thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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 ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
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contradiction ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is Cauchy ;
q in X ;
V in W ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `2 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A ` in B ` ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in R^1 ;
let p be set ;
let A be set ;
let G be _Graph , a , b be set ;
let G be _Graph , a , b be set ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
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 = which ;
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 ( x , a ) 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 Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , f be FinSeq-Location ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in dom f1 ;
cluster uparrow x -> be finite ;
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 ;
- - s >= - s ;
G . y <> 0 ;
let X be RealNormSpace , x be Point of X ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , f be Function ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
IT c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
pp c= PI / 2 ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq , seq1 , seq2 be Real_Sequence ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x << sup D ;
b1 >= Z1 ;
assume w = 0. V ;
assume x in A . i ;
g in the carrier of Comput ( X , 1 ) ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
Let Let Let Let Let Let Let Let Let Let Let Let Let C be f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
IT is non-decreasing ;
IT is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , A be non-empty MSAlgebra over S ;
assume P [ n ] ;
assume that union S is independent and card S = card S ;
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 set ;
b `1 c= b9 `1 & b9 `2 c= b9 `2 ;
assume not x in NAT + Q ;
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 ;
Observe : sqrt I is left ideal ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { n , m } < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g is_differentiable_in x0 ;
g is_continuous_in x0 & g . x0 in dom f ;
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 , P2 = P +* stop I ;
d , c // a , b ;
let t , u be set ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
let Y be Subset of S ;
let X be non empty TopSpace , A be Subset of X ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , C be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( mn (#) F ) ;
h2 . a = y ;
P [ n + 1 ] ;
Observe : G * F is pre| ;
let R be non empty multMagma , a be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
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 `1 ;
let M be LSeg mamaid ;
let N be non empty m1 be multiplicative be Element of M ;
let R be RelStr with finite finite Anumber ;
let n , k be Nat ;
let P , Q be be relational structure ;
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 FinSeq-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < ( v + u ) ;
x <= c2 . x ;
x in F ` & y in F ` ;
Observe S --> T is such that S --> T is such that S is such
assume t1 <= t2 & t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F1 <> F2 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A2 <> A1 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & rng g1 c= A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom ( f1 (#) f2 ) ;
x in dom ( sec * f ) ;
assume [ x , y ] in R ;
set d = ( x / y ) / ( x / y ) ;
1 <= len g1 & 1 <= len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 (#) f2 ) ;
1 in dom ( D2 | Seg len D2 ) ;
( 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 ;
lcm ( i , i ) = i ;
X1 c= dom f & X2 c= dom g ;
h . x in h . a ;
let G be thesis ;
cluster m * n -> square ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive implies field R = field R
set F = <* u , w *> ;
p-2 c= P3 & p`2 c= Int L~ f ;
I is_halting_on t , Q ;
assume [ S , x ] is vertical ;
i <= len f2 & j <= width f2 ;
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= XX & BX c= BY ;
n2 <= ( 2 |^ ( n2 + 1 ) ) ;
A /\ [#] ( TOP-REAL 2 ) c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , a be set ;
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 ;
S be SigmaField of X , T , f be Function of S , T ;
i . y in rng i ;
R^1 c= dom f & f | X is bounded ;
f . x in rng f ;
mt <= ( r / 2 ) ;
s2 in r-5 & s2 in r-5 ;
let z , z be complex number ;
n <= Nm . m ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = |[ S , T ]| ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 . n ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT , a be Element of REAL ;
rng ( F * g ) c= Y
dom f c= dom x & rng f c= dom y ;
n1 < n1 + 1 & n2 + 1 < n2 ;
n1 < n1 + 1 & n2 + 1 < n2 ;
cluster (0). T -> non empty ;
[ y2 , 2 ] `1 = z ;
let m be Element of NAT , n be Nat ;
let S be Subset of R ;
y in rng ( S . k ) ;
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 & n in dom h2 ;
w + 1 = ( a + 1 ) ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p & Support u c= Support p ;
assume X is complete complete ;
assume that f = g and p = q ;
n1 <= n1 + 1 & n2 <= len p1 ;
let x be Element of REAL ;
assume x in rng s2 & y in rng s2 ;
x0 < x0 + 1 & x0 < x0 + 1 ;
len ( L (#) F ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n & rng q c= Seg n ;
j <= width M ^ M ;
let r8 be real-valued Real_Sequence ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT , x be set ;
assume z in atat-> of at_ _ _ 0 ( A ) ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( nu ) = n ;
\cal Z ( Z ) c= F
assume x in X or x = X ;
x is element of b , c ;
let A , B be non empty set , f be Function ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & dom q = Seg len q ;
let s be Element of E ^ omega ;
let B1 be Basis of x , B2 be Basis of x ;
L3 /\ L2 = {} ;
L1 /\ L2 = {} implies L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f . -129 ) ;
set n8 = n + j ;
let DA be non empty set , F be Function of D , REAL ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , M be Matrix of K ;
assume that f `2 = f and h `1 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 is open ;
assume a , b are_maximal in the InternalRel of 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 -> ne\mathop sthen -> n] ;
not u in { \hbox { \boldmath $ g } } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster for non empty Str over L ;
r (#) H is PartFunc of X , REAL ;
s . intloc 0 = 1 & s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U0 be strict non-empty MSAlgebra over S , A be Subset of U0 ;
[ x , Bottom T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
r-35 in iff r( y ) in .. ( x ) ;
let x , y be Element of X ;
let A , I be such that I is seq of X ;
[ y , z ] in O ;
( } ( Macro i ) ) = 1 ;
rng Sgm A = A & rng Sgm A c= A ;
q |- All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b ;
p . 2 = Z / Y ;
( D . ( j - 1 ) ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: the carrier of Al , the carrier of Al :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f3 + f3 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `1 , y `2 in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty for multMagma ;
x in support ( support t ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y9 `2 ) ;
assume p divides b1 + b2 & p divides b2 ;
M . x0 <= upper_bound M1 & M . x0 <= upper_bound M1 ;
assume x in W-min ( X ) /\ L~ f ;
j in dom ( z | ( i + 1 ) ) ;
let x be Element of D ( ) ;
IC s4 = l1 .= l1 .= IC s .= 0 ;
a = {} or a = { x } ;
set uR = Vertices G , uR = Vertices G ;
seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcarrier c= h-14 ( F , i ) ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> real-valued for Relation of INT ;
ex v st C = v + W ;
let IT be non empty zero Nat , n be Nat ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper & sup B is upper ;
let L be non empty reflexive RelStr , x be Element of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= ( the_left_argument_of H ) '&' ( the_right_argument_of G ) ;
dom G / y = a ;
( 1 - 4 ) / 4 >= - r / 4 ;
G . p0 in rng G & G . O in rng G ;
let x be Element of ( F . i ) * ;
D [ ( P . O ) `1 , 0 ] ;
z in dom id ( B ) & z in dom id ( B ) ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng ( f | [: Seg n , Seg n :] ) c= NAT ;
j `2 + 1 in dom s1 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k2 in rng P ;
M = ( A +* {} ) +* ( {} .--> {} ) ;
let p be FinSequence of REAL , r be Real ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
h . [. a , b .[ = b-a ;
assume that the distance of V , Q and the distance of V , Q ;
let a be Element of ^ ( V ) ;
let s be Element of PS , a be Element of S ;
let Pa be non empty \rbrace ;
n be Nat ;
the carrier of g c= B & the carrier of g c= B ;
I = halt SCM R & I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K , BK = BCS K ;
l <= ( -> y2 ) . j ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in *> ( JumpParts T , 1 ) \ { {} } ;
let h be Morphism of c , a ;
Y c= 1. ( K , card Y ) ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b ;
b , c // d1 , e2 ;
x1 , x2 , x3 , x4 , x5 , x5 be Element of Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[. a , b .] = [. a , b .] ;
cluster -> -> -> -> -> -> -> -> -> -> -> -> closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear & q2 , q2 , q2 is_collinear ;
dom ( M1 * M2 ) = Seg n & dom ( M1 * M2 ) = Seg n ;
x = [ x1 , x2 ] & y = [ x1 , x2 ] ;
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 ( ( L * R ) * R ) ;
let b be Element of the carrier of T ;
dist ( e , z ) - r-r > r-r ;
u1 + v1 in W2 & v1 in W2 + W3 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M ;
0 <= 2 * PI & 0 < 2 * PI ;
o9 , a9 // o9 , y & o9 , b9 // o9 , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume D2 . k in rng D & D1 . k in rng D1 ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) c= REAL ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of f ;
conv @ A c= conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the topology of T ;
J , v |= ( P ! l ) . ( m , v ) ;
cluster J . i -> non empty for TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is_\! to field ( W1 + W2 ) implies W1 + W2 is_field ( W2 + W3 )
assume x in the carrier of R & y in the carrier of R ;
dom ( nI . i ) = Seg n & rng ( nI . i ) c= Seg n ;
s4 misses s4 \/ s5 \/ s6 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices f ;
assume that that } c= J and Reloc J c= K and I c= J ;
Im ( ( lim seq ) ^\ k ) = 0 ;
( sin . x ) ^2 <> 0 & ( sin . x ) ^2 <> 0 ;
sin * ( cos * ( sin - cos ) ) is_differentiable_on Z ;
t3 . n = t3 . n .= s . n ;
dom ( ( element ) (#) ( F | A ) ) c= dom F ;
W1 . x = W2 . x & W2 . x = W2 . x ;
y in W .vertices() \/ W .vertices() \/ W .vertices() ;
( k <= len ( v | ( k -' 1 ) ) ) ;
x * a divides y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: S c= proj2 .: P ;
h . p4 = g2 . I .= ( g2 . I ) `1 ;
IT = ( U /. 1 ) `1 .= ( U /. 1 ) `1 ;
f . rp1 in rng f & rp2 in rng f ;
i + 1 + 1 <= len - 1 ;
rng F = rng ( F . n ) .= rng ( F . n ) ;
mode multiplicative non empty multMagma ;
[ x , y ] in A ~ ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of support ( m ) c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
( seq ^\ k1 ) ^\ ( k + 1 ) is lower ;
len ( F . i ) = len I & len ( F . i ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number , p be Real ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of \HM { the InternalRel of T : Y is empty } ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
cluster J => y -> total for Function of J , J ;
K c= 2 |^ the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
attr a <> {} means a = 1 & a = 1 ;
assume that exp 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 FS2 be non empty thesis ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x as Subset of m ;
let A , B , C be Element of R ;
cluster non empty strict for be a9 of b9 ;
rng c `1 misses rng ( e `1 ) \/ rng ( e `2 ) ;
z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( ( - cot (#) ( sin * f ) ) `| Z ) ;
the component of Q c= UBD ( A ) & the component of Q c= UBD ( A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) & g2 in dom ( 1 / 2 ) ;
redefine pred f = u * a ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b be Nat ;
assume that S = S2 and p = p2 and S = p1 and S = p2 ;
gcd ( n1 , n2 , n3 ) = 1 & gcd ( n2 , n1 , n2 ) = 1 ;
set os = a * ( - 1 / 2 ) , os = - 1 / 2 ;
seq . n < |. r1 .| & |. seq . n .| < r1 ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a ;
ex c being Nat st P [ c ] ;
set g = { n / 1 : n in NAT } ;
k = a or k = b or k = c ;
a9 , b9 , c9 is_collinear & b9 , c9 , a9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
Ip1 . x = f . x .= 0 .= 0 ;
W3 .last() = W3 . 1 & W . 2 = W . 1 ;
cluster -> trivial for subgraph of G , finite _Graph ;
reconsider u = u as Element of Bags X ;
A in B |^ implies A , B |^ C |^
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is_\in the _ of L~ f2 implies f1 = f2
( f /. 2 ) `2 <= ( q `2 ) ^2 ;
h is_dom Cage ( C , n ) & h is_width Gauge ( C , n ) ;
( b `2 ) ^2 / ( b `1 ) ^2 <= ( p `2 ) ^2 / ( p `2 ) ^2 ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom max ( - f , - f ) ;
p2 in Na . p1 & p2 in Na . p2 ;
len ( the_left_argument_of H ) < len ( H ) & len ( the_right_argument_of H ) < len ( H ) ;
F [ A , FF . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def3 : A c= C |^ A ;
assume r1 <> 0 or r2 <> 0 or r1 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) & rng q1 c= rng C1 ;
A1 , L , A2 , A3 , A3 is_collinear & A1 , A2 , p3 is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in being element ( p , Sp ) & c in A ( ) ;
then S is negative & P-2 [ S ] ;
Cl ( [#] T ) = [#] T & Cl ( [#] T ) = [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of V ;
v , v `2 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 ~ ;
consider H1 such that H = 'not' H1 and H1 in F ;
1_ G c= ( REAL * ( p2 - 1 ) ) * ( ( p2 - 1 ) * ( p2 - 1 ) ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A |^ 2 ;
set vas = ( vseq /. n ) `1 , vR = ( vseq /. n ) `1 ;
r = 0. ( \langle TOP-REAL n , \Vert \cdot \Vert *> ) ;
( f . p4 ) `1 >= 0 & ( f . p4 ) `1 >= 0 ;
len W = len ( W | ( len W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 / ( W + 1 ) does not destroy b1 , T ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i - 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 ] -> non pair for set ;
downarrow a /\ downarrow t is Ideal of T & downarrow a /\ downarrow t is Ideal of T ;
let X be with_NAT -defined non empty set , f be Function ;
rng f = \rm \rm \rm \rm \rm *> ( S , X ) ;
let p be Element of B , x be the bound Element of S ;
max ( N1 , 2 ) >= N1 & max ( N2 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume that i in I and R2 . i = R ;
i = j1 & p1 = q1 & p2 = q2 & p1 = p2 ;
assume gR in the right of g & gR in the carrier of g ;
let A1 , A2 be Point of S , x be Point of S ;
x in h " P /\ [#] ( T1 | P ) ;
1 in Seg 2 & 1 in Seg 3 & 2 in Seg 3 ;
reconsider X9 = X as non empty Subset of Tsuch that X9 is non empty and x in X9 ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 <= len g2 implies i1 <= i2 + len g2
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & u in the carrier' of G1 ;
y = Re y + ( Im y ) * i ;
( width ( ( - 1 ) |^ p ) ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & x2 is_differentiable_on ]. a , b .[ ;
rng ( M | ( Seg len M ) ) c= rng ( D2 | ( Seg len D2 ) ) ;
for p being Real st p in Z holds p >= a
( cn ) * ( ( cn ) * ( ( cn ) ) * ( ( cn ) ) ) = proj1 ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path M ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h divides gg . ( mod P , T ) ;
reconsider i1 = i-1 - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , a be Real ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i9 = i - 1 as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the Element of B ) . n ;
len , i in Seg len ( f3 | ( 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 Element of T2 ;
G * ( B * A ) = id o1 & G * ( B * A ) = id o2 ;
assume that p , u , v , w is_collinear and p , q , w , y ;
[ z , z ] in union rng ( F . m ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , S = S . $1 ;
LIN a1 , a3 , b1 & LIN a1 , a2 , b2 ;
f " ( f .: x ) = { x } ;
dom ( w2 ) = dom ( r12 ) .= dom ( r12 ) ;
assume that 1 <= i and i <= n and j <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I * ( i , j ) = 0. ( K , n , m ) ;
|. f . ( s . m ) -g - g .| < g1 ;
qx . x in rng ( qx | ( Seg n ) ) ;
Carrier ( L7 ) misses Carrier ( L7 ) \/ Carrier ( - 1 ) ;
consider c being element such that [ a , c ] in G ;
assume that & N5 = o8 and o6 = o7 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ ( C + 1 ) ) * ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= x ;
p `2 - q `2 <> 0. TOP-REAL 2 & p `2 <= 0 ;
cluster being aa] ( S , T ) -> non empty ;
let x be Element of S ~ ;
the Arrows of F is one-to-one & the Arrows of F is one-to-one ;
|. i .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom ( P * ( f | X ) ) ;
exp_R * ( n + 1 ) ! > 0 * th ;
S c= ( A1 /\ A2 ) /\ ( A2 /\ A3 ) ;
a3 , a4 // a3 , a4 & a3 , a4 // a3 , a4 ;
then dom A <> {} & dom A <> {} & rng A c= dom B ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y & y in X implies x = y & y = x
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = v /. ( i + 1 ) ;
x = r . n .= ( r . n ) * ( r . n ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & rng g c= the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: A2 , A1 :] & dom d2 = [: A2 , A1 :] ;
0 < ( p `1 / |. 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 ) ;
cluster O := F -> \HM { \HM { a } : a in X } -> operation ;
let U1 , U2 be non-empty MSAlgebra over S , a be Element of U1 ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x as Subset of V ;
x in the carrier of Lin ( A ) & y in the carrier of Lin ( B ) ;
let I , J be parahalting Program of SCM+FSA , a be Real ;
assume that - a is lower and b is lower and a is lower ;
Int Cl ( Int Cl A ) c= Cl Int Cl ( Int Cl A ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 / p2 `1 ) ^2 / ( p2 `2 ) ^2 <= ( p2 `1 ) ^2 / ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( T | ( [#] T ) ) ;
set S = the carrier of T , T = the carrier of T ;
set I8 = being Element of NAT , f = f |^ n ;
len p -' n = len ( p -' n ) ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n6 - n7 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( ( s | u ) | v ) ;
let q\mathopen ( q , m ) , q\mathopen ( p , m ) ) be State of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 & c9 in the carrier of S2 ;
c1 /. n1 = c1 . n1 & c2 /. n2 = c2 . n2 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( f * SS ) . x .= ( f * SS ) . x ;
consider x being element such that x in q1 q1 in _ in P ;
assume r in ( dist ( o ) ) .: P ;
set i2 = \mathopen { - ( u `1 ) / 2 } ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i ;
reconsider m = ( x - 1 ) / 2 as Element of REAL ;
let U1 , U2 be Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 < len p1 + 1 ;
let T1 , T2 be Scott Scott topological \looparrowleft of L , x be Element of T ;
then x <= y & .. ( x ) c= .. ( y ) ;
set M = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of a9 ) . ( a , b ) ) c= dom H & rng ( ( the_arity_of o ) . ( a , b ) ) c= dom H ;
z1 " = ( z " ) * ( z " ) .= ( z " ) * ( z " ) ;
x0 - r / 2 in L /\ dom f /\ dom f ;
then w is being such that rng w /\ ( S \ { w } ) <> {} ;
set x-10 = xZ ^ <* Z *> ^ <* Z *> ^ ( x , Z ^ <* Z *> ) ;
len w1 in Seg ( len w1 + len w2 ) & len w1 in Seg ( len w1 + len w2 ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of \times \mathopen { k } , k ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( G * ( i1 , 1 ) `1 ) ^2 <= ( G * ( i1 , 1 ) `1 ) ^2 ;
rng g c= L~ ( g | ( L~ g ) ) \/ LSeg ( g , 1 ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider xm = xM , xm = xM as VECTOR of M ;
dom ( f | X ) = X /\ dom f /\ X .= dom f ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x as Element of REAL m ;
assume i in dom ( a (#) p ) ;
m . ( k + 1 ) = p . ( k + 1 ) ;
a / ( s . m - n ) / ( s . m - n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ C1 = B2 \/ C2 \/ C2 & B2 \/ C2 = B2 \/ C2 ;
X . i = { x1 , x2 } . i .= { x1 , x2 } . i ;
r2 in dom ( h1 + h2 ) /\ dom ( h2 + h2 ) ;
- 0. R = a & b-0 = b ;
FF is_halting_on t2 , Q8 & which is_halting_on t2 , Q8 ;
set T = -> in in -> InInInIn0 ( X ) ;
Int Cl ( Int Cl R ) c= Int Cl R & Int Cl R c= Cl R ;
consider y being Element of L such that c . y = x ;
rng ( FF . x ) = { F . x } & rng ( F . x ) c= rng ( F . x ) ;
G-23 ( { c } ) c= B \/ S \/ S ;
f[#] ( [: X , Y :] , X ) is Relation of [: X , Y :] , Y ;
set Rx = the Point of ( ( TOP-REAL 2 ) | P ) ;
assume that n + 1 >= 1 and n + 1 <= len M ;
k2 be Element of NAT , k be Nat ;
reconsider pu = u as Element of ( ( TOP-REAL n ) | R ) ;
g . x in dom f & x in dom g implies f . x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , N ) ;
len ( ( P . i ) `1 ) <= len ( ( P . i ) `1 ) ;
x " in the carrier of A1 & x in the carrier of A1 & x in the carrier of A1 ;
[ i , j ] in Indices ( ( A @ ) * ( i , j ) ) ;
for m being Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= a1 . k ;
let f be PartFunc of REAL , REAL , x be Real ;
rng f = the carrier of ( \bf Y ) | ( the carrier of A ) ;
assume s1 = sqrt ( 2 * p ) - sqrt ( 2 * p ) ;
attr a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of as Subset of I1 ;
reconsider X0 = X , Y0 = Y as Point of X ;
let f be PartFunc of REAL , REAL , x be Real ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 be Relation of T , such that t-3 = tt1 and t-3 is binary ;
Q [ e-14 \/ { v-5 } , f ] ;
g \circlearrowleft ( W-min L~ z ) = z & g /. ( E-max L~ z ) = z ;
|. |[ x , v ]| - |[ x , y ]| .| = vy0 ;
- f . w = - ( L * w ) ;
z - y <= x iff z <= x + y & z <= y + x ;
( 7 / p1 ) ^2 > 0 & ( 1 / p1 ) ^2 > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 & ( f | X ) . x2 = f . x2 ;
( ( ( tan (#) sec ) `| Z ) . x ) in dom ( sec (#) sec ) ;
i2 = ( f /. ( len f ) ) `1 & i2 = ( f /. ( len f ) ) `1 ;
X1 = X2 \/ ( X1 \ X2 ) & X2 = X1 \/ X2 ;
[. a , b , 1_ G .] = 1_ G & a = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be Function of V , W ;
dom g2 = the carrier of I[01] & rng g2 c= the carrier of I[01] ;
dom f2 = the carrier of I[01] & rng f2 c= the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X & ( proj2 | X ) .: Y = proj2 .: Y ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 < x0 + r ;
|. ( f /* s ) . k - G . ( n + 1 ) .| < r ;
len Line ( A , i ) = width A & len Line ( A , i ) = width A ;
SFinSequence / ( g , m ) = ( S . g ) / ( g , m ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
( intloc 0 ) in dom Initialized ( p +* I ) ;
i1 , i2 i2 & 3 , 4 does not contradiction & ( a , b ) := ( a , b ) ;
arccos r + arccos r = ( cos r ) + 0 & cos r = ( cos r ) + 0 ;
for x st x in Z holds f2 is_differentiable_in x & f2 is_differentiable_in x ;
reconsider q2 = ( q - x ) / 2 as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 + 1 ;
assume f in the carrier of [: X , Y :] ;
F . a = H / ( x , y ) . a ;
( true T ) at ( ( C , u ) at ( C , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] & 1 in dom f ;
( p2 `1 - x1 ) / 2 > - g / 2 & ( p2 `1 - x1 ) / 2 > - g / 2 ;
|. r1 - `2 .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b ;
D0W .order() = DW .2 ( n ) + 1 ;
i1 = ma + n & i2 = 0. TOP-REAL 2 & i1 = 0. TOP-REAL 2 ;
f . a [= f . ( f ^ O1 ) ;
attr f = v & g = u & f + g = 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 * ( len R4 * ( len R4 * ( len R4 * ( len R4 * ( I , 1 ) ) ) ) ) ) ;
set h = the continuous Function of X , R ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is negative holds P7 [ H ] ;
set b9 = S5 ^\ ( i + 1 ) , S = ( S ^\ ( i + 1 ) ) ;
Hom ( a , b ) c= Hom ( a opp , b ) ;
( 1 / ( n + 1 ) ) < ( 1 / s ) " ;
( [ l , cod l ] ) `1 = [ dom l , cod l ] `1 .= dom l ;
y +* ( i , y /. i ) in dom g ;
let p be Element of QC-WFF ( Al ) , x be Element of D ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ p1 ) \/ rng ( f /^ p2 ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 - 1 ;
assume x in ( ( 2 /\ K0 ) \/ ( ( 2 /\ K0 ) /\ ( ( 2 /\ K0 ) \/ ( ( 2 /\ K0 ) /\ ( ( 2 /\ K0 ) /\ ( 2 /\ K0 ) ) ) ) ) /\ ( ( 2 /\ K0 ) /\ ( (
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . O ) `2 ;
f , g be Function of I[01] , TOP-REAL 2 , a , b be Real ;
k1 -' k2 = k1 - k2 & k2 -' k2 = k2 -' k2 + 1 ;
rng ( seq ^\ k ) c= ]. x0 - r , x0 .[ & ( seq ^\ k ) . n < x0 ;
g2 in ]. x0 - r , x0 .[ & g2 in ]. x0 - r , x0 .[ ;
sgn ( p `1 , K ) = - 1. K & sgn ( p `2 , K ) = - 1 ;
consider u being Nat such that b = p |^ y * u ;
ex A being non empty as normal NAT st a = Sum A & A is limit_ordinal ;
Cl ( union ( ( Cl H ) \ ( Cl H ) ) ) = union ( ( Cl H ) \ ( Cl H ) ) ;
len t = len t1 + len t2 & len t1 = len t1 + len t2 ;
v-29 = v + w |-- ( A , B ) ;
v <> DataLoc ( t3 . GBP , 3 ) & v <> DataLoc ( t3 . GBP , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot { y } ) . s = s . ( \dot { y } ) ;
{ s : s < t } in INT & t = {} implies t = {}
s ` \ s = s ` \ 0. X .= ( 0. X ) \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A & $1 in B ;
( 329 + 1 ) ! = 3329 ! * ( 329 + 1 ) ;
1. ( A , A ) = 1. ( A , A ) .= 1. ( A , A ) ;
reconsider y = y as Element of ( len y ) -tuples_on REAL ;
consider i2 being Integer such that y0 = p * i2 and i2 in dom f ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { z } ;
consider Z being set such that ( for n being Nat holds Z in F iff n in F ) ;
let f be Function of I[01] , TOP-REAL n , a be Real ;
( SAT M ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , x be Element of REAL n ;
reconsider l = 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 is not empty & 'not' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. ( x - g ) - ( g - x ) .|| < r2 & ||. ( x - g ) - x .|| < r2 ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & b9 , c9 // b9 , c9 ;
1 <= k2 -' k1 & k2 + 1 <= k2 & k2 + 1 <= len f & k2 + 1 <= len f ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( ( q `2 / |. q .| - sn ) / ( 1 + sn ) ) ^2 < 0 ;
( E-max C ) `1 in RightComp ( ( R /. 1 ) ) `1 & ( E-max L~ f ) `1 = ( W-bound L~ f ) `1 ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) . m ) ;
LIN b , a , c or LIN b , c , a ;
p `1 , a // a `1 , b or p `1 , a // b `1 , a `1 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is being set ;
F | ( N2 , S ) = CircleMap * ( F * ( N2 , S ) ) ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r0 ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } & the carrier of V = { 0. V } ;
rng ( ( - 1 ) (#) ( cos | [. - 1 , 1 .] ) ) = [. - 1 , 1 .] ;
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( vseq . n ) - ( vseq . m ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t11 , t2 = t22 as 0 string of S2 , t2 = S S ;
reconsider x-29 = seq . n , xp = seq . n as sequence of REAL n ;
assume that that that that that that C meets L~ go and L~ go /\ L~ pion1 = L~ go /\ L~ pion1 and L~ pion1 /\ L~ pion1 = { pion1 /. len pion1 } ;
- ( ( - 1 ) (#) ( F . n ) ) < F . n - ( F . n ) ;
set d1 = being element , d2 = dist ( x1 , z1 ) , d2 = dist ( x2 , z2 ) ;
2 |^ ( q -' 1 ) = 2 |^ ( q -' 00 ) - 1 ;
dom ( ( len d6 ) | ( Seg len d6 ) ) = Seg len d6 ;
set x1 = - k2 + |. k2 .| + 4 * ( - 1 ) ;
assume for n being Element of X holds 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( LT ) ) c= I2 & the carrier of ( Carrier ( LT ) ) 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 w.r.t. over {} ;
Z c= dom ( ( - 1 / 2 ) (#) ( ( sin * f1 ) ^ ) ) ;
|. 0. TOP-REAL 2 - ( q `1 / |. q .| - cn ) / ( 1 + cn ) .| < r ;
ConsecutiveSet2 ( A , L ) c= ConsecutiveSet2 ( A , ConsecutiveSet ( d , L ) ) ;
E = dom ( L . m ) & L . m is_measurable_on E & L . m is_measurable_on E ;
C / ( A + B ) = C / B * C ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of W2 implies W1 is Subspace of W2
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) ;
pred x > 0 means : Def3 : ( 1 - x ) |^ 2 = x |^ ( - 1 ) ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) \/ LSeg ( f , i ) ;
consider p being Point of T such that C = [. p , R .] ;
b , c are_connected & - C , - C are_Path - a , b implies - C , - C are_Path
assume that f = id the carrier of O and f is Function of the carrier of O , the carrier of O ;
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 ;
A1 in the carrier of ( G . k ) & A2 in the carrier of ( G . k ) ;
|. - x .| = - x .= - x .= - x .= - x ;
set S =
Fib ( n ) * ( 5 * ( 5 + 1 ) ) >= 4 * ( 5 * ( n + 1 ) ) ;
( vseq /. ( k + 1 ) ) = ( vseq . ( k + 1 ) ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) mod i ;
Indices M1 = [: Seg n , Seg n :] & Indices M1 = [: Seg n , Seg n :] ;
Line ( S\mathopen { - } j , j ) = Sj . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| Y. implies |. ( Re f ) * ( ( card b ) * h ) .| is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ ( b1 ^ <* x1 *> ) ;
MS is_halting_on IExec ( I , P , s ) , P , s ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x .| = - x + y & |. x - y .| = - x ;
LIN c , q , b & LIN c , q , c & LIN c , q , a ;
fsuch . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + ( y1 + z1 ) ;
fbeing . a = fbeing . a & f`2 in InputVertices S & f`2 in InputVertices S ;
( p `1 ) ^2 / ( ( E-max C ) `1 ) ^2 <= ( ( E-max C ) `1 ) ^2 / ( ( E-max C ) `1 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E7 = Cage ( C , n ) \circlearrowleft E8 ;
( ( E-max C ) `1 ) ^2 >= ( ( E-max C ) `1 ) ^2 + ( E-max C ) `1 ;
consider p such that p = p-20 and s1 < p and p < s2 and s2 <= p ;
|. ( f /* ( s * F ) ) . l - G .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len ( Line ( N , k + 1 ) ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = lim ( f2 /* s1 ) ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = REAL m & rng s c= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * t ) ) ;
dom B = 2 \ { {} } & rng B c= the carrier of V ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of [: Y1 , Y2 :] as Subset of [: X1 , X2 :] ;
1 in the carrier of [. 1 / 2 , 1 .] & 1 / 2 <= 1 ;
for L being complete LATTICE holds \mathbb L , L are_isomorphic ( L ) is isomorphic ;
[ gi , gj ] in Ii \ ( 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 f2 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 and f3 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F . ( len F ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , ( 1 - g ) ) / ( f . c ) ) <= h . c ;
set G3 = the subgraph of G , a = the Vertex of G , b = the Vertex of G , c = the carrier of G , d = the carrier of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m / p .| < d / p . m / p . m ;
for x being element st x in in in in in ( B * u ) holds x in in in in in in in in in in in in B ;
P = the carrier of ( TOP-REAL n ) | K1 & Q = the carrier of ( TOP-REAL n ) | K1 ;
assume that p11 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) and p11 in LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , cod f ) ;
2 * a * b + ( 2 * c ) <= 2 * C1 * C2 + ( 2 * c ) ;
f , g , h be Point of the carrier of X , g be Point of X , h be Point of Y ;
set h = Hom ( a , g (*) f ) ;
then idseq ( n ) | Seg m = idseq ( m ) | Seg m & m <= n ;
H * ( g " * a ) in the carrier of H * ( g " * a ) ;
x in dom ( ( ( id Z ) (#) ( ( id Z ) ^ ) ) `| Z ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p2 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is BDD of A means : Def4 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set ) = union rng ( $2 | $1 ) ;
n + - n < len ( pT + - n ) + - n ;
attr a <> 0. K means : Def3 : the_rank_of M = rk ( a * M ) ;
consider j such that j in dom \mathbb Y. and I = len TOP-REAL m + j ;
consider x1 such that z in x1 and x1 in P8 and x = [ x1 , x1 ] ;
for n ex r being Element of REAL st X [ n , r ]
set CS1 = Comput ( P2 , s2 , i + 1 ) , CS2 = Comput ( P2 , s2 , i + 1 ) , CS2 = P2 ;
set cv = 3 / ( 2 * a , c ) , cv = - 1 / ( 2 * a , c ) , cv = - 1 / ( 2 * a , c ) , cv = - 1 / ( 2 * a , b ) , cv = - 1 / ( 2 *
conv @ W c= union ( F .: ( E " ( W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( #Z 2 ) * ( ( arccot ) ) * ( 1 + 1 ) ) ;
r3 <= s0 + ( r2 - |. v2 - v1 .| ) / 2 * ( 1 - r ) ;
dom ( f (#) f4 ) = dom f /\ dom ( f (#) f3 ) .= dom f /\ dom ( f (#) f3 ) ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= Seg ( k + 1 ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider gg = gp , gp = gp as Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s ) ;
I . ( L . ( J . x ) ) = ( I * L ) . x ;
y in dom *> *> <* *> *> <* *> -: ( ( Frege ( A ) ) . o ) ;
for I being non degenerated commutative commutative commutative commutative commutative commutative commutative commutative associative commutative commutative distributive non empty doubleLoopStr holds I is commutative
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 in the carrier of [. a , b .] ;
v . ( lpp . i ) = ( v *' ( lpp ) ) . i ;
consider n being element such that n in NAT and x = ( sn ) . n ;
consider x being Element of c such that F1 . x <> F2 . x and F1 . x <> 0 ;
Funcs ( X , 0 , x1 , x2 , x3 ) = { E } & card X = card { E } ;
j + ( 2 * k9 ) + m1 > j + ( 2 * k9 ) + ( 2 * k9 ) ;
{ s , t } on A3 & { s , t } on B2 & { s , t } on B2 ;
n1 > len crossover ( p2 , p1 , n1 , n2 ) & n2 <= len crossover ( p2 , p1 , n1 , n2 ) ;
( for m2 being bag of n holds HT ( m2 , T ) ) . HT ( m2 , T ) = 0. L ) ;
then H1 , H2 are_: for H being Subgroup of H1 st H , H1 are_commutative holds ( H , H1 ) / ( x , H2 ) is that H1 , H2 / ( x , y ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m )
( ( N-min L~ f ) .. f ) .. f > 1 & ( ( N-min L~ f ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL , x be Point of S ;
DigA ( t-23 , z9 ) is Element of k -tuples_on REAL , ( k + 1 ) -tuples_on REAL ;
I \mathop { \rm 223j , I \mathop { - } = k2 & I \mathop { - 1 , - 1 } = k2 ;
u9 ~ = { [ a , u9 ] } & u9 in { [ a , u9 ] } ;
( w | p ) | ( p | ( w | ( w | ( w | ( w | ( w | ( w | w ) ) ) ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u1 in W2 and u2 in W2 and u2 in W2 ;
for y st y in rng F ex n st y = a |^ n & F . n in rng F ;
dom ( ( g * ( ( ( ( g * ( {} , V ) ) } ) | K ) ) ) | K ) = K ;
ex x being element st x in ( ( U0 U1 ) \/ A ) . s & x in ( ( U0 U1 ) \/ B ) . s ;
ex x being element st x in ( ( ( ( ( O ) \/ A ) . s ) . x ) . x ) . s ;
f . x in the carrier of [. - r , 1 .] & f . x in [. - r , 1 .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 meet X2 ) <> {} implies X1 is SubSpace of X2 & X2 is SubSpace of X1
L1 /\ LSeg ( p11 , p2 ) c= { p11 } /\ LSeg ( p10 , p2 ) \/ { p10 } /\ LSeg ( p10 , p2 ) ;
( b + ( be - be ) ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of IT such that z = y and P [ z ] and P [ z ] ;
( the sequence of ( iff the sequence of X ) ) . ( x - x0 ) <= e ;
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 2 + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q in the carrier of ( ( TOP-REAL 2 ) | K1 ) ;
f | ( E-4 ` ) = g | ( E-4 ` ) .= g | ( EK ` ) ;
reconsider i1 = x1 , i2 = x2 , z = x3 as Element of NAT ;
( a * A ) @ = ( a * ( A @ ) ) @ ;
assume ex n0 being Element of NAT st f / ( n0 , n ) is min ;
Seg len ( ( ( the multF of G ) * ( f2 , i ) ) | ( Seg len ( f2 , i ) ) ) = dom ( ( the multF of G ) * ( f2 , i ) ) | ( Seg len ( f2 , i ) ) ;
( Complement ( Complement A1 ) ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . m c= ( Complement A1 ) . m ;
f1 . p = p8 & g1 . p = d & g1 . p = d & g1 . p = c ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( ( |. x .| ) to_power n ) <= ( ( r2 ) to_power n ) to_power ( n + 1 ) ;
Sum ( F ) = Sum f & dom ( F ) = dom g & for i st i in dom F holds F . i = f . i ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W2 ;
||. ( ( t . x ) - ( t . x ) ) .|| = lim ( ||. ( ( t . x ) - ( t . x ) ) ) ;
assume that i in dom D and f | A is lower and g | A is lower and g | A is lower ;
( ( p `2 / |. p .| - cn ) / ( 1 + cn ) ) ^2 <= ( ( - ( 1 + cn ) ) / ( 1 + cn ) ) ^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 ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies T is countable countable & T is countable
width B |-> 0. K = len ( B * i ) .= len ( B * i ) .= len B ;
attr a <> 0 means a <> 0 implies ( A -- a ) -- a = ( A Y. a ) Y. ( a \in A ) ;
then f is_differentiable on pdiff1 ( f , 1 ) & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and b <> 1 and c <> 0 and c <> 0 ;
w1 , w2 in Lin { w1 , w2 : w1 in Lin { w1 , w2 } } ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= p2 . IC Comput ( p2 , s2 , k ) ;
ind ( T-10 | b ) = ind b .= ind b .= ind b .= ind b ;
[ a , A ] in the carrier of Line ( 2 , card A9 ) & [ a , A ] in the carrier of Line ( 2 , card A9 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in ( the Arrows of C ) . ( o2 , o2 ) ;
( ( a 'imp' CompF ( PA , G ) ) ) . z = FALSE & ( a 'imp' ( a 'imp' ( b 'imp' ( a 'imp' ( b ) ) ) ) . z = FALSE ;
reconsider phi = phi /. 11 , phi = phi /. 11 as Element of being Element of from ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 1 - 1 ;
( delta ( D ) ) * ( f . ( upper_bound A ) - lower_bound A ) < r ;
[ f21 , f22 ] in the carrier of [: A , B :] ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 ;
consider z being element such that z in dom g2 and p = g2 . z and x = g2 . z ;
[#] ( V1 ) = { 0. V1 } .= the carrier of ( ( the carrier of V1 ) \ { 0. V1 } ) ;
consider P2 being FinSequence such that rng P2 = M 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 < x0 and s < x0 ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ <* p3 *> ^ <* p3 *> .= h ;
c / ( |[ b , c ]| ) = c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) .= c / ( |[ a , c ]| ) ;
reconsider t1 = p1 , t2 = p2 as Term of C , V , s be State of C ;
( 1 - ( 2 * ( 1 - r ) ) / 2 ) * ( 1 - r ) in the carrier of ( TOP-REAL 2 ) | K1 ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D * ( p2 `2 ) ;
R . ( b , a ) = 2 * ||. b , b .|| .= 2 * b .= b ;
consider D1 such that B = 1- 1 * C + ( - 1 ) * A and 0 <= D1 & D1 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( a , b ) ) & rng g c= the carrier of S ;
[ P . ( l6 , P . ( l6 , P . ( l6 , Q . ( l6 , Q . ( l6 , Q . ( l6 , Q . ( l6 , Q . ( ' 6 , Q . ( ' 6 ) ) ) ) ) ) ) ) , Q . ( l6 , Q . ( l6 , Q . (
set s2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) as Matrix of len z , len z , len z , K ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / ( |[ 0 , 1 ]| ) = 1 & 1 / ( |[ 0 , 1 ]| ) = 0 ;
assume x in the left of g or x in the left of g or x in the left of g ;
consider M being strict Subspace of A9 such that a = M and T is Subspace of M ;
for x st x in Z holds ( ( ( 1 / 2 ) (#) f ) `| Z ) . x <> 0 ;
len ( W1 + ( len W2 ) ) = 1 + ( len W3 + m ) .= len ( W1 + ( len W2 + m ) ) ;
reconsider h1 = ( vseq . n ) - ( vseq . n ) as Lipschitzian Lipschitzian Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the |= of s1 and F in the |= of s2 and F in the |= of s2 ;
( ( for x , y being Element of NAT holds x , y ) gcd ( x , y ) = gcd ( x , y )
for u being element st u in Bags n holds ( p *' m ) . u = p . u
for B being Subset of u9 st B in E holds A = B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree ( p ) \/ ( dom <* p *> ) ;
x in { X where X is Ideal of L : X is Ideal of L & X is directed } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 implies W1 /\ W2 = W2 /\ W3
( for a , b holds a * ( id a + b ) ) * ( a * b + b * a ) = 1. ( K , n )
( ( X --> f ) . x ) . x = ( X --> dom f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( - ( / ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( ( f1 (#) f2 ) | X ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and b2 . r = c2 . r ;
ex P st a1 on P & a2 on P & a3 on P & a4 on P & a2 on P & a3 on P & a4 on P & a4 on P & a2 on P & a3 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 / |. p .| >= cn & p `2 <= 0 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) . ( O1 , O1 ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = SubFrom ( a , intloc 0 ) , I2 = [ a , intloc 0 ] , I1 = [ a , intloc 0 ] ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) /\ ( the carrier of L2 ) & the carrier of X c= the carrier of L1 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a ;
reconsider ez = ez , fw = fw , fw = f-5 , fw = f-5 as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( ( Cl f ) . O ) ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 and S . m in U1 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) * g ) . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = ( A + $1 ) + ( $1 + 1 ) ;
the left of - g = the left of g & the left of - g = the strict strict strict Subgroup of g implies g = f
reconsider pp = x , pp = y , pp = z as Point of ( TOP-REAL 2 ) | K1 ;
consider g2 such that g2 = y and x <= g2 and 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 ( x2 ^ y2 ) = len x2 + len y2 ;
for x being element st x in X holds x in the set of holds x in the set of the set of m & m <= n implies x = m
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} & LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func for X be set means : Def3 : for x being set holds x in it iff x is thesis & x is Function of X , Y ;
len ( such that ( CR ( C , n ) ) /. ( len ( C , n ) ) ) <= len ( ( C , n ) /. ( len ( C , n ) ) ) ;
attr K is Field means : Def3 : a <> 0. K & v . ( a |^ i ) = a * v . a ;
consider o being OperSymbol of S such that t `1 . {} = [ o , the carrier of S ] and t `1 = [ o , the carrier of S ] ;
for x st x in X ex y st x c= y & y in X & y is element & f . x = f . y ;
IC Comput ( P-6 , k ) in dom ( ( n + 1 ) .--> ( n + 1 ) ) ;
attr q < s & r < s implies ]. r , s .[ c= ]. p , q .[ & ]. p , q .[ c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . c ) `1 and c in X ;
func the ResultSort of S2 -> ResultSort of S2 means : Def4 : for x being set holds it . x = id the carrier of S2 & x is Function of the carrier of S2 , the carrier of S2 ;
set y9 = [ <* y , z *> , f2 ] , z9 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( f1 + #Z 2 ) ) ) ) ) `| Z ) ) ;
r-7 in Int cell ( GoB f , i , GoB f ) \ L~ f & r-7 in L~ f \ L~ f & r-7 in L~ f \ L~ f ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) * ( i , 1 ) ) `2 ;
set Y = { a "/\" a ` : a in X } ;
i -' len f <= len f + ( len f1 -' len f + 1 ) - len f + ( len f -' len f + 1 ) ;
for n ex x st x in N & x in N1 & h . n = x- x0 & h . n > x0 ;
set s0 = ( ( > a , I , p , s ) +* ( a , I ) ) . i ;
( p . k = 1 or p . k = - 1 ) or p . k = 1 & p . k = - 1 & p . k = - 1 ;
u + Sum ( L ) in ( U \ { u } ) \/ ( L \ { u } ) ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x9 in V1 and x = f . x9 ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( : len p = ( len p ) + ( len p ) ) ;
g + h = gg + h1 & ||. g + h .|| = g + h + h ;
L1 is distributive & L2 is distributive implies L1 is distributive & L2 is distributive & L1 is distributive & L2 is distributive & L2 is distributive
pred x in rng f & y in rng ( f | x ) & f . x = f . y implies f . x = f . y ;
assume that 1 < p and ( 1 + p ) * q = 1 and 0 <= a and a <= b and b <= 1 ;
F* ( f , <* <* 0. F_Complex *> *> ) = rpoly ( 1 , <* the carrier of F_Complex , 1 ) *> + 1 ;
for X being set , A being Subset of X , B being Subset of X holds A ` = {} implies B = {} & B = {}
( ( ( ( ( ( ( ( ( ( ( ( ( X X ) ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( X X ) ) ) ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( X X ) ) ) ) ) ) ) )
for c being Element of the \rbrack of the Sorts of A , a being Element of the Sorts of A holds c <> a implies c <> a
s1 . GBP = ( Exec ( i2 , s2 ) ) . GBP .= Exec ( i2 , s2 ) . GBP .= Exec ( i2 , s2 ) . GBP .= s . GBP ;
for a , b being Real holds |[ a , b ]| in ( y >= 0 implies a >= 0 ) & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y = ( x \ y ) ` ;
mode BCK-algebra of i , j , m , n , m , n , m , m , n , m , n , m , n , m , n , m , n , m , n , m ) is BCK-algebra implies n + m in dom m
set x2 = |( ( Re y ) , ( Im x ) )| ;
[ y , x ] in dom u9 & u9 . ( y , x ) = g . y & u9 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & A c= divset ( D , k ) ;
0 <= \delta ( S2 . n ) & |. \delta ( S2 . n ) - 0 .| < e / 2 ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) / ( 2 * a ) ;
for x , y being set st x in R" & y in R" holds x , y are__let x , y
deffunc FF ( Nat ) = b . ( $1 * ( M * ( G * ( $1 , 1 ) ) ) ) . ( ( M * ( $1 , 1 ) ) . ( ( M * ( $1 , 1 ) ) ) . ( ( M * ( $1 , 1 ) ) . ( ( M * ( $1 , 1 ) ) . ( ( $1 , 1 ) . ( ( $1 , 1 ) . (
for s being element holds s in contradiction ( f 'or' g ) iff s in -> Element of -> Element of -> Element of contradiction : s in -> Element of contradiction } ;
for S being non empty non void holds S is non void holds S is connected iff S is connected
max ( degree ( z ) , degree ( z ) ) >= 0 & degree ( z ) >= 0 ;
consider n1 being Nat such that for k holds seq . ( n1 + k ) < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( B ) ;
set n-15 = npp '&' ( M . x qua Element of BOOLEAN ) , n-15 = ( M . x ) . ( n + 1 ) ;
f " V in cluster ( the carrier of X ) & f " V in D & f " V in D implies f " V in D & f " V in D
rng ( ( a , c ) *> +* ( 1 , b ) ) c= { a , c , b } ;
consider y being as Point of G1 such that y `1 = y and dom y `1 = WW\mathopen ( x , y ) `1 ;
dom ( ( 1 / 2 ) (#) ( f | ]. - r , x0 .[ ) ) /\ ]. - r , x0 .[ c= ]. - r , x0 .[ ;
as Element of as Element of as Element of as Element of as Element of as Element of as ( ( i , j , n , - r ) ) ;
v ^ ( ( n-3 |-> 0 ) ^ ( ( Bm | c1 ) | c1 ) ) in Lin ( ( ( Bm | c1 ) | c2 ) ) ;
ex a , k1 , k2 st i = a := k1 & i = b := k2 & i = k2 & k2 <> k2 ;
t . NAT = ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT .--> ( i1 , i2 ) ) . NAT .= ( NAT .--> ( i1 , i2 ) ) . NAT ;
assume that F is bbfamily and rng p = F and rng p = Seg ( n + 1 ) and for i be 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 a , a9 , c ;
( L1 \HM { L1 \HM { O } \HM { O } ) \HM { , where O is Element of L1 : not contradiction } c= ( L1 \HM { O } ) \HM { O } }
consider F be ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( --> w ) = b * ( -w ) and 0 < a and a < b & b < 1 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos / ( x , y ) * x + cos ( x , y ) * y + cos ( x , y ) * y .= v ;
dist ( ( seq . n ) + x , g + x ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| : p <> {} & ( not p in the Sorts of A ) & not p in the Sorts of A )
consider X being Subset of Al such that X c= Y and X is finite and X is finite and inininLet X ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) + 1 ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= g & l1 <= h } ;
( Partial_Sums ( ( G . n ) vol ) ) . m <= ( Partial_Sums ( ( G . n ) " ) ) . m ;
f . y = x .= x * 1. L .= x * 1. L .= x * ( power L ) . ( y , 0 ) ;
NIC ( halt SCM+FSA , ( - ( i1 + 1 ) ) + ( - ( i1 + 1 ) ) ) = { i1 , succ ( i1 + 1 ) } ;
LSeg ( p11 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } /\ LSeg ( p1 , p2 ) .= { p1 , p2 } ;
product ( ( the support of I1 ) +* ( i , { 1 } ) ) in ( Z +* ( i , { 1 } ) ) . i ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) | ( the carrier of S2 ) .= Following ( s1 , n ) | ( the carrier of S2 ) ;
W-bound Qb <= ( q1 `1 ) / 2 & W-bound Qb <= ( W-bound Qb ) / 2 & W-bound Qb <= ( W-bound Qb ) / 2 ;
f /. i2 <> f /. ( len f + len g -' 1 ) & f /. ( len f + 1 ) = f /. ( len f + 1 -' 1 ) ;
M , f / ( x. 3 , a ) / ( x. 4 , a ) / ( x. 0 , a ) / ( x. 4 , a ) |= H ;
len ( ( P ^ ( Q ^ ( P ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( P ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^
A |^ ( m , n ) 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 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds X in Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( ( ||. v .|| ) . v ) .| & ||. v .|| <= ||. ( ||. v .|| ) . v .|| ;
for phi st phi in X holds phi in X & not phi in X & phi in X implies phi in X & phi in X ;
rng ( ( Sgm 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 & for a st a in dom c holds a = c . a ;
( the_arity_of o ) . ( <* a , b *> , <* c , d *> ) = <* ( o , b ) . ( <* a , c *> , <* c , d *> ) *> ;
consider f1 be Function of the carrier of X , R^1 such that f1 = |. f .| and f1 is continuous ;
a1 = b1 & a2 = b2 or a1 = b2 & a2 = b2 or a1 = b1 & a2 = b2 & a3 = b3 & a4 = b2 & a4 = b3 & a4 = 6 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) & D1 . ( len D1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. |[ r , s ]| .|| ) = ||. |[ r , s ]| .|| .= <* r *> . 1 .= x ;
consider n be Nat such that for m be Nat st n <= m holds C-25 . m = C-25 . m ;
consider d be Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= b ;
||. L /. h - ( K * |. h .| ) + ( K * |. h .| ) + ( K * |. h .| ) <= p0 + ( K * |. h .| ) ;
attr F is commutative means : Def3 : for b being Element of X holds F is commutative & F is commutative & F is one-to-one ;
p = - 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= 1 * p0 + 0. TOP-REAL 2 .= ( 1 - 1 ) * p2 + ( 1 - 1 ) * p1 .= p2 ;
consider z1 such that b `1 , x3 , x3 is_collinear and o , x1 , x2 is_collinear and not o , x1 , x3 is_collinear and not o , x1 , x2 is_collinear ;
consider i such that Arg ( Rotate ( s ) ) = s + Arg ( q ) and 2 * PI < r + ( 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 ;
assume that A = P2 \/ Q2 and Q1 <> {} and Q2 <> {} and Q1 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q2 <> {} and Q1 <> {} and Q2 <> {} ;
attr F is associative means : Def3 : F .: ( F .: ( f , g ) ) = F .: ( f , g ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x in { i } or m in { i } & x in { i } ;
consider k2 being Nat such that k2 in dom P-2 and l in dom P-2 and ( for k st k in dom P-2 holds P [ k , k2 ] ) ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) . [ a , a ] , ( id a ) . [ b , a ] ] = f * ( id a ) ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } ;
consider z being element such that z in dom ( ( the Sorts of F ) . ( s . i ) ) and ( ( the Sorts of F ) . ( s . i ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y ;
cell ( G , i , 1 ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F ` * b1 ) . x = ( Mx2Tran ( J , T , i ) ) . ( Y. , T ) . ( T . j ) ;
- 1 / ( - ( - 1 / 2 ) ) = mm (#) D | n .= ( - 1 ) (#) D .= ( - 1 ) (#) D .= ( - 1 ) (#) D .= ( - 1 ) (#) D ;
attr for x being set st x in dom f /\ dom g holds g . x <= f . x & g . x <= f . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' a , A , G ) 'imp' Ex ( 'not' a , B , G ) ;
LSeg ( E . i0 , F . i0 ) c= Cl RightComp Cage ( C , k + 1 ) \/ RightComp Cage ( C , k + 1 ) ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ a ) |^ k * a ;
k = ( ( commute th ) -ininininin ( ( commute I ) --> ( x , y ) ) ) . k .= ( ( commute I ) --> ( x , y ) ) . k .= ( ( commute I ) --> ( x , y ) ) . k ;
for s being State of A\mathopen ( s , n ) + ( n + 2 ) * ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a |^ x & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ;
support ( ( support max ( n ) ) \/ support ( ( support ( m ) ) ) ) c= support ( max ( n ) ) \/ support ( ( support ( m ) ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( b ) = f . ( g . a ) & phi . ( b ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i <> j ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 } = { x1 } \/ { x2 , x3 , x4 , x4 , 8 } \/ { x4 , 8 , 7 , 8 , 8 } ;
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 c= the Sorts of U2 implies the Sorts of U1 = the Sorts of U2
( - ( 2 * a ) + b ) / ( 2 * a ) + b / ( 2 * a ) - b / ( 2 * a ) > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ & P [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the ResultSort of S ) . o = r and ( the ResultSort of S ) . o = r ;
Z = dom ( ( exp_R * ( ( ( #Z n ) * ( f1 + #Z n ) ) ) ) / ( f1 + #Z n ) ) ) ;
sum ( f , SS1 ) is convergent & lim ( sum ( f , SS1 ) ) = integral ( f , S ) & lim ( ( lim ( f , S ) ) ) = integral ( f , S ) ;
( X , a9 ) => ( ( a , b ) => ( ( x , b9 ) => ( x , x9 ) ) ) in iff ( x , a9 ) => ( x , b9 ) in J
len ( M2 * M3 ) = n & width ( M2 * M3 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n & width ( M2 * M1 ) = n ;
attr X1 \/ X2 is open SubSpace of X means : Def3 : X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated & X2 , X1 are_separated & X1 , X2 are_separated implies X1 , X2 are_separated ;
for L being lower-bounded antisymmetric non empty RelStr , X being non empty Subset of L holds X "\/" { Bottom L } = { Bottom L }
reconsider f-129 = F2 . ( ( b - a ) / ( b - a ) ) as Function of ( ( the carrier of X ) \ ( b - a ) ) , M ;
consider w being FinSequence of I such that the InitS of M = <* s *> ^ w ^ <* s *> ^ w ^ w ^ w ^ ( <* s *> ^ w ) ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= ( g . ( a |^ 0 ) ) |^ ( a , a ) .= g . ( a |^ 0 ) ;
assume for i be 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 L = L & for K being Subset of X st K in C holds L /\ K <> {} & L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 & the carrier' of C2 = the carrier' of C2 & the carrier' of C2 = the carrier' of C2 ;
reconsider o-21 = o `1 , op = o `2 , op = o `1 , op = o `2 as Element of ( ( the Sorts of A ) . ( ( the Sorts of A ) . ( the_arity_of o ) ) . ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 + x3 * x3 ) + ( 0 * x3 + x4 * x4 ) = x1 + ( 0 * x2 + x3 * x4 ) .= x1 + ( 0 * x3 + x4 * x4 ) .= x1 + ( 0 * x2 + x3 ) .= x1 + ( 0 * x3 + x4 * x4 ) ;
( ( ( E . 1 ) qua Function ) " ) . 1 = ( ( E . 1 ) qua Function ) . 1 .= ( ( E . 1 ) " ) . 1 .= ( ( E . 1 ) " ) . 1 .= ( ( E . 1 ) " ) . 1 .= ( ( E . 1 ) " ) . 1 .= ( ( E . 1 ) " ) . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" ( x "\/" y ) ) ;
|. f . ( s1 . ( l1 + 1 ) - 1 ) - 1 ) - ( s1 . ( l1 + 1 ) - 1 ) .| < ( 1 - M ) + 1 ;
LSeg ( ( ( Rev Cage ( C , n ) ) * ( i , j ) ) , ( ( Rev Cage ( C , n ) ) * ( i + 1 , j ) ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - g . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) ;
assume that ColVec2Mx f in the set of Indices ( A ) and ColVec2Mx f = ( the carrier of A ) and len ( f ) = width ( A ) and width ( f ) = width ( A ) ;
len ( - M1 ) = len M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 & width ( - M1 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of \mathop { n + 1 }
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 1 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f1 , 2 ) is_partial_differentiable_in z0 , 2 & pdiff1 ( f2 , 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 ( f2 , 2 ) is_partial_differentiable_in z0 , 2 implies pdiff1 ( f2 , 2 ) is_partial_differentiable_in z0
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b ;
for c being set st not c in [. a , b .] holds not c in Intersection ( ( the open empty empty set ) , ( the topology of a ) , ( the topology of b ) ) . c
assume that V1 is closed and V2 is closed and V2 = { v + u : u in V1 & v in V1 & u in V1 & u in V1 } and V1 = { v + u : u in V1 & v in V1 & u in V1 & u in V1 } ;
z * x1 + ( 1 - z ) * x2 in M & z * x1 + ( 1 - z ) * x2 in N & z * x1 + ( 1 - z ) * x2 in N ;
rng ( ( ( P1 - ( 1 - ( x + 1 ) ) ) * ( S . ( n + 1 ) ) ) " ) = Seg card ( ( ( ( 1 - ( x + 1 ) ) * ( S . ( n + 1 ) ) ) ) ) ;
consider s2 being Integer such that s2 is convergent and b = lim s2 and for n holds s2 . n <= b . n and s2 . n <= b . n ;
h2 " . n = h2 . n & 0 < h2 . n implies 0 < ( - 1 / 2 ) / ( n + 1 ) & 0 < ( - 1 / 2 ) / ( n + 1 ) & ( - 1 / 2 ) < ( - 1 / 2 ) / ( n + 1 )
( Partial_Sums ( ||. seq .|| ) ) . m = ||. ( ( ||. seq .|| ) . m ) - ( ||. seq .|| ) . m .|| .= ||. ( ||. seq .|| ) . m - ( ||. seq .|| ) . m .|| .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b ;
- v = - 1_ G & - w = - 1_ G implies - w = - ( - 1_ G ) * ( - w ) & - w = - ( - 1_ G ) * ( - w )
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= ( k .: D ) . ( sup D ) .= ( sup D ) . ( sup D ) .= sup D ;
A |^ ( k , l ) = ( A |^ ( n , l ) ) |^ ( k , l ) .= ( A |^ ( n , l ) ) |^ ( k , l ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) & ( f . p ) `2 = ( p `1 ) / sqrt ( 1 + ( p `2 / p `1 ) ^2 ) ;
for a , b being non zero Nat st a , b are_relative_prime holds ( for n being Nat holds Q [ n ] ) & ( for n being Nat holds Q [ n ] implies k <= n ) implies k = max ( a ) + ( n + 1 )
consider A5 being countable Nat such that r is Element of CQC-WFF ( Al ) and A5 is Element of D and A5 is ( n + 1 ) -element ;
for X be non empty addLoopStr , M , N being Subset of X , x , y being Point of X st x in M & y in N holds x + y in x + y
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { x1 , y1 , y2 } & { x1 , x2 } c= { x1 , x2 } ;
h . ( f . O ) = |[ A * ( f . O ) `1 + B , C * ( f . O ) `2 + D ]| ;
( Gauge ( C , n ) * ( k , i ) ) * ( i , j ) in L~ Upper_Seq ( C , n ) /\ L~ Lower_Seq ( C , n ) ;
cluster m , n are_relative_prime -> prime for Nat means : Def3 : for Nat st m divides n & m divides n & m divides n holds it divides m & m divides n & m divides n & m divides n ;
( f * F ) . x1 = f . ( F . x1 ) & ( f * F ) . x2 = f . ( F . x2 ) & ( f * F ) . x1 = f . ( F . x2 ) ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a \ b <= c & a \ c <= c
consider b being element such that b in dom ( H / ( x , y ) ) and z = ( H / ( x , y ) ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 & W . 1 = W . 3 & e = W . 4 & W . 1 = W . 4 ;
( ] (#) ( ' ( h ) ) ) . ( 2 * n ) = ( ] (#) ( ( h h ) . n ) ) . ( 2 * n ) ;
j + 1 = ( i - len h11 + 2 ) + 2 .= i + 1 - len h11 + 2 .= i + 1 - len h11 + 2 - len h11 + 2 .= i + 1 - len h11 + 2 ;
S ^ ( S /* ( f /* s ) ) . f = S *' ( ( f ^ ) . s ) .= S *' ( f . s ) .= S *' ( f . s ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is >= >= `2 means : Def3 : for p , q st p in R & q <> p & p <> q holds ex P st P = R & P is closed & q in P & P c= R ;
dom ( product ( X --> f ) ) = meet ( dom ( X --> f ) ) .= meet ( X --> f ) .= meet ( X --> f ) .= meet ( X --> f ) .= dom f ;
upper_bound ( proj2 .: ( Lower_Arc C /\ Lower_Arc C ) /\ holds ( proj2 .: ( Lower_Arc C /\ Lower_Arc C ) ) . ( w + 1 ) <= upper_bound ( proj2 .: ( Lower_Arc C /\ Vertical_Line ( w ) ) ) )
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - p .| < r
i * f-28 - f<> = i * f-28 - ( i * f99 - ( i * f99 ) ) .= i * f99 - ( i * f-32 ) .= i * f99 - ( i * f-32 ) ;
consider f being Function such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g = [ g1 , g2 ] ;
func d |-count n -> Nat means : Def3 : d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n & d |^ ( n + 1 ) divides n ;
felement . [ 0 , t ] = f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = h . M or t = h . N ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . m ) ) < 1 / ( ( seq . m ) + ( seq . m ) ) ;
( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 <= ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ;
h0 . ( i + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 -' 1 ) + 2 - 1 .= len h21 + 2 - 1 ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S : a = [ o , x2 ] } and a = [ o , x1 ] ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a <= b & b <= a & a <= b & b <= a ;
||. h1 .|| . n = ||. h1 . n .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| .= ||. ( h . n ) .|| ;
( - ( ( - 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) ) `| Z ) = f . x - ( ( 1 / 2 ) (#) ( f1 + f2 ) ) . x .= ( - 1 / 2 ) * ( f1 + f2 ) . x ;
attr r = F .: ( p , q ) means : Def4 : len r = min ( len p , len q ) ;
( ( r / 2 ) |^ 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 ( (
then a <> 0. R & a " * ( a * v ) = 1 & a " * ( a * v ) = 1 & a " * ( a * v ) = 1 ;
p . ( j -' 1 ) * ( q *' r ) . ( i + 1 ) = Sum ( p . ( j -' 1 ) ) * ( q . ( j -' 1 ) ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* ( h ^\ n ) ) * ( h ^\ n ) ) " ) . ( ( h ^\ n ) . ( h ^\ n ) ) " ) . ( h . n ) ;
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 = f .: the carrier of H2 and the carrier of H1 = f .: the carrier of H2 and the carrier of H1 = f .: the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o ;
H1 = n + 1 & n + 1 <= len ( 2 to_power ( n + 1 ) ) + 1 .= n + 1 + 1 ;
( O = 0 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O = 1 & O =
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( ( n + 2 ) /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & d <> 0 & b <> d & ( a = ( - e ) / ( b - d ) ) implies ( a = ( - b ) / ( d - b ) ) / ( d - c ) ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= ( ( f +* g ) | D ) /\ D .= ( f +* g ) | D .= f | D \/ g | D .= f | D ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & g /. i = a * v & g /. i = a * v
g `2 * P `2 * g `2 = g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 and not ( ex s1 st f . i = s1 & s1 . i <> s2 & s1 . i <> s2 . i ) & s1 . i <> s2 . i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] connected connected & [ s2 , t2 ] , [ s2 , t2 ] connected & [ s2 , t2 ] , [ s2 , t2 ] connected & [ s2 , t2 ] , [ s2 , t2 ] connected & [ s2 , t2 ] <= [ s2 , t2 ] ;
then H is non negative & H is non negative implies H is not negative implies H is not implies H is non empty -g\mathopen ( x , H ) ;
attr f1 is total means : Def3 : ( 1 / 2 ) (#) ( f2 * f1 ) is total & ( 1 / 2 ) (#) ( f2 * f1 ) = ( f1 . c ) (#) ( f2 * f1 ) ) " ;
z1 in W2 ` iff z1 = z2 & ( z1 = z2 & z2 = z2 & ( z1 = z2 & z2 = z2 ) & ( z1 = z2 ) & ( z1 = z2 ) & z2 = z2 ) & ( z1 = z2 ) & z2 = z2 ) ;
p = 1 * p .= a " * a * p .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a " * ( b " * p ) .= a ;
for seq1 be Real_Sequence , seq2 be Real_Sequence , K be Real st for n be Nat holds seq1 . n <= K holds upper_bound ( ( seq ^\ k ) ^\ n ) <= K * ( seq1 ^\ k )
( for E-max C being Element of TOP-REAL 2 holds E-max C meets L~ go \/ L~ pion1 ) or ( E-max C in L~ pion1 or E-max C in L~ pion1 & E-max C in L~ pion1 ) & E-max C in L~ pion1 \/ L~ pion1 ) implies ( E-max C in L~ go \/ L~ pion1 or not E-max C in L~ pion1 & not E-max C in L~ pion1 & E-max C in L~ pion1 & E-max ( C , n ) c= L~ pion1 )
||. f . ( g . ( k + 1 ) - g . ( k + 1 ) ) - f . ( g . ( k + 1 ) - g . ( k + 1 ) ) .|| <= ||. g . ( k + 1 ) - g . ( k + 1 ) .|| ;
assume h = ( ( B .--> ( B .--> ( C .--> C ) ) +* ( D .--> E ) ) +* ( F .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) +* ( M .--> N ) +* ( M .--> N ) ) +* ( M .--> N ) . ( M .--> N ) . ( M .--> N ) = ( M .--> N ) . ( M .--> N ) . ( M .--> N ) . ( M .--> N )
|. ( ( ( ( delta ( H ) ) . n ) || ( A , T ) ) . k - ( ( ( lower ( H ) . n ) || ( A , T ) ) . k ) ) . k ) .| <= e * ( ( b-a ( H ) . n ) . k - ( ( element ( H ) . n ) . k ) ) ;
(
{ x1 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 } = { x1 , x1 , x2 , x3 , x4 , 8 } .= { x1 , x2 , x3 , x4 } ;
consider A such that A = [. 0 , 2 * PI .] and integral ( ( exp_R (#) sin ) , A ) = 0 and ( ( exp_R (#) cos ) | A ) . ( x , A ) = 0 ;
p `2 is Permutation of dom f1 & p `1 " = ( ( Sgm Y ) " ) * p & p `2 " * ( Sgm Y ) = ( ( Sgm Y ) " ) * p ;
for x , y st x in A holds |. ( 1 / 2 ) * ( f . x - 1 / 2 ) .| <= 1 * |. f . x - 1 / 2 .|
( p2 `2 / |. p2 .| ) ^2 = |. q2 .| * ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ) .= ( ( q2 `2 / |. q2 .| - sn ) / ( 1 + sn ) ) ^2 / ( 1 + sn ) ^2 ;
for f be PartFunc of the carrier of CNS , REAL st dom f is compact & f is continuous holds rng f is compact & f | X is continuous & f | X is continuous
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FF such that dom FF = n1 and for k be Nat st k in n1 holds Q [ k , F . k ] ;
ex u , u1 st u <> u1 & u , u1 , u2 , u1 , v1 , v2 & u , u1 , v1 , u1 is_collinear & u , v1 , u1 , v1 is_collinear & u , u1 , v1 , u1 is_collinear & u , v1 , u1 is_collinear & v , v1 , v1 is_collinear & u , u1 , u1 is_collinear & v , v1 , u1 is_collinear & u , v1 , u1 is_collinear & v , v1 , u1 is_collinear implies u , v , v1 is_collinear
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N N , B ) ` * ( N , B ) = N ` A * N ` B
for s be Real st s in dom F holds F . s = integral ( R , ( R + e ) (#) ( f `| Z ) ) . s * ( ( f `| Z ) . s ) ^2 )
width AutMt ( f1 , b1 , b2 ) = len b2 .= len b2 .= len b1 .= len b1 .= len b1 .= len b1 .= len b1 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 .= len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ implies f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[
assume that X is closed and a in X and a in X and y in a and a in f .: { [ n , x ] } \/ f .: { x } \/ f .: { x } and x in X ;
Z = dom ( ( ( ( 1 / 2 ) (#) ( ( ( #Z 2 ) * ( f1 + f2 ) ) ) * ( f1 + f2 ) ) ) `| Z ) /\ dom ( ( #Z 2 ) * ( f1 + f2 ) ) ;
func -> Subset of V means : Def1 : for l being Subset of V st 1 <= l & l <= len l holds it . l in V & l . l in V ;
for L being non empty TopSpace , N being net of L , M being net of N , N being net of L st c is Point of N for x being Point of L st x in N holds x is Point of N holds x is Point of N
for s being Element of NAT holds ( ( ( id the carrier of ^ ( G . v ) ) + ( id the carrier of ^ ( G . v ) ) ) . s = ( ( ( id the carrier of ^ ( G . v ) ) ) . s ) . s ) . s
then z /. 1 = ( N-min L~ z ) .. z & ( N-min L~ z ) .. z < ( ( N-min L~ z ) .. z ) .. z & ( N-min L~ z ) .. z < ( ( E-max L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) * ( ( 0 qua Real ) * ( 1 / 2 ) ) *> ) = len p + len <* ( 0 qua Real ) * ( 1 / 2 ) *> .= len p + 1 ;
assume that Z c= dom ( ( - ( ( 1 / 2 ) (#) f ) (#) f ) `| Z ) and for x st x in Z holds f . x = x and f . x > 0 ;
for R being add-associative right_zeroed right_complementable commutative associative well-unital distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J & I /\ ( I + J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B2 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 x + ( len z ) ) .= Seg ( len ( x2 + y2 ) ) .= Seg ( len ( x + z ) ) .= Seg ( len ( x + z ) ) .= Seg ( len ( x + z ) ) ;
for S being holds for C being holds card S = id C iff card S = id ( ( Obj S ) . ( id c ) ) & S is one-to-one implies S is one-to-one
ex a st a = a2 & a in f /\ ( dom f \/ f /\ g ) & a in f /\ ( dom f \/ g /\ h ) & a in f /\ ( dom f \/ g /\ h ) & a in f /\ ( dom f \/ g /\ h ) implies a = b & a = f . ( a \/ g )
a in Free ( H2 / ( x. 4 , x. k ) ) '&' ( H2 / ( x. 4 , x. k ) ) ;
for C1 , C2 being f1 , C2 being stable Function , f being Function of C1 , C2 st f = g holds f = g iff f = g & f = g
( W-min L~ go \/ L~ pion1 ) `1 = W-bound L~ pion1 \/ W-bound L~ pion1 & ( W-min L~ pion1 ) `1 = E-bound L~ pion1 \/ E-bound L~ pion1 & ( E-max L~ pion1 ) `1 = E-bound L~ pion1 \/ E-bound L~ pion1 & ( E-max L~ pion1 ) `1 = E-bound L~ pion1 \/ E-bound L~ pion1 & ( E-max L~ pion1 ) `1 = E-bound L~ pion1 ;
consider u , y0 , z0 being Point of REAL-NS 2 such that u = <* x0 , y0 *> & f is partial & y0 = y0 & f . 3 = z0 & f . 1 = z0 & f . 2 = z0 & f . 3 = z0 & f . 2 = z0 & f . 3 = z0 & f . 1 = z0 & f . 2 = z0 & f . 3 = z0 & f . 2 = z0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = ( t . {} ) `1 & t . {} = ( t . {} ) `1 ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b <= a ;
func Class R -> Subset-Family of R means : Def3 : for A being Subset of R holds A in it iff ex a being Element of R st a in it & R = Class ( R , a ) ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) \ ( ( ( ( ( ( B ) \ ( ( B ) ) \ ( ( B ) ) \ ( ( B ) \ ( B ) ) ) ) ) ) . ( n + 1 ) ) ) ) . ( n + $1 ) ) ) . ( n + $1 ) ) ) . ( n + $1 ) = G . ( n + $1 ) ;
assume that dim ( W1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 and dim ( U1 ) = 0 ;
mamam . ( m . t ) = ( m . t ) . {} .= ( ( m . t ) . {} ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= ( m . t ) `1 .= m . t ;
d11 = ( x9 , d11 ) /^ ( m + 1 ) .= f . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= ( f | ( y9 , d22 ) ) . ( y9 , d22 ) .= d22 ;
consider g such that x = g and dom g = dom f and for x being element st x in dom f holds g . x in f . x and g . x in f . x ;
x + 0. F_Complex = x + len x .= x + len x .= ( x + 0. F_Complex ) * ( x , ( len x ) |-> 0. F_Complex ) .= x `1 * ( x , ( len x ) |-> 0. F_Complex ) .= x `1 * x `1 ;
( ( k -' ( k9 -' 1 ) ) + 1 ) in dom ( f /. ( ( \downharpoonright ( k9 -' 1 ) ) + ( k -' 1 ) ) ) /\ dom ( f | ( ( \downharpoonright ( -' 1 ) ) + 1 ) ) ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 is_an_arc_of p1 , p2 and P1 = P1 \/ P2 and P1 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p1 , p2 } and P1 /\ P2 = { p2 , p3 } = { p1 , p3 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p2 , p3 } and P1 /\ P2 = { p2 , p3 , p3 /\ p2 , p3 /\ p2 , p2 /\ p2 , p3 /\ p2 /\ P2 /\ P2 \/
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c1 = c , c2 = d , c2 = d , c1 = c , c2 = d , c1 = d , c2 = c , c1 = d , c2 = e , c2 = d , c1 = c , c2 = d , c2 = e , c1 = d , c2 = e , c2 = c , c2 = d , M = ( a , c ) , c2 = d , c2 = c , c1 = d , c2 = d , c1 = d , c2 = d , c2 = d , c2 = d , c2 = d , c1 = d , c2 = d , c2 = d , c2 = d , c2 = e , c1 = e , c2 = d , c2 =
reconsider thesis thesis thesis Gtbf = G1 . t * F1 . b as Morphism of ( G1 * F1 ) . a , ( G1 * F2 ) . b ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 + 1 ) ) \/ LSeg ( f /. ( i + 1 -' 1 ) , f /. ( i + 1 -' 1 + 1 ) ) ;
Integral ( P , ( P . m ) | dom ( P . n ) ) <= Integral ( M , ( P . m ) | dom ( P . m ) ) ;
assume that dom f1 = dom f2 and for x , y being element st [ x , y ] in dom f1 & [ x , y ] in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( - G * ( i , 1 ) ) , ( - G * ( i + 1 , 1 ) ) ) / 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
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] ;
reconsider K1 = { p0 where p0 is Point of TOP-REAL 2 : P [ p0 ] & ( for q being Point of TOP-REAL 2 st q in P holds ( ( f | q ) . q ) `1 <= ( f | q ) . q ) `1 } as Subset of TOP-REAL 2 ;
( ( ( ( ( ( N ) - 2 ) / 2 ) |^ m ) / 2 ) |^ ( m + 1 ) ) / 2 |^ ( m + 1 ) ) <= ( ( ( ( ( N - 2 ) / 2 ) |^ m ) / 2 ) |^ ( m + 1 ) ) / 2 |^ ( m + 1 ) ) / 2 |^ ( m + 1 ) ;
for x be 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 ( @ z ^ <* 0 *> ) = len ( ( @ z ^ <* 1 *> ) | ( len <* 2 *> ) ) + len <* 2 *> .= len ( ( @ z ^ <* 1 *> ) | ( len <* 1 *> ) ) + len <* 2 *> .= len ( ( @ z ) | ( len <* 1 *> ) ) + len ( <* 1 *> ) ;
v / ( x. 3 , m1 ) / ( 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. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) ) = m3 ;
consider r being Element of M such that M , v / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3
func w1 \ w2 -> Element of Union ( G , R8 ) equals ( ( the PKK6 of G , R8 ) . ( w1 , w2 ) ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= ( s1 . b2 ) . b2 .= ( s2 . b2 ) . 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 + k ) + ( Partial_Sums ( |. seq .| ) ) . n ) ;
set F = S -\mathop { 0 } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( n + 1 ) ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x0 = L . ( x- x ) + R . ( x- x ) * ( x- x ) ;
func the closed non empty Subset of TOP-REAL 2 equals ( the distance of TOP-REAL 2 ) | ( LSeg ( a , b ) \/ LSeg ( b , c ) ) ) ` ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 + ( c * b ) ^2 >= 6 * a * a * b + ( c * a ) ^2 ;
v / ( x1 , m1 ) / ( x2 , m2 ) / ( x1 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) ) = v / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) / ( x2 , m2 ) /
mid ( Q ^ <* x *> , M , 0 ) = ( iff ( Q +* ( Q +* ( M +* ( x , TRUE ) ) ) +* ( x , TRUE ) ) +* ( x , TRUE ) ) +* ( x , TRUE ) ) +* ( x , TRUE , TRUE ) ) ;
Sum ( F |^ n1 ) = ( r |^ n1 ) * Sum ( C |^ n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) .= C . ( n1 + n1 ) ;
( GoB f ) * ( len GoB f , 1 ) `1 = ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 .= ( GoB f ) * ( len GoB f , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * ( ( $1 + 1 ) * ( 1 / ( $1 + 1 ) ) ) * ( 1 / ( $1 + 1 ) ) ) + ( a * ( $1 + 1 ) ) * ( 1 / ( $1 + 1 ) ) ;
( the_arity_of g ) . ( ( the Arity of S ) . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . ( g . (
( X ~ ) tolerates X ~ & card ( X ~ ) = card Y ~ & card ( X ~ ) = card ( X ~ ) & card ( Y ~ ) = card ( Y ~ ) ;
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & a = N . n holds b = N . n \ G . n ;
E , f |= All ( x. 2 , ( x. 1 ) '&' ( x. 2 , x. 1 ) ) '&' ( x. 1 , ( x. 1 ) '&' ( x. 1 ) ) ) '&' ( x. 1 , ( x. 1 ) '&' ( x. 1 , x. 1 ) ) ) ;
ex R2 being 1-sorted st R2 = ( p | ( n2 -' 1 ) ) . i & ( the carrier of p ) . i = the carrier of R2 & ( the carrier of p ) . i = the carrier of R2 ;
[. a , b + ( 1 / ( k + 1 ) ) .[ is Element of the REAL , the carrier of ( REAL ) , the carrier of ( ( and _ f ) . ( k + 1 ) ) ) is Element of the carrier of ( ( the REAL ) . ( k + 1 ) ) , the carrier of ( ( the _ of ( ( the REAL ) . ( k + 1 ) ) ) ) ) ;
Comput ( P , s , 2 + 1 ) = Exec ( ( P . 2 ) , Comput ( P , s , 2 ) ) . ( 2 + 1 ) .= Exec ( a3 , Comput ( P , s , 2 ) ) . ( 2 + 1 ) .= Exec ( a3 , s ) . ( 2 + 1 ) ;
card ( h1 ) . k = power ( K , n ) . ( - 1_ K , k ) * Sum u .= ( ( - 1_ K ) * u ) . k * Sum u .= ( ( - ( - 1_ K ) * u ) ) . k * Sum u .= ( ( - ( - 1_ K ) * u ) ) . k ;
( f / g ) / c = f / c * ( g / c ) .= ( f / c ) / c .= ( f / c ) / c .= ( f / c ) / c .= ( f / c ) / c .= ( f / c ) / c .= ( f / c ) / c ;
len ( ( ( ( the carrier of ( ( carrier of ( 2 ) ) ) ) - ( the carrier of ( 2 ) ) ) ) * ( ( ( the carrier of ( 2 ) ) - ( the carrier of ( 2 ) ) ) ) ) = len ( ( ( ( the carrier of ( 2 ) ) - ( the carrier of ( 2 ) ) ) ) * ( ( the carrier of ( 2 ) ) - ( the carrier of ( 2 ) ) ) ) ) ;
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 ) .= dom ( ( r (#) f ) | X ) /\ X ) .= dom ( ( r (#) f ) | X ) .= dom ( ( r (#) f ) | X ) /\ X ) .= dom ( r (#) f ) /\ X ) /\ 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 + $1 ) ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) * Fib ( n + $1 ) ) ;
consider f being Function of [: INT , INT :] , INT such that f = f `1 and f is onto and for n being Nat holds f . n = n + 1 & f " { f . n } = { n } ;
consider vs be Function of S , BOOLEAN such that vs = chi ( S , B ) and ( for A being Element of S holds E . A = Prob . A ) & ( for A being Element of S holds E . A = Prob . A ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , L ( ) ) and Q [ y , x ] ;
assume that A c= Z and f = f and Z = dom f and f = ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) * ( ( 1 / 2 ) ) ) ) `|
( f /. i ) `2 = ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 .= ( GoB f ) * ( 1 , j2 ) `2 ;
dom Shift ( Seq q2 , len Seq q2 ) = { j + len Seq q2 where j is Nat : j in dom Seq q2 & j in dom Seq q2 } & len Seq q2 = len q2 + len Seq q2 } ;
consider G1 , G2 , G3 being Element of V such that G1 <= G2 and G2 <= G2 and f in G1 and g in G2 & g in G2 & f in G1 & g in G2 & f in G2 & g in G2 & f in G2 & g in G2 & f in G2 & g in G2 & g in G2 & f in G2 & g in G2 ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & for c st c in dom it holds it /. c = - f /. c & for c st c in dom it holds it /. c = - f /. c ;
consider phi such that phi is increasing and for a st phi . a = a & for v st v <> {} holds ( for a st a in sup L holds L , a |= H iff L , a |= H ) iff L , a |= 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 ) and f /. ( i + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) * ( ( i - n ) * ( i - n ) ) and for n1 being Integer st n1 <> 0 & n2 <> 0 holds ( i <= n1 & n1 <= n ) & n2 <= n ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) `| Z ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) `| Z ) . x > - 1 & ( ( 1 / 2 ) (#) ( f1 + f2 ) ) `| Z ) . x < 1 ;
cell ( G1 , i1 -' 1 , 2 |^ ( m -' 1 ) ) \ ( ( Y -' 1 ) * ( ( Y -' 1 ) + ( Y -' 1 ) * ( i1 -' 1 ) ) ) \ ( ( Y -' 1 ) * ( i1 -' 1 ) ) ) c= BDD L~ f \/ ( ( Y -' 1 ) * ( i1 -' 1 ) ) \ ( ( Y -' 1 ) * ( i1 -' 1 ) ) ) ;
ex Q1 being open Subset of X st s = Q1 & ex Q1 being Subset-Family of Y st Q1 c= F & Q1 is finite & ( for a being set st a in Q1 holds Q1 . a is finite ) & ( for b being set st b in Q1 holds b is finite ) & ( ex a being set st a in Q1 & b in Q1 holds a is finite ) implies a is finite )
gcd ( Au , r1 , s1 ) = 1 & gcd ( fu , s1 , s2 ) = 1 & gcd ( fu , s1 , s2 ) = 1 & gcd ( fu , s1 , s2 , Amp ) = 1 & gcd ( fu , s2 , Amp ) = 1 & gcd ( fu , s1 , Amp ) = 1 ;
R8 = ( ( ( the InitS of s2 ) . ( m2 + 1 ) ) . m2 ) . m2 .= ( ( the _ of s2 ) . m2 ) . m2 .= ( ( the _ 3 ) . m2 ) . m2 ) . m2 .= ( ( the _ 3 3 ) . m2 ) . m2 .= ( ( the _ 3 ) . m2 ) . m2 .= ( ( the _ 3 ) . m2 ) . m2 ;
CurInstr ( P3 , Comput ( P3 , s3 , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m2 ) ) .= halt SCMPDS .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p1 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) c= ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) \/ LSeg (
func not -> Subset of the bound of Al means : Def1 : ex a , b st a in it & b in it & a in it & b in it & a in it & b in it & a in it & b in it & a in it & b in it & a in it & b in it & a in it & b in it implies it is not empty ;
for a , b being Element of F_Complex st |. a .| > |. b .| & |. a .| >= 1 holds a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) ) is >= and a * ( f | ( len f ) = f | ( len f ) = f . ( len f ) = f . ( len f ) = f . ( len f ) = f . ( len f ) + b * ( f | ( len f ) = f . ( f . ( len f ) ) implies f . ( len
defpred P [ Nat ] means 1 <= $1 & $1 <= len g & for i , j st [ i , j ] in Indices G & G * ( i , j ) = g . ( $1 , j ) holds G * ( i , j ) = g . ( $1 , j ) ;
assume that C1 , C2 are_every MSAlgebra over C1 and g being State of C1 , C2 st g = s2 and for s1 being State of C1 , s2 being State of C2 st s1 = s2 holds s1 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 & s1 is stable & s2 is stable holds s1 is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. ( f | X ) . c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| .= ||. ( f | X ) /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 < ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `1 ) ^2 ;
for F being Subset-Family of TT st F is open & {} in F & {} in F & for A being Subset of T st A in F & A is open & A is open & F is open holds card A c= card F & card F c= card A & card F c= card A & card F c= card 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 F . k = g . k and for k st k in dom F holds F . k = g . k ;
i |^ ( ( Let ( Let ( ' ) |^ n ) - i ) |^ s ) * ( i |^ s ) ) = i |^ ( s + k ) - ( i |^ k ) * ( i |^ s ) .= i |^ ( s * ( i |^ k ) - i ) * ( i |^ k ) .= i |^ ( s * ( i |^ k ) - i ) ;
consider q being oriented oriented oriented Chain of G such that r = q and q <> {} and ( for n st n < len F holds F . ( q . n ) = v1 ) & ( for n st n < len F holds F . ( q . n ) = v2 ) & ( for n holds F . ( q . n ) = v2 ) ;
defpred P [ Element of NAT ] means $1 <= len ( I , Z , I ) . ( ( f , Z ) . ( len g + $1 ) ) = ( ( f , Z ) . ( len f + $1 ) ) . ( len g + $1 ) ) . ( len f + $1 ) ;
for A , B being Matrix of n , REAL holds len ( A * B ) = len A & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n & width ( A * B ) = n ;
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 in dom s ex a , b being Element of R st s . i = a * b & a in I & b in J & a in J & b in J & a in J & b in K & b in K & a in J & b in K & a in K & b in J & b in K & a in K ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x , Re y ) , ( Re x ) , ( Im y ) )| + ( ( Re x ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im x ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 + ( Im y ) ^2 ) ;
consider g2 being FinSequence of F such that g2 is continuous & rng g2 c= A & g2 is continuous & rng g2 c= A & g2 . 1 = x1 & g2 . len g2 = x2 & g2 . len g2 = x2 & g2 . len g2 = x2 & g2 . len g2 = x1 & g2 . len g2 = x2 & g2 . len g2 = x2 & g2 . len g2 = x2 ;
then n1 >= len p1 & crossover ( p1 , p2 , n1 , n2 , n3 ) = crossover ( p1 , p2 , n1 , n3 ) & crossover ( p1 , p2 , n1 , n2 , n3 ) = crossover ( p1 , p2 , n1 , n2 ) & crossover ( p1 , p2 , n1 , n2 , n3 ) = crossover ( p1 , p2 , n1 , n3 ) ;
( ( q `1 ) * a ) ^2 <= ( q `1 ) ^2 & - ( q `1 ) * a <= ( q `1 ) ^2 & - ( q `1 ) * a <= ( q `1 ) ^2 & - ( q `1 ) * a <= - ( q `1 ) * a & - ( q `1 ) * a <= - ( q `1 ) * a ;
( F . ( p9 . ( len p9 ) ) ) . ( ( len p9 ) + 1 ) ) = ( ( F . ( p9 . ( len p9 ) ) ) . ( ( len p9 ) + 1 ) ) . ( ( len p9 ) + 1 ) ) .= ( ( F . ( len p9 ) ) . ( ( len p9 ) + 1 ) ) . ( ( len p9 ) + 1 ) .= ( ( F . ( len p9 ) + 1 ) ) . ( len p9 ) ) . ( len p9 ) .= ( ( len p9 ) + 1 ) . ( len p9 ) . ( len p9 ) . ( len p9 ) . ( len p9 ) ) .= ( ( len p9 ) + 1 ) .= ( ( len p9 ) + 1 ) . ( len p9 ) . ( len p9 ) + 1 ) .= ( ( len p9 ) + 1 ) + 1 ) +
consider k1 being Nat such that k1 + 1 = 1 and a := ( k + 1 ) = ( ( <* a := intloc 0 *> ) +* ( ( a := intloc 0 ) .--> ( ( a := intloc 0 ) ) + ( a := intloc 0 ) ) ) . ( ( a := intloc 0 ) + ( a := intloc 0 ) ) . ( ( a := intloc 0 ) + ( a := intloc 0 ) ) . ( ( a intloc 0 ) + ( a intloc 0 ) ) ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , B2 such that B1 is finite and D1 = the carrier of B1 and B1 = the carrier of B1 and B1 is finite and B1 is finite and B2 is finite and B1 is finite and B1 is finite and B2 is finite ;
v2 . b2 = ( curry ( F2 , g ) ) * ( ( curry ( F2 , g ) ) . b2 ) .= ( ( curry ( F2 , g ) ) . b2 ) . b2 .= ( ( curry ( F2 , g ) ) . b2 ) . b2 .= ( ( curry ( F2 , g ) ) . b2 ) . b2 .= ( ( ( curry ( F2 , g ) ) . b2 ) . b2 .= ( ( ( curry ( F2 , g ) ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( F2 , g ) . b2 ) . b2 .= ( ( ( F2 , g ) . b2 ) . b2 .= ( ( ( curry ( F2 , g ) ) . b2 ) . b2 .= ( ( ( ( curry ( F2 , g ) ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 .= (
dom IExec ( I , P , Initialize s ) = the carrier of SCMPDS .= dom ( IExec ( I , P , Initialize s ) +* ( 1 , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* ( 1 , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* ( 1 , SCMPDS ) ) .= dom ( IExec ( I , P , Initialize s ) +* ( 1 , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & d-32 > 0 & for h be Real st h <> 0 & |. h .| < d\overline ( R1 ) & |. h .| < R holds |. h .| " * ||. ( R1 + R2 ) . h .|| < r
LSeg ( G * ( len G , 1 ) + |[ 1 , - 1 ]| , G * ( len G , 1 ) + |[ 1 , 0 ]| ) c= Int cell ( G , len G ) \/ { G * ( len G , 1 ) + |[ 1 , 0 ]| } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h , i ) ;
A = { q where q is Point of TOP-REAL 2 : LE LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 } c= P & LE p1 , p2 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 } c= P & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p1 , p2 & LE p2 , p3 , P , p2 & LE p3 , p4 , P , p1 , p2 & LE p3 , p4 , P , p1 , p2 & LE p3 , p4 , P , p1 , p2 & LE p3 , p4 , P , p1 , p3 , K & LE p3 , p4 , P , p1 , p3 , p3 , K & LE p3 , p4 , P , p1 , p3 , K & LE p3 , p4 , P ,
( ( - x ) .|. y ) = - ( ( - 1 ) .|. y ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / |. p .| - cn ) / ( 1 + cn ) ^2 ) = ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 + ( p `2 / |. p .| - cn ) ^2 ) * ( 1 + cn ) ;
( ( U . n ) (#) ( W . n ) ) * ( ( W . n ) * ( W . n ) ) = ( ( U . n ) (#) ( W . n ) ) * ( W . n ) .= ( ( U . n ) (#) ( W . n ) ) * ( W . n ) .= ( ( W . n ) (#) ( W . n ) ) * ( W . n ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom f & for x be Element of REAL , y be Element of REAL n st x in dom it holds it . x = - h . x & it . y = - h . y & for x be Element of REAL n holds it . x = - h . x + h . y ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and f /. ( i + 1 ) = G * ( i + 1 , j ) and f /. ( i + 1 ) = G * ( i + 1 , j ) and f /. ( i + 1 ) = G * ( i + 1 , j ) ;
assume that not y in Free H and not x in Free H and not x in Free H and not y in Free H and not x in Free H and not y in Free H and not x in Free H and not x in Free H and y in Free H and x in Free H and y in Free H and x in Free H ;
defpred P11 [ Element of NAT , Element of NAT ] means ( ( p |-count $1 ) |^ ( 2 * $1 ) ) * ( ( p |-count $1 ) |^ ( 2 * $1 ) ) < ( p |-count $1 ) |^ ( 2 * $1 ) ) & ( p |-count $1 ) |^ ( 2 * $1 ) = ( p |-count $1 ) |^ ( 2 * $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def4 : for A being Subset of X holds A in it iff for W being Subset of X st W in it holds W is non empty & for A being Subset of X st A in it holds it . A is non empty ;
[#] ( ( dist ( such that ( dist ( P ) ) ) .: Q ) ) = ( ( dist ( P ) ) .: Q ) .: 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 , S :] \ { 0 } ) ) = {} or rng ( F | ( [: S , S :] \ { 0 } ) ) = { 1 } or rng ( F | ( [: S , S :] \ { 0 } ) ) = { 1 , 2 } or rng ( F | ( [: S , S :] \ { 0 } ) ) = { 1 , 2 } ;
( f " ( rng ( 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 ) ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 } and C = { p1 , p2 } and C = { p2 , p1 , p2 } and C = { p1 , p2 } and C = { p2 , p1 , p3 } and C = { p2 , p3 } and C = { p1 , p2 } and C = { p2 , p3 } ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 + ( p2 `2 ) ^2 * ( p2 `1 ) ^2 ;
( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( AffineMap ( a , X ) ) * ( ( a , X ) ) ) ) ) ) ) ) ) " ) ) . x ) ) = ( ( ( AffineMap ( a , X ) ) ) ) . x ) ) " ) . x ) ) . x ) ) " .= ( ( a , X ) ) . x ) " ) * ( (
for T being non empty normal TopSpace , A being closed Subset of T , B being closed Subset of T , A being Subset of T st A <> {} & A misses B & B is closed & A is closed & B is closed holds A is non empty & B is non empty implies A is non empty & B is non empty & B is non empty & B is non empty
for i , j being strict Subgroup of G1 st i + 1 in dom F for G1 , G2 being strict normal Subgroup of G1 st G1 = F . i & G2 = F . ( i + 1 ) & G1 is strict Subgroup of G2 holds G1 is strict Subgroup of G2 & G2 is strict Subgroup of G1 & G2 is strict Subgroup of G2
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) `| Z ) . x = ( ( 1 / 2 ) (#) ( ( #Z 2 ) * ( f1 + f2 ) ) ) `| Z ) . x
synonym f is right & f /* a = lim ( f /* a ) & ( for n st n in dom f holds f . n = ( f /* a ) . n ) & ( for n st n <= m holds f . n < x0 ) & ( for n st n <= m holds f . n < ( f /* a ) . n ) implies f . ( n + 1 ) = ( f /* a ) . n ) ;
then X1 , X2 are_separated or X1 misses X2 or X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated implies X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 are_separated & X1 , X2 are_separated & X2 , Y2 an implies X2 , Y2 an
ex N be Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L be Real st for x st x in N holds ( SVF1 ( 1 , f , u ) ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - x0 ) + R . ( x - x0 ) ;
( ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) * sqrt ( 1 + ( p2 `1 ) ^2 ) ) ^2 + ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) * sqrt ( 1 + ( p2 `1 ) ^2 ) >= ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) * sqrt ( 1 + ( p2 `1 ) ^2 ) ;
( ( 1 / ( t1 * t2 ) ) to_power ( n + 1 ) ) / ( ( 1 / ( t1 * t2 ) ) to_power ( m + 1 ) ) = ( ( 1 / ( t1 * t2 ) ) to_power ( m + 1 ) ) ) / ( ( 1 / ( t1 * t2 ) ) to_power ( m + 1 ) ) ) & ( ( 1 / ( t1 * t2 ) ) to_power ( m + 1 ) ) = ( ( 1 / ( t1 * t2 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ) to_power ( m + 1 ) ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin - cos ) ) . x and for x st x in dom ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z holds ( ( ( - 1 / 2 ) (#) ( sin - cos ) ) `| Z ) . x = ( 1 / 2 ) * ( sin . x ) and ( ( - 1 / 2 ) (#) ( cos - sin ) ) . x = 1 ;
consider Xi1 being Subset of Y , Y1 being open Subset of X such that Y1 = [: Y1 , Y2 :] and Y1 is open and ex Y1 being Subset of Y , Y2 being Subset of Y st Y1 = Y1 & Y1 is open & Y1 is open & Y2 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y2 is open ;
card ( S . ( ( d |^ n ) + 1 ) ) = card { [: [: d , d :] + b , d :] where d is Element of GF ( p ) : d in 3 } \/ { 0 } } .= { 3 } \/ { 4 , 5 } \/ { 4 , 6 } \/ { 4 , 5 } } .= { 4 , 5 } ;
( W-bound D - E-bound D ) / ( ( W-bound D - E-bound D ) / ( m + 1 ) ) * ( i + 1 ) = ( W-bound D - E-bound D ) / ( m + 1 ) * ( i + 1 ) .= ( W-bound D - E-bound D ) / ( m + 1 ) * ( i + 1 ) ;