ( f . ( n + 1 ) ) and x in a & y in a & x in X ;
Z = dom ( ( ( ( ( #Z 2 ) * ( arctan + arccot ) ) ) (#) ( ( #Z 2 ) * ( arctan + arccot ) ) ) ) /\ dom ( ( ( #Z 2 ) * ( arctan + arccot ) ) (#) ( ( #Z 2 ) * ( arctan + arccot ) ) ) ;
func TAUT ( V ) -> Subset of V means : Def1 : for k st 1 <= k & k <= len l holds it . k in V & l . k in V ;
for L being non empty TopSpace , N being net of L , M being net of L , c being Point of N st c is net of M & c is cluster cluster cluster cluster for net of N for net of N holds c is net of N
for s being Element of NAT holds ( ( ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) ) (#) ( f - 1 ) ) ) ) ) `| REAL ) . s = ( ( ( ( - 1 ) (#) ( f - 1 ) (#) ( f - 1 ) ) `| REAL ) . 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 ) * ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( - ( ln * f ) ) and f . x = x and for x st x in Z holds f . x = x & f . x > 0 ;
for R being add-associative right_zeroed right_complementable commutative associative non empty doubleLoopStr , I being non empty Subset of R , J being Subset of R , I being Subset of R holds ( I + J ) *' ( I /\ J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B1 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x2 (#) z ) .= Seg len ( x2 (#) z ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) ;
for S being Functor of C , B for c being Object of C holds card S = id ( ( Obj S ) . c ) & ( Obj S ) . c = id ( ( Obj S ) . c )
ex a st a = a2 & a in f6 /\ f5 & for x st x in f6 holds \rbrace ( f , a ) = f2 ( f , a ) & { a } c= dom ( f , a ) & { a } c= dom ( f , a ) ;
a in Free ( ( H / ( x. 4 , x. 0 ) ) '&' ( H2 / ( x. 4 , x. 0 ) ) ) & b in Free ( ( H / ( x. 0 , x. 0 ) ) '&' ( H2 / ( x. 4 , x. 0 ) ) ) ;
for C1 , C2 being f1 , f , g being stable Function of C1 , C2 st ( for x being set st x in C1 holds f . x = g . x ) & ( for x being set st x in C1 holds f . x = g . x ) holds f = g
( W-min ( L~ go \/ L~ pion1 ) ) `1 = W-bound ( L~ go \/ L~ pion1 ) & ( W-min ( L~ go \/ L~ pion1 ) ) `1 = E-bound ( L~ go \/ L~ pion1 ) & ( W-min ( L~ go \/ L~ pion1 ) ) `2 = N-bound ( L~ go \/ L~ pion1 ) ;
assume that u = <* x0 , y0 , z0 *> and f is_assume u , 3 , 2 & u is_not N & SVF1 ( 3 , f , u ) is_differentiable_in z0 & SVF1 ( 3 , f , u ) is_differentiable_in z0 & SVF1 ( 3 , f , u ) . z = SVF1 ( 3 , f , u ) . z ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & ( t . {} ) `2 = y & ( t . {} ) `2 = y & ( t . {} ) `2 = y ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( q , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b >= a ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in Class ( R , A ) & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( b ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) c= G ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and V = the carrier of W1 & V = the carrier of W2 & W = the carrier of W2 & V = the carrier of W1 & V = the carrier of W2 & V = the carrier of W2 & W = the carrier of W1 & V = V & V = V ;
mamas ( m . t ) = ( m . t ) `1 .= ( [ m . t , the carrier of C ] `1 ) `1 .= ( [ m , the carrier of C ] `1 ) `1 .= m . t ;
d11 = x9 ^ d22 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( x9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( x9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( x9 , d22 ) . ( x9 , d22 ) .= ( f | ( y9 , d22 ) .= ( f | ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( x9 , d22 ) .= ( f | ( y9 , d22 ) . ( y9 , d22
consider g such that x = g and dom g = dom ( f | X ) and for x being element st x in dom ( f | X ) holds g . x in ( f | X ) . x ;
x + 0. F_Complex = x + len x |-> 0. F_Complex .= ( ( x + len x ) |-> 0. F_Complex ) ^ ( x |-> 0. F_Complex ) .= ( x + ( len x ) |-> 0. F_Complex ) ^ ( x |-> 0. F_Complex ) .= x ` + ( x * ( x * ( x * ( x * 0. F_Complex ) ) ) .= x ` ;
( k -' ( k + 1 ) ) + 1 in dom ( f | ( 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 = P \/ P2 and P1 = P \/ P2 and P2 = P \/ { p1 , p2 } and P1 = P \/ { p1 , p2 } and P2 = P \/ { p2 , p1 } and P1 = P \/ { p1 , p2 } and P2 = P \/ { p2 , p1 } ;
reconsider a1 = a , b1 = b , b1 = c , c1 = d , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c1 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , c2 = p `2 , _ { p `2 , p `2 } , _ { p `2 , p `2 } , 7 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , M = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , 8 = p `2 , M = p `2 , 8 = p `2
reconsider set set set set set set set set Ft1f = G1 . ( t . b * F1 . f ) , FFf = G1 . ( t . b * F2 . f ) , FFf = G2 . ( t . b * F2 . f ) , FFf = G2 . ( t . b * F2 . f ) , FFf = G2 . ( t . b * F2 . f ) , FFf = G2 . ( t . f * F2 . f * F2 . f ) , FFf = G2 . ( t . f * F2 , FFf = G2 . ( t . f * F2 , FFf * F1 . ( t . f * F2 . ( t . f * F2 . f ) ,
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 + 1 ) , f /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( f , i + i1 -' 1 + 1 ) ;
Integral ( M , P . m ) | dom ( ( P . n ) -P ) <= Integral ( M , P . m ) | dom ( ( P . n ) -P ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( - ( G * ( i , 1 ) ) `1 ) , ( G * ( i + 1 , 1 ) `2 ) / 2 ) ;
for G being Group , H being Subgroup of G , a being Integer st a = b holds for i being Integer st i in H holds a |^ i = b |^ i & a |^ i = b |^ i holds a = b
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where 7 is Point of TOP-REAL 2 : P [ 7 ] & for p being Point of TOP-REAL 2 st p in P & p <> 7 holds P [ p ] } , K1 = { p : p <> 7 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 8 & p <> 7 & p <> 7 & p <> 8 & p <> 8 & p <> 7
( ( N-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 ) / 2 + ( ( S-bound C ) - ( S-bound C ) ) / 2 ) / 2 + ( ( S-bound C ) - ( S-bound C ) / 2 ) / 2 ) ;
for x being Element of X , n be Nat st x in E holds |. Re ( F . n ) .| <= P . x & |. Im ( F . n ) .| <= P . x & |. Im ( F . n ) .| <= P . x
len ( @ ( @ p ^ @ q ) ) = len ( @ p ^ <* 0 *> ) + len <* [ 2 , 0 ] *> .= len ( @ p ^ <* 0 *> ) + len ( @ q ^ <* 1 *> ) .= len ( @ p ^ <* 1 *> ) + len ( @ q ^ <* 1 *> ) .= len ( @ q ^ <* 1 *> ) + len ( @ q ^ <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x.
consider r being Element of M such that M , v2 / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m ) / ( x. 0 , m ) / ( x. 4 , m ) / ( x. 4 , m )
func w1 \ w2 -> Element of Union ( G , R6 ) equals ( ( ( ( ( ( ( ( ( ( G , R6 , R6 ) * ( G , R7 ) ) ) * ( H , R8 ) ) | [: G , R8 :] ) ) | [: G , R8 :] ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s1 . b2 .= s2 . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . b2 .= ( s2 . b2 ) . 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 + Partial_Sums ( seq ) . ( K + 1 ) >= ( Partial_Sums ( seq ) ) . K + ( 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- ( f . x - x0 ) ) + R . ( x- ( f . x - x0 ) ) ;
func the closed ( a , b , c , d ) -> Subset of ( the carrier of \HM { a , b , c , d } ) | ( the carrier' of ( ( TOP-REAL 2 ) | P ) | P ) & ( the NAT of ( ( TOP-REAL 2 ) | P ) | Q ) = ( the carrier' of ( ( TOP-REAL 2 ) | P ) | Q ) ;
a * b ^2 + ( a * c ^2 + ( b * a ^2 + c ) ) + ( b * c ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( a * c ) ^2 >= 6 * a * a * b * c + ( b * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * c ) ^2 + ( a * b ) ^2 + ( a * c ) ^2 ) >= 6 * a * c + ( a * b * c ) * ( a * c ) ^2 + ( a * b * d ) ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m1 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) = v / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m1 ) ;
Rotate ( Q ^ <* x *> , M ) = ( ( ( Q +* ( M , 0 ) ) +* ( M +* ( { x } --> FALSE ) ) +* ( M +* ( M , 0 ) ) ) +* ( M +* ( M +* ( { x } --> FALSE ) ) ) +* ( M +* ( M --> ( { x } --> FALSE ) ) ) ) +* ( M --> ( M --> ( x , 0 ) ) ) ;
Sum ( FM ) = r |^ ( n1 + 1 ) * Sum ( CM ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . n1 ) .= C . ( n1 + 1 ) * ( C . 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 ) * ( $1 + 1 ) ) + b * ( ( $1 + 1 ) * ( $1 + 1 ) ) / ( ( $1 + 1 ) * ( $1 + 1 ) ) ;
the_arity_of g = ( the Arity of S ) . g .= ( [ ( the Arity of S ) . g , ( the Arity of S ) . g ] ) `1 .= ( ( the Arity of S ) . g ) `1 .= ( ( the Arity of S ) . g ) `1 .= ( ( the Arity of S ) . g ) `1 .= ( the Arity of S ) . g ;
( X [: Y , Z :] ) |^ Z tolerates X |^ Z & card ( ( X [: Y , Z :] ) |^ Z ) = card ( X [: Z , Z :] ) & card ( X [: Z , Z :] ) = card ( X [: Z , 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 ) \ G . ( s . n ) ;
E , f |= All ( All ( x. 2 , ( x. 2 ) ) , ( x. 2 ) ) => ( ( x. 2 , ( x. 2 ) ) '&' ( x. 0 ) ) '&' ( ( x. 2 ) '&' ( ( x. 2 ) '&' ( x. 1 ) ) '&' ( ( x. 2 ) '&' ( ( x. 2 ) '&' ( x. 1 , ( x. 2 ) '&' ( x. 2 ) ) ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n + ( n + 1 ) ) ) . i & ( ( p | ( n + 1 ) ) | ( n + 1 ) ) . i = the carrier of R2 & ( p | ( n + 1 ) ) . i = the carrier of R2 & ( p | ( n + 1 ) ) . i = the carrier of R2 ;
[. a , b + ( 1 / ( k + 1 ) ) .[ is Element of the _ of the carrier of G & ( the partial F of f ) . k is Element of the carrier of G & ( the partial F of f ) . k is Element of REAL & ( the \overline of f ) . k is Element of REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , Comput ( P , s , 2 ) ) .= Exec ( a3 := ( s . a ) , Comput ( P , s , 2 ) ) ;
card ( h1 ) . k = power ( F_Complex ) . ( ( - 1_ F_Complex ) * u , k ) * Sum u .= ( ( f *' ) . ( - 1_ F_Complex ) ) * Sum u .= ( ( f *' ) . ( - 1_ F_Complex ) ) * u .= ( ( f *' ) . ( - 1_ F_Complex ) ) * u .= ( ( f *' ) . ( - 1_ F_Complex ) ) * u ;
( f (#) g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( g /. c ) ) .= ( ( 1 - g ) * ( g /. c ) ) * ( ( g /. c ) " ) .= ( ( 1 - g ) * ( g /. c ) ) * ( ( g /. c ) " ) .= ( ( 1 - g ) * ( g /. c ) ) * ( ( g /. c ) " ) ;
len ( C - ( len ( the carrier of ( ( ( the carrier of ( ( the carrier of ( C ) ) ) ) ) ) ) ) ) = len ( ( C - ( len ( the carrier of ( C ) ) ) ) ) ) .= len ( ( the carrier of ( ( the carrier of ( C ) ) ) ) ) .= len ( ( the carrier of ( C ) ) --> ( the carrier of ( C ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom f /\ X /\ X .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n ) + ( 5 * Fib ( n ) * Fib ( n ) + 5 * Fib ( n ) * Fib ( n ) * Fib ( n ) * Fib ( n ) + 5 * Fib ( n ) * Fib ( n ) * Fib ( n ) + 5 * Fib ( n ) * Fib ( n ) * Fib ( n ) ;
consider f being Function of INT , INT such that f = f `2 and f is onto and f is onto and n < n + 1 & f | n is increasing & f | n is increasing ;
consider c9 be Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and ( for A , B being Element of S holds c . ( A \/ B ) = Prob . ( A \/ B ) ) & ( for A being Element of S holds c . ( A \/ B ) = Prob . ( A \/ B ) ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y ] & P [ y ] ;
assume that A c= Z and Z = dom f and f = ( exp_R (#) ( ( cos - cos ) / ( sin ) ) (#) ( cos - sin ) ) and Z = dom f and f | A is continuous 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 q2 & q1 . ( len Seq q2 ) = q1 . ( len Seq q2 ) & q2 . ( len Seq q2 ) = q2 . ( len Seq q2 ) } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f is Morphism of G1 , G2 and g is Morphism of G2 , G3 and g is Morphism of G2 , G3 and f = G1 * f and g = G2 * f and g = G2 * f ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c + f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & {} <> a & for v holds union ( L , v ) |= ( L , ( union L ) | ( union L ) ) iff L , ( v ) |= H ;
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 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) / ( n + 1 ) and for n1 being Integer , n being Nat st n <> 0 & n < 0 & n <= len p holds sqrt p = ( i - n ) / ( n + 1 ) and n <= len p ;
assume that not 0 in Z and Z c= dom ( ( ( - 1 ) (#) ( arccot ) ) (#) ( f1 - f2 ) ) and for x st x in Z holds ( ( ( - 1 ) (#) ( f1 - f2 ) ) (#) ( f2 - f3 ) ) . x > - 1 & ( ( ( - 1 ) (#) ( f1 - f2 ) ) (#) ( f2 - f3 ) ) . x < 1 ;
cell ( G1 , i1 -' 1 , 2 |^ ( m -' 1 ) ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 & ( ( Y -' 1 ) * ( Y -' 1 ) ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 & ( ( Y -' 1 ) * ( Y -' 1 ) ) \ ( ( Y -' 1 ) * ( Y -' 1 ) ) c= BDD L~ f1 ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st Q1 c= F & Q1 is open & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & ( [#] Y ) c= Q1 & Q1 c= Q1 & Q1 c= Q1 ) ;
gcd ( ( gcd ( A , B , r ) , ( gcd ( A , B , r ) , 1 ) , gcd ( A , B , 1 ) , 1 ) = 1 / ( ( gcd ( A , B , 1 ) , 1 ) , ( gcd ( A , B , 1 ) , 1 ) ) .= 1 / ( ( gcd ( A , B , 1 ) , 1 ) ;
R8 = ( ( the let s2 ) . ( m2 + 1 ) ) . ( m2 + 1 ) .= ( ( the let s of s3 ) . m2 ) . ( m2 + 1 ) .= [ 3 , m2 ] . ( m2 + 1 ) .= [ 3 , m2 ] . ( m2 + 1 ) .= [ 3 , m2 ] . ( m2 + 1 ) .= [ 3 , m2 ] ;
CurInstr ( P-6 , Comput ( PE , E , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m3 ) ) .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ ( { p2 } /\ LSeg ( p1 , p2 ) ) \/ ( { p2 } /\ LSeg ( p1 , p2 ) ) .= ( { p1 } \/ { p2 } \/ ( { p2 } \/ { p1 } ) \/ { p2 } .= { p1 } \/ { p2 } ;
func not ( ex f being Subset of the Sorts of A2 ( ) ) st a in dom f & f = f . i & ex a , b st a in dom f & b in dom f & f . a = f . b & f . b = f . a & f . b = g . b & f . c = g . c ;
for a , b being Element of F_Complex st |. a .| > |. b .| & for f being Polynomial of F_Complex st f >= 1 & f is non zero holds f * ( - b ) is >= f & f * ( - b ) = f * ( - b ) implies f is \cup
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & G * ( i , j ) = g * ( i , j ) & G * ( i , j ) = G * ( i , j ) & G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 are_<* f , g *> and for s1 , s2 being State of C1 , s2 being State of C2 st s1 = s2 & s2 = f & s1 = g holds s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s2 is stable & s1 is stable & s2 is stable & s2 is stable & s2 is stable & s1 is stable & s1 is stable & s1 is stable & s2 is stable & s2 is stable & s1 is stable & s2
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. c .|| .= ||. c .|| .= ||. c .|| .= ||. c .|| .= ||. c .|| .= ||. c .|| ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) & 0 + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) < ( ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) ;
for F being Subset-Family of TT st F is open & not {} in F & for A , B being Subset of TT st A in F & B in F & A <> B & B <> {} holds Cl ( A , B ) c= Cl ( B , A ) & Cl ( A , B ) c= Cl ( B , A )
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 in dom F holds H . k = g . ( F . k , G . k ) ;
i |^ ( ( mod ( n , m ) - i ) ) |^ s = i |^ ( ( s + k ) - i ) .= i |^ ( ( s + k ) - i ) * ( ( s + k ) - i ) .= i |^ ( ( s + k ) - i ) * ( ( s + k ) - i ) .= i |^ ( ( s + k ) - i ) * ( ( s + k ) - i ) .= i |^ ( ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) * ( ( s + 1 ) *
consider q being oriented oriented Chain of G such that r = q and q <> {} and F8 . ( q . ( len q ) ) = v1 and rng q c= rng ( p | ( len p ) ) and rng q c= rng ( p | ( len p ) ) and rng q c= rng ( p | ( len p ) ) ;
defpred P [ Element of NAT ] means $1 <= len I & I <= len I implies ( ( g , Z , I ) . $1 = ( ( g , Z , I ) ^ I ) . ( len I + $1 ) ) & ( g , I ) . $1 = ( ( g , Z , I ) . ( len I + $1 ) ) * ( I . ( len I + $1 ) ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & s . i = I & s . ( len s ) = J & a * b = J ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) ^2 , ( Re y ) ^2 , ( Re y ) ^2 , ( Im y ) ^2 , ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 ) ;
consider \mathop be FinSequence of Fq such that g is continuous and rng ( g | A ) c= A & g . 1 = x1 & g . len g = x2 & g . len g = y1 & rng ( g | A ) = A & g . len g = y1 & rng g c= B & g . len g = y2 & rng g c= A ;
then n1 >= len p1 & n2 >= len p1 implies crossover ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , F , 5 , 6 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 ) = crossover ( p1 , p2 , n1 , n2 , n3 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 9 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 9 , 7 , 7 , 8 , 7 , 7 , 7
( q `1 ) * a <= ( q `1 ) * a & - ( q `2 ) * a <= ( q `1 ) * a or q `1 >= ( q `2 ) * a & - ( q `2 ) * a <= ( q `2 ) * a or - ( q `2 ) * a >= 0 & - ( q `2 ) * a <= - ( q `2 ) * a ;
Fv . ( p9 . ( len p9 ) ) = Fv . ( p . ( len p9 ) ) .= ( v . ( len p9 ) ) * ( v . ( len p9 ) ) .= ( v . ( len p9 ) ) * ( v . ( len p9 ) ) .= ( v . ( len p9 ) ) * ( v . ( len p9 ) ) .= v . ( len v ) ;
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ^ ( ( intloc 0 ) --> ( a := intloc 0 ) ) ) ^ ( ( intloc 0 ) --> ( a := intloc 0 ) ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , y2 such that B8 is finite and D8 = the carrier of ( A1 \/ A2 ) and D8 = the carrier of ( A1 \/ A2 ) and B8 = the carrier of ( A1 \/ A2 ) and B8 = the carrier of ( A1 \/ A2 ) \/ the carrier of ( A2 \/ B2 ) ;
v2 . b2 = ( ( curry F2 ) * ( g * ( id B ) ) ) . b2 .= ( ( curry F2 ) * ( g * ( id B ) ) ) . b2 .= ( ( curry F2 ) * ( g * ( id B ) ) . b2 ) . b2 .= ( ( ( curry F2 ) * ( g * ( id B ) ) ) . b2 ) . b2 .= ( ( ( curry F2 ) * ( g * ( id B ) ) . b2 ) . b2 ) . b2 .= ( ( ( id B ) ) . b2 .= ( ( ( ( ( id B ) * ( g * ( id B ) ) . b2 ) ) . b2 .= ( ( ( id B ) * ( g * ( g * ( g * ( g * ( g * ( id B ) ) . b2 ) * ( g * ( id B ) ) . b2 ) .=
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) .= dom ( IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < d-32 holds |. h .| " * ||. ( R + R1 ) /. h .|| < e / ( ( L + R1 ) * ( L + R1 ) ) * ||. h .|| < e / ( ( L + R1 ) * ( L + R1 ) ) ;
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + i1 -' 1 + 1 ) , h /. ( i + i1 -' 1 + 1 ) ) .= LSeg ( h , i ) /\ LSeg ( h , i + i1 -' 1 + 1 ) .= LSeg ( h , i + 1 ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q2 , q1 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } , P , p1 , p2 & LE q1 , q2 , P , p1 , p2 } ;
( ( - x ) .|. y ) = - ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( - 1 ) * ( ( - 1 ) * ( x .|. y ) ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) ) * ( x .|. y ) .= ( ( ( - 1 ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) * ( x .|. y ) .= ( ( - 1 ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) * ( x .|. y ) .= ( ( - 1 ) * ( x
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
( ( U * W ) * ( W * ( \llangle W , W \rrangle ) ) ) = ( ( U * W ) * ( W * ( W * ( W , L ) ) ) ) * ( W * ( W , L ) ) .= ( ( W * ( W * ( W , L ) ) ) * ( W * ( W , L ) ) ) * ( W * ( W , L ) ) .= ( W * ( W * ( W , L ) ) * ( W , L ) ) * ( W , L ) .= ( W * ( W , L ) ) * ( W , L ) ) * ( W , L ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom ( f + h ) & for x st x in dom it holds it . x = ( f + h ) . x + ( h + h ) . x * ( f . x ) + ( h + h ) . x * ( f . x ) = ( f + h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i , j ) ;
assume that not y in Free H and not x in Free H and not ( x in Free H & not x in Free H & not x in Free H & not x in Free H & not x in Free H & not x in Free H & x in Free H & not x in Free H & x in Free H & not x in Free H & x in Free H & not x in Free H ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT ] means ( P [ $1 ] implies ( $1 = p |^ ( p |^ ( p |^ ( p |^ ( p |^ ( p |^ ( p |^ ( p -' 1 ) ) ) ) ) ) , ( $1 |^ ( p |^ ( p |^ ( p |^ ( p -' 1 ) ) ) ) ) ] ) & ( $1 |^ ( p |^ ( p |^ ( p -' 1 ) ) ) = 1 implies $2 = 1 ) ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A being Subset of X holds A in it iff for W being Subset of X holds W in it iff for Z being Subset of X st Z in it & Z c= W holds it . ( W \/ Z ) = C . W \/ C . Z ;
[#] ( ( dist ( ( dist ( P ) ) ) .: Q ) = ( ( dist ( P ) ) .: Q ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 , 2 } ;
( f " ( rng f ) ) . i = f . i " . ( ( f . i ) " . ( f . i ) ) .= ( f . i ) " . ( ( f . i ) " . ( f . i ) ) .= ( f . i ) " . ( f . i ) .= ( f . i ) " . ( f . i ) .= ( ( f . i ) " ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * sqrt ( 1 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 * sqrt ( 1 + ( p2 `1 ) ^2 ) ]| .= |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * sqrt ( 1 + ( p2 `2 ) ^2 ) ]| ;
( ( ( ( ( ( a , X ) " ) ) * ( ( ( ( a , X ) qua Function ) * ( ( - 1 ) / ( a - 1 ) ) ) * ( 1 - 1 ) ) ) ) " ) . x = ( ( ( ( ( a , X ) " ) * ( 1 - 1 ) ) * ( 1 - 1 ) ) ) * ( ( ( a , X ) " ) * ( 1 - 1 ) ) ) .= ( ( ( a , X ) " ) * ( 1 - 1 ) ) * ( ( ( a , X ) " ) * ( 1 - 1 ) ) .= ( ( ( a , X ) " ) * ( 1 - 1 ) ) * ( 1 - 1 ) * ( 1 - 1 ) * ( 1 - 1 ) * ( ( ( ( ( a , X ) " ) * ( ( a , X ) " ) * ( ( a , X ) " ) * ( ( a , X ) " ) * ( ( a , X ) .= ( ( ( ( ( a , X ) " ) * ( ( a , X ) " )
for T being non empty normal TopSpace , A , B being closed Subset of T , r being Real st A <> {} & A misses B for p being Point of T , r being Point of A st p in B & A = ( in in G ) holds ( for p being Point of T st p in A & A = B holds ( in G ) implies ( for p being Point of T holds p in A )
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = G . i & ( for i being Element of NAT st i in dom G1 holds G1 . i = F . ( i + 1 ) ) & ( for i being Element of NAT st i in dom G1 holds G1 . i = F . ( i + 1 ) ) holds G1 is strict Subgroup of G
for x st x in Z holds ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x = ( ( ( #Z 2 ) * ( arctan - arccot ) ) `| Z ) . x / ( 1 + x ^2 )
synonym f is right continuous means : Def1 : x0 in dom ( f /* a ) & ( for x st x in dom f & x in ]. x0 , x0 + r .[ holds f . x = ( f /* a ) . x ) & ( for x st x in dom f holds f . x = ( f /* a ) . x ) & ( for x st x in dom f holds f . x = ( f /* a ) . x ) ;
then X1 , X2 are_separated & X2 misses X1 & Y1 misses Y2 or ex Y2 being non empty SubSpace of X st Y2 misses Y1 & Y2 misses Y2 & ( Y1 misses Y2 & ( Y2 misses Y2 & Y1 misses Y2 or Y2 misses Y2 & ( Y1 misses Y2 & Y2 misses Y2 & Y2 misses Y2 ) & ( Y1 misses Y2 & Y2 misses Y2 & Y2 misses Y2 ) & ( Y1 misses Y2 implies Y1 = Y2 ) & ( Y1 = Y2 ) & Y2 = Y2 ) ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x- ( 1 , f , u ) ) + R . ( x - x0 )
( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) >= ( ( p2 `1 ) ^2 * sqrt ( 1 + ( p3 `2 ) ^2 ) ) * sqrt ( 1 + ( p3 `2 ) ^2 ) ;
( ( ( 1 / t1 ) (#) ||. f1 .|| ) to_power n ) to_power m = ( ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n ) to_power m & ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n = ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n & ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n = ( ( 1 / t2 ) (#) ||. f1 .|| ) to_power n ;
assume that for x holds f . x = ( ( sin (#) ( cos * sin ) - cos ) / ( sin * sin ) ) & x + h / 2 in dom ( ( sin * ( cos * sin ) - sin ) / ( sin * ( cos * ( sin * ( cos * sin ) ) ) ) ) and h in dom ( ( cos * ( sin * ( cos * ( sin * ( sin * ( sin * ( cos * ( sin * ( sin * ( sin * ( sin * ( cos * ( sin , h ) ) ) / ( sin * ( cos * ( cos * ( cos , h ) ) ) ) ) ) and h ) ) / ( sin * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( cos * ( sin * ( cos * ( sin * ( cos * ( cos * ( sin * ( sin * ( sin * ( sin * ( sin * ( cos * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( cos * ( sin * ( sin * ( sin * ( cos *
consider X-23 being Subset of Y , Y1 being Subset of X such that t = [: [: Y1 , Y1 :] , Y1 :] and Y1 is open and ex Y1 being Subset of [: Y , Y :] st Y1 = [: Y1 , Y1 :] & Y1 is open & Y1 is open & Y2 = [: Y1 , Y2 :] & Y1 is open & Y1 is open & Y2 is open ;
card ( S . n ) = card { [: d , Y :] + ( a * d ) / ( 2 * a ) where d is Element of GF ( p ) : [ d , Y ] in R & [ d , Y ] in R } .= ( R * a ) * ( a * d ) + ( a * b ) / ( 2 * a ) * ( a * b ) ;
( ( E-bound D ) / 2 ) * ( i1 - 1 ) / 2 * ( i - 1 ) = ( ( W-bound D ) / 2 ) * ( i - 1 ) / 2 * ( i - 1 ) .= ( ( W-bound D ) / 2 ) * ( i - 1 ) / 2 * ( i - 1 ) .= ( ( W-bound D ) / 2 ) * ( i - 1 ) / 2 * ( i - 1 ) ;