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contradiction ;
thesis ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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contradiction ;
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thesis ;
contradiction ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `2 is convergent
q in X ;
V in X ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a is_>=_than X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D is_<=_than E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y `2 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 `2 = b `2 ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is from squares ;
assume m > 0 ;
assume A c= B ;
X is lower
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A `2 in B `2 ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCI-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , A be Subset of G ;
let G be _Graph , A be Subset of G ;
let a be Complex ;
let x be element ;
let x be element ;
let C be FormalContext , a , b , c be Element of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= ma ;
let y be element ;
r2 in dom f ;
let x be element ;
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 = Set ;
n >= 0 + 1 ;
k c= k + 1 ;
R1 c= R ;
k + 1 >= k ;
k c= k + 1 ;
let j be Nat ;
o , a // Y ;
R c= Cl G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
arccot is_differentiable_on Z ;
exp_R is_differentiable_in x ;
j < i0 ;
let j be Nat ;
n <= n + 1 ;
k = i + m ;
assume C meets S ;
n <= n + 1 ;
let n be Nat ;
h1 = {} ;
0 + 1 = 1 ;
o <> b3 ;
f2 is one-to-one ;
support p = {}
assume x in Z ;
i <= i + 1 ;
r1 <= 1 ;
let n be Nat ;
a "/\" b <= a ;
let n be Nat ;
0 <= r0 ;
let e be Real , x be Element of REAL ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , I be Program of SCM+FSA ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E
Cl R is boundary ;
let i be Nat ;
R2 in X ;
cluster downarrow x -> \mathclose { \rm c } ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 to_power x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
2 >= TOP-REAL s ;
G . y <> 0 ;
let X be RealNormSpace , x be Element of X ;
a in X ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , A be Subset of V ;
assume x in - M ;
k < s . a ;
not t in { p } ;
let Y be set , X be set ;
M , L are_equipotent ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded ;
rng f = Y ;
G8 c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
still_not-bound_in p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= i-32 ;
1 <= i-32 ;
pp c= PI ;
1 <= ii ;
1 <= ii ;
LMP C in L ;
1 in dom f ;
let seq , B ;
set C = a * B ;
x in rng f ;
assume f is_continuous_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 : x in A2 ;
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 Sorts of X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
c= c= f ;
x9 is increasing & x9 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 , X be non-empty ManySortedSet of S ;
assume P [ n ] ;
assume union S is independent & A is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume ex_inf_of X , L ;
y in rng f ;
let s , I be set , x be set ;
b `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 & cos . x <> 0 ;
assume y in rng S ;
let x , y be element ;
i2 < i1 & i2 < i2 ;
a * h in a * H ;
p , q in Y ;
redefine func sqrt I ;
q1 in A1 & q2 in A2 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A1 c= A2 ;
an < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m be element ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 & g . x0 = 0 ;
g is_continuous_in x0 & g . x0 = 0 ;
assume O is symmetric transitive ;
let x , y be element ;
let j0 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s , m ;
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 A ;
let S be non empty ManySortedSign ;
let x be variable , f be Function ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` = x ` .= x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom mn ;
h2 . a = y ;
P [ n + 1 ] ;
redefine func G * F -> preJ ;
let R be non empty multMagma , a , b be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng co ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be be be be be mamaid ;
let N be non empty Subset of M ;
let R be RelStr with finite and card R = n ;
let n , k be Nat ;
let P , Q be be be be in ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as Int-Location ;
assume I does not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v8 ;
x <= c2 . x ;
x in F ` ;
redefine func S --> T -> such that S is such that T is such that S is such
assume that t1 <= t2 and t2 <= t2 ;
let i , j be even Integer ;
assume F1 <> F2 & F2 <> F1 ;
c in Intersect ( union R ) ;
dom p1 = c & dom p2 = c ;
a = 0 or a = 1 ;
assume A1 : A <> 6 ;
set i1 = i + 1 ;
assume a1 = b1 & a2 = b2 ;
dom g1 = A & dom g2 = A ;
i < len M + 1 ;
assume not -infty in rng G ;
N c= dom f1 /\ dom f2 ;
x in dom sec & y in dom sec ;
assume [ x , y ] in R ;
set d = ( x - y ) / 2 ;
1 <= len g1 & g1 < len g2 ;
len s2 > 1 & len s2 > 1 ;
z in dom ( f1 - f2 ) ;
1 in dom ( D2 | 1 ) ;
( p `2 ) ^2 = 0 ;
j2 <= width G & j2 <= width G ;
len PI > 1 + 1 ;
set n1 = n + 1 ;
|. q-35 .| = 1 ;
let s be SortSymbol of S ;
gcd ( i , i ) = i ;
X1 c= dom f & X2 c= dom g ;
h . x in h . a ;
let G be \it cluster cluster cluster -> functor ;
cluster m * n -> invertible ;
let k9 be Nat , k be Nat ;
i - 1 > m - 1 ;
R is transitive in field R ;
set F = <* u , w *> ;
p-2 c= P3 & p-2 c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is vertical ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 - f2 ) ;
assume [ X , p ] in C ;
B9 c= ( X0 \/ X2 ) ;
n2 <= ( 2 + 1 ) - 1 ;
A /\ cP c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , A be Subset of S ;
assume n in dom g2 & n < len g2 ;
let a be Element of R ;
t `2 in dom e2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , T be non empty set ;
i . y in rng i ;
REAL c= dom f & dom f = dom g ;
f . x in rng f ;
mt <= ( r / 2 ) * ( r / 2 ) ;
s2 in r-5 ;
let z , z be complex number ;
n <= ( N . m ) `2 ;
LIN q , p , s ;
f . x = waybelow x /\ B ;
set L = [ S \to T ] ;
let x be non positive Real ;
let m be Element of M ;
f in union rng ( F1 ) ;
let K be add-associative right_zeroed right_complementable non empty doubleLoopStr , p be Polynomial of K ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x & dom f = dom y ;
n1 < n1 + 1 & n1 + 1 < n2 ;
n1 < n1 + 1 & n1 + 1 < n2 ;
cluster ( T | X ) -> On ;
[ y2 , 2 ] `2 = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng S29 ;
b = sup dom f & a = sup dom f ;
x in Seg len q ;
reconsider X = D ( ) as set ;
[ a , c ] in E1 ;
assume n in dom h2 & n < len h2 ;
w + 1 = ma ;
j + 1 <= j + 1 + 1 ;
k2 + 1 <= k1 + 1 ;
let i be Element of NAT ;
Support u = Support p & Support u = Support q ;
assume X is complete \frac m ;
assume f = g & p = q ;
n1 <= n1 + 1 & n1 + 1 <= n2 ;
let x be Element of REAL ;
assume x in rng ( s2 | X ) ;
x0 < x0 + 1 & x0 + 1 < x0 + 1 ;
len ( L ) = W ;
P c= Seg len A & P c= Seg len A ;
dom q = Seg n & dom q = Seg n ;
j <= width M *' ;
let r8 be real-valued sequence of S ;
let k be Element of NAT ;
Integral ( M , P ) < +infty ;
let n be Element of NAT ;
assume z in z := ) . 0 ;
let i be set ;
n - 1 = n-1 - 1 ;
len ( n-27 ) = n ;
\mathop { Z } c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be Function of A , B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q & i in dom q ;
let s be Element of E -tuples_on E ;
let B1 be Basis of x , B2 ;
L2 /\ L2 = {} ;
L1 /\ L2 = {} implies L1 /\ L2 = {} ;
assume downarrow x = downarrow y ;
assume b , c // b , c ;
LIN q , c , c ;
x in rng ( f | dom f ) ;
set n8 = n + j ;
let D7 be non empty set , f be FinSequence of D ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , p be Polynomial of K ;
assume f `2 = f & h `2 = h ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K1 ` is open & K1 ` is open ;
assume a , b ] is maximal in C ;
let a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = Integral ( M , f ) ;
cluster -> ne] for
not u in { ag } ;
the carrier of f c= B ;
reconsider z = x as Vector of V ;
cluster \rm Str over L -> non-empty ;
r (#) H is as as as as as as as as as as \upharpoonright of X ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U0 be strict strict strict universal algebra , 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 : ex y st y in W & x = [ y , r ] ;
let x , y be Element of X ;
let A , I be such that A is such such that A is such that I is |^ ;
[ y , z ] in [: O , O :] ;
( card Macro i ) = 1 & ( card Macro i ) = 1 ;
rng Sgm A = A & rng Sgm A = A ;
q |- \! \! \setminus All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = |[ x , y , z ]| ;
LIN o , a , b & LIN o , b , b ;
p . 2 = Z |^ Y ;
( D ) `2 = {} & ( D ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: the carrier of A , the carrier of A :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Ideal of L ;
set g = f1 + f2 , h = f1 + f2 ;
a <= max ( a , b ) ;
i-1 < len G + 1-1 ;
g . 1 = f . i1 ;
x `2 , y `2 ] in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster associative for non empty multMagma ;
x in support ( Sgm ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= len ( y `2 ) ;
assume p divides b1 + b2 ;
M1 <= sup M1 & M1 <= sup M2 ;
assume x in W-min ( X ) & y in L~ f ;
j in dom ( z | n ) ;
let x be Element of D ( ) ;
IC s4 = l1 & IC s4 = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , uG = Vertices G ;
seq " is non-zero & seq " is non-zero implies seq " is non-zero
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hcn c= h-14 & hcn c= dom h ;
]. a , b .[ c= Z ;
X1 , X2 are_separated & X2 , X1 are_separated ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 - 1 = k - 1 ;
cluster -> real-valued for Relation ;
ex v st C = v + W ;
let IT be non empty zero RelStr , x be Element of IT ;
assume V is Abelian add-associative right_zeroed right_complementable ;
X-21 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper Subset of B ;
let L be non empty reflexive transitive RelStr , X be Subset of L ;
R is reflexive & X is transitive implies R is transitive
E , g |= ( the_right_argument_of H ) ;
dom G `2 = a & dom G = b ;
( 1 - 4 ) / ( 2 * r ) >= - 4 ;
G . p0 in rng G & G . O in rng G ;
let x be Element of FF , y be Element of FF ;
D [ P-6 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N & y in the carrier of N ;
g in the carrier of H & h in the carrier of G ;
rng fbeing c= NAT & rng fA2 c= NAT ;
j `2 + 1 in dom s1 & j + 1 in dom s2 ;
let A , B be strict Subgroup of G ;
let C be non empty Subset of R^1 ;
f . z1 in dom h & f . z2 in dom h ;
P . k1 in rng P & P . k1 in rng P ;
M = A9 +* {} & M = A +* {} ;
let p be FinSequence of REAL , n be Nat ;
f . n1 in rng f & f . n2 in rng f ;
M . ( F . 0 ) in REAL ;
diameter [. a , b .[ = b-a ;
assume the distance of V , Q ;
let a be Element of ^ ( V ) ;
let s be Element of PE ( X ) ;
let PA be non empty \rm RelStr ;
let n be Nat ;
the carrier of g c= B & the carrier of g c= A ;
I = halt SCM R .= ( the InstructionsF of SCM ) ;
consider b being element such that b in B ;
set BM = BCS K , BM = BCS M ;
l <= ( a1 . j ) `2 ;
assume x in downarrow [ s , t ] ;
( x `2 ) ^2 in uparrow t ;
x in ( <* 1 *> , <* T *> ) . 1 ;
let h be Morphism of c , a ;
Y c= 1. ( Y , the_rank_of Y ) & Y c= A ;
A2 \/ A3 c= Carrier ( L ) \/ Carrier ( L ) ;
assume LIN o , a , b & LIN o , b , c ;
b , c // d1 , e2 ;
x1 , x2 in Y & x2 in Y ;
dom <* y *> = Seg 1 & dom <* y *> = Seg 1 ;
reconsider i = x , j = y as Element of NAT ;
set l = |. ar s .| ;
[ x , x `2 ] in X ~ ;
for n being Nat holds 0 <= x . n
[' a , b '] = [. a , b .] ;
cluster -> -> -> -> -> -> sqrt closed for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 in P & q1 <> q2 ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
let R , Q be ManySortedSet of A ;
set d = ( 1 / n + 1 ) / n ;
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 , A be Subset of V ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( ( id R ) * ( id S ) ) ;
let b be Element of the lattice of T ;
dist ( e , z ) > r-r ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume the carrier of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric transitive RelStr ;
assume [ x , y ] in a9 ;
dom ( A * e ) = NAT ;
let a , b be Vertex of G ;
let x be Element of Bool M , i be Element of I ;
0 <= 2 * PI ;
o9 , a9 // o9 , y & o9 , c9 // o9 , y ;
{ v } c= the carrier of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) & y in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X
assume that D2 . k in rng D and D . k = x ;
f " . p1 = 0 & f " . p2 = 0 ;
set x = the Element of X , y = the Element of Y ;
dom Ser ( G ) = NAT & rng Ser ( G ) = NAT ;
let n be Element of NAT ;
assume LIN c , a , e1 ;
cluster -> natural for FinSequence ;
reconsider d = c , e = d as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the carrier of f & y in the carrier of g ;
conv @ A c= conv @ A & conv @ A c= conv @ A ;
reconsider B = b as Element of the open domains of T ;
J , v |= P \lbrack l , P \rbrack ;
redefine func J . i -> non empty TopSpace ;
ex_sup_of Y1 \/ Y2 , T & ex_sup_of Y1 , T ;
W1 is well transitive & R is well transitive implies R + R is connected
assume x in the carrier of R & y in the carrier of R ;
dom n-16 = Seg n & dom n-16 = Seg n ;
s4 misses s4 & s4 misses s4 ;
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in [: C2 , C2 :] ;
assume that function function function function I c= J and dom function function function I = K ;
Im ( lim seq ) = 0 & Im ( lim seq ) = 0 ;
( sin . x ) <> 0 & ( sin . x ) <> 0 ;
sin * cos is_differentiable_on Z & for x st x in Z holds cos . x <> 0 ;
t3 . n = t3 . n .= s . n ;
dom ( ( - 1 ) (#) F ) c= dom F ;
W1 . x = W2 . x .= W1 . x ;
y in W .vertices() \/ W .vertices() ;
( for k st k <= len ( vM ) holds vM . k = v ) implies k = len ( vM ) ;
x * a \equiv y * a . ( mod m ) ;
proj2 .: S c= proj2 .: P & proj2 .: P c= P ;
h . p4 = g2 . I .= g . I ;
Gij `1 = U * ( 1 , 1 ) `1 .= G * ( 1 , 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 seq is well unital associative non empty multMagma ;
[ x , y ] in A ~ { a } ;
x1 . o in L2 . o & x2 . o in L2 . o ;
the carrier of ( m + 1 ) c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k1 is lower & seq ^\ k1 is lower ;
len ( F-12 ) = len I + len <* a *> ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be Complex , p be Point of TOP-REAL 2 ;
Comput ( P , s , n ) = s ;
k <= k + 1 & k + 1 <= len p ;
reconsider c = {} T as Element of L ;
let Y be Element of of of of of of of of \mathbin { X } ;
cluster -> directed-sups-preserving for Function of L , L ;
f . j1 in K . j1 & f . j2 in K . j1 ;
redefine func J => y -> total Function equals J * ( J * f ) ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 & F . b2 = F . b2 ;
x1 = x or x1 = y or x1 = z ;
pred a <> {} means : Def1 : ( a - a ) / 2 = 1 ;
assume that a c= b and b in a ;
s1 . n in rng s1 & s1 . n in rng s1 ;
{ o , b2 } on C2 ;
LIN o , b , b9 & LIN o , b , b9 ;
reconsider m = x , n = y as Element of Hom ( V , C ) ;
let f be non constant FinSequence of D ;
let FG2 be non empty element , f be Function ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in F-8 ;
reconsider pp = x , pp = y as Subset of m -tuples_on REAL ;
let A , B , C be Element of R ;
redefine func strict non empty for sqrt A ;
rng c `2 misses rng ee `2 & rng e misses rng ee `2 ;
z is Element of gr { x } & z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) & Z c= dom ( cot * cot ) ;
the component of Q c= UBD A & the component of Q c= UBD A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( 1 / 2 ) (#) ( f ^ ) ) ;
redefine pred f = u means : Def1 : a (#) f = a (#) u ;
for n holds P1 [ \mathop { \rm prop n } ]
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
let a , b be Nat ;
assume that S = S2 and p = p2 and q = q2 ;
gcd ( n1 , n2 , n1 , n2 , n3 ) = 1 ;
set oo = ( a * _ ) * ( b * _ ) ;
seq . n < |. r1 .| & |. seq . n .| < r ;
assume that seq is increasing and r < 0 and seq is increasing ;
f . ( y1 , x1 ) <= a & f . ( y1 , x1 ) <= b ;
ex c being Nat st P [ c ] ;
set g = { n |^ 1 : n in dom f } ;
k = a or k = b or k = c ;
a9 , b9 , c9 is_collinear & a9 , b9 , c9 is_collinear & a9 , b9 , c9 is_collinear ;
assume that Y = { 1 } and s = <* 1 *> ;
Ip1 . x = f . x .= 0. ( K , f . x ) .= 0. K ;
W3 .last() = W3 . 1 & W2 .last() = W3 . 1 ;
cluster trivial -> trivial for _Graph ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B |^ n are_relative_prime
x in { [ 2 * n + 3 , k ] } ;
1 >= ( q `1 / |. q .| - cn ) / ( 1 + cn ) ;
f1 is__ _ - f2 implies f1 - f2 is__ _ _ - ( f1 - f2 )
( f . q ) `2 <= ( q `2 ) ^2 ;
h is in the carrier of Cage ( C , n ) ;
b `2 / |. b .| <= ( p `2 / |. p .| - cn ) ;
let f , g be Function of X , Y ;
S * ( k , k ) <> 0. K ;
x in dom ( max ( - f , - f ) ) ;
p2 in ( N . p1 ) . ( p1 , p2 ) ;
len ( the_right_argument_of H ) < len ( H ) ;
F [ A , F-14 . A ] ;
consider Z such that y in Z and Z in X ;
redefine pred 1 in C means : Def1 : A c= C |^ A ;
assume that r1 <> 0 or r2 <> 0 and r1 < r2 ;
rng q1 c= rng C1 & rng q1 c= rng C2 ;
A1 , L , A3 , A3 , A2 , A1 , A2 be Element of L ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in let ( p , SS ) . b ;
then S is negative & P-2 [ S ] ;
Cl Int [#] T = [#] T .= [#] T ;
f12 | A2 = f2 | A2 & f12 | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of W ;
let v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ Z ;
let X be Subset of S ~ ;
consider H1 such that H = 'not' H1 and H1 in D ;
1_ 1 c= ( 1 - 1 ) * ( ( 1 - 1 ) * ( 1 - 1 ) ) ;
0 * a = 0. R .= a * 0. R ;
A |^ ( 2 , 2 ) = A ^^ A ;
set v\rbrace = ( vseq /. n ) `1 , v4 = ( vseq /. n ) `1 ;
r = 0. ( REAL-NS n ) * ||. x - x0 .|| .= 0 ;
( f . p4 ) `1 >= 0 ;
len W = len ( W Exec ( W , W ) ) ;
f /* ( s * G ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t16 . ( n + 7 ) does not destroy b1 . ( n + 7 ) ;
reconsider Y1 = X1 , Y2 = X2 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c , d be Real ;
reconsider i = i , j = j as non zero Element of NAT ;
c . x >= id ( L . x ) ;
\sigma ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 ] -> pair ;
downarrow a /\ downarrow t is Ideal of T ;
let X be with NAT with NAT , N be non empty set ;
rng f = S2 -*> ( S , X ) ;
let p be Element of B , x be Element of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N2 ;
0. X <= b |^ ( m * mm1 ) ;
assume i in I & R0 . i = R ;
i = j1 & p1 = q1 & p2 = q2 implies p1 = p2
assume gRRR1 in the carrier of g & ggR2 in the carrier of g ;
let A1 , A2 be Point of S , A be Subset of T ;
x in h " P /\ [#] T1 & x in h " P ;
1 in Seg 2 & 1 in Seg 3 implies 1 in Seg 3
reconsider X-5 = X , X] as non empty Subset of Tsuch that X = T] ;
x in ( the Arrows of B ) . i ;
cluster E-32 . n -> ( the Source of G ) -valued ;
n1 <= i2 + len g2 & i2 + len g1 < len g2 + 1 ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume that v in the carrier' of G2 and v <> 0. G2 ;
y = Re y + ( Im y ) * i ;
( ( - 1 ) |^ p ) gcd ( - 1 ) = 1 ;
x2 is_differentiable_on ]. a , b .[ & ( for x st x in ]. a , b .[ holds x - b < x ) implies ( for x st x in ]. a , b .[ holds ( x - b ) (#) ( x - b )
rng M5 c= rng ( D2 | Seg 1 ) & rng M5 c= rng ( D2 | Seg 1 ) ;
for p being Real st p in Z holds p >= a
( cn ) * f = proj1 * f .= f * f ;
( seq ^\ m ) . k <> 0 & ( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p -Path ( M ) ) . 2 = d ;
A \oplus ( B \ominus C ) = ( A \oplus B ) \ominus C
h \equiv gg . ( mod P ) & g . ( mod P ) = g . ( mod P ) ;
reconsider i1 = i-1 , i2 = - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V , v be VECTOR of V ;
for V being Subspace of V holds V is Subspace of [#] V
reconsider i-7 = i , is = j as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the sequence of B ) . n ;
len that len that len there f2 st f1 in Seg len f2 & f2 in Seg len f1 ;
pp1 c= the topology of T & pp1 c= the topology of T ;
]. r , s .[ c= [. r , s .[ ;
let B2 be Basis of T2 , x be Element of T2 ;
G * ( B * A ) = ( id o1 ) * ( id o2 ) ;
assume that p , u ] and u , q , v is_collinear ;
[ z , z ] in union rng ( F . n ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S , $1 = $1 .. S ;
LIN a1 , a3 , b1 & LIN b1 , b2 , c1 ;
f " ( f .: x ) = { x } ;
dom w2 = dom r12 & dom r12 = dom r12 ;
assume that 1 <= i and i <= n and j <= n ;
( ( g2 ) . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ii * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q7 . x in rng ( q7 | rng q7 ) ;
Carrier ( LLet ) misses Carrier ( L7 ) ` ;
consider c being element such that [ a , c ] in G ;
assume that N|[ o , o ]| = obeing Element of A and o <> o ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F |^ C-1 ) \/ ( F |^ C-1 ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x ^2 <= 1 ;
p `2 - q `2 <> 0. TOP-REAL 2 - q `2 ;
redefine func \subseteq [: S , T :] ;
let x be Element of S ~ ;
the ObjectMap of F is one-to-one & the ObjectMap of F is one-to-one ;
|. i .| <= - ( 2 |^ n ) / ( 2 |^ n ) ;
the carrier of I[01] = dom P & the carrier of I[01] = dom P ;
} * ( n + 1 ) > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 \/ A2 ) ;
a3 , a4 // b3 , b3 & a3 , a4 // b3 , b3 ;
then dom A <> {} & dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G & y in X implies x = y
set v2 = ( vseq /. ( i + 1 ) ) `1 , v2 = ( vseq /. ( i + 1 ) ) `2 ;
x = r . n .= r4 . n .= r4 . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom g = the carrier of I[01] ;
p in Lower_Arc ( P ) /\ Lower_Arc ( P ) ;
dom d2 = [: [: A , A :] , [: A , A :] :] ;
0 < ( p - ||. z .|| ) + 1 ;
e . ( m3 + 1 ) <= e . ( m3 + 1 ) ;
B \ominus X \/ B \ominus Y c= B \ominus X
-infty < Integral ( M , Im ( g | B ) ) ;
cluster O := F -> being Ordinal for OperSymbol of X ;
let U1 , U2 be non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X & g is_differentiable_on X ;
let x , y , z be Point of X , p be Point of X ;
reconsider pp = p . x , pp = p . y as Subset of V ;
x in the carrier of Lin ( A ) & x in the carrier of Lin ( A ) ;
let I , J be parahalting Program of SCM+FSA , a be Int-Location ;
assume that - a is lower and - a is lower and - a is lower ;
Int Cl A c= Cl Int Cl A & Int Cl A c= Cl Int Cl A ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( |[ x , y ]| , r ) ;
( p2 `2 ) ^2 / ( |. p2 .| ) ^2 <= ( p2 `2 ) ^2 / ( |. p2 .| ) ^2 ;
Cl Q ` = [#] TT .= [#] TT ;
set S = the carrier of T , T = the carrier of T ;
set I8 = ' ( f |^ n ) , I8 = ' ( f |^ n ) ;
len p - n = len ( thesis - n ) + 1 ;
A is Permutation of Swap ( A , x , y ) ;
reconsider n6 = n7 , n6 = n8 as Element of NAT ;
1 <= j & j + 1 <= len ( s | X ) ;
let q\mathopen be Let of M , qbe Let of M , q be State of M ;
a9 in the carrier of S1 & b9 in the carrier of S1 ;
c1 /. n1 = c1 . n1 & c2 . n1 = c2 . n1 ;
let f be FinSequence of TOP-REAL 2 , p be Point of TOP-REAL 2 ;
y = ( ( f * S8 ) . x ) . ( f . x ) ;
consider x being element such that x in an " A ;
assume r in ( dist ( o ) ) .: P ;
set i2 = ( n + 1 ) - 1 , h = ( n + 1 ) - 1 ;
h2 . ( j + 1 ) in rng h2 & h2 . ( j + 1 ) in rng h2 ;
Line ( M29 , k ) = M . i .= Line ( M29 , k ) ;
reconsider m = ( x - 2 ) / 2 as Element of ( - 2 ) / 2 ;
let U1 , U2 be strict Subspace of U0 , a be Element of U1 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 + 1 < len p2 + 1 ;
let T1 , T2 be Scott Scott TopAugmentation of L , X be Subset of L ;
then x <= y & ( x + y ) c= ( x + y ) ;
set M = n -tuples_on the carrier of K ;
reconsider i = x1 , j = x2 as Nat ;
rng ( ( the_arity_of a9 ) * ( the_arity_of o ) ) c= dom H ;
z1 " = z9 " & z1 " = z1 " * z1 " .= z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 "
x0 - r / 2 in L /\ dom f & x0 - r / 2 in L /\ dom f ;
then w is that rng w /\ L <> {} & rng w /\ L <> {} ;
set x9 = x9 ^ <* Z *> , y9 = y9 ^ <* Z *> , x9 = Z ^ <* Z *> ;
len w1 in Seg len w1 & len w1 = len w2 & len w1 = len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = ( |. a . n .| ) * ( |. b . n .| ) ;
( p `1 ) ^2 / ( G * ( len G , 1 ) ) ^2 <= ( G * ( len G , 1 ) ) `1 ;
rng ( g ) c= L~ ( g ) \/ L~ ( g ) ;
reconsider k = i-1 * ( l + j ) as Nat ;
for n being Nat holds F . n is \HM { -infty } ;
reconsider x9 = x9 , y9 = y9 as VECTOR of M ;
dom ( f | X ) = X /\ dom f .= dom f /\ X ;
p , a // p , c & b , a // c , c ;
reconsider x1 = x , y1 = y , y2 = z as Element of REAL m ;
assume i in dom ( a (#) p ^ q ) ;
m . ag = p . ag .= s . ag ;
a / ( s . m - n ) / ( s . m - n ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume that B1 \/ C1 = B2 \/ C2 and C2 = {} \/ C2 ;
X . i = { x1 , x2 } . i .= x1 ;
r2 in dom ( h1 + h2 ) & r1 < r2 & r2 < x0 + r2 ;
- - 0. R = a & b-0 = b ;
F8 is_closed_on t3 , Q & F8 is_halting_on t , Q ;
set T = for X being in for x0 being Element of X holds x0 in X ;
Int Cl Int Cl R c= Int Cl R & Int Cl R c= Int Cl R ;
consider y being Element of L such that c . y = x ;
rng ( F{} ) = { F{} . x } .= { F . x } ;
G-23 " { c } c= B \/ S ;
f[#] is Relation of [: X , X :] , X & f\HM of X , X ;
set RP = the Point of ( P ) | R ;
assume that n + 1 >= 1 and n + 1 <= len M ;
let k2 be Element of NAT , k be Nat ;
reconsider pcj = u , pj = v , pj = w as Element of ( TOP-REAL n ) | ( i + 1 ) ;
g . x in dom f & x in dom g implies x = g . x
assume that 1 <= n and n + 1 <= len f1 ;
reconsider T = b * N as Element of G / ( N , G ) ;
len P\mathopen { x } <= len P-35 & len P\mathopen { x } = len P-35 ;
x " in the carrier of A1 & x " in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple ;
f . x = a . i .= a1 . k .= b1 . k ;
let f be PartFunc of REAL i , REAL , x be Element of REAL i ;
rng f = the carrier of \bf 1 & rng f c= the carrier of \bf 1 ;
assume s1 = sqrt ( 2 * p ) - sqrt ( 2 * p ) ;
pred a > 1 & b > 0 & a / b > 1 ;
let A , B , C be Subset of Ik ;
reconsider X0 = X , Y0 = Y as RealNormSpace , X0 = Z as Point of X ;
let f be PartFunc of REAL , REAL , x be Element of REAL ;
r * ( v1 |-- I ) . X < r * 1 ;
assume that V is Subspace of X and X is Subspace of V ;
let t-3 , t-3 , t-3 be Relation of the carrier of S ;
Q [ e-14 \/ { v-5 } , f ] & f . v-5 = f . e-5 ;
g \circlearrowleft ( L~ z ) = z implies ( g /. 1 ) .. z = ( g /. len z ) .. z
|. |[ x , v ]| - |[ x , y ]| .| = vfunction ;
- f . w = - ( L (#) w ) .= - ( L (#) w ) ;
z - y <= x iff z <= x + y & y <= z + x ;
( 7 / p1 ) |^ ( 1 / e ) > 0 ;
assume X is BCK-algebra & 0 , 0 , 0 , 0 & 0 , 0 , 0 , 0 ;
F . 1 = v1 & F . 2 = v2 & F . 3 = v2 ;
( f | X ) . x2 = f . x2 .= ( f | X ) . x2 ;
( ( tan - cot ) `| Z ) . x = dom ( sec . x ) ;
i2 = ( f /. len f ) `2 .= ( f /. len f ) `2 .= ( f /. 1 ) `2 ;
X1 = X2 \/ ( X1 \ X2 ) .= X2 \/ ( X1 \ X2 ) ;
[. a , b , 1_ G .] = 1_ G ;
let V , W be non empty VectSpStr over F_Complex , f be FinSequence of V ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] & dom f2 = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: X .= proj2 .: ( X /\ Y ) ;
f . ( x , y ) = h1 . ( x `1 , y `2 ) ;
x0 - r < a1 . n & x0 - r < x0 + r ;
|. ( f /* s ) . k - x0 .| < r ;
len Line ( A , i ) = width A & width Line ( A , i ) = width A ;
Sbeing Element of S5 * , g being Element of S5 * ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom Initialized s & ( Initialized s ) . intloc 0 in dom Initialized s ;
i1 , i2 , i3 , i3 , a5 , 8 , 7 , 8 , 9 , 8 , 8 , 9 , 8 be Nat ;
arccos r + arccos r = ( PI / 2 ) + 0 ;
for x st x in Z holds f2 is_differentiable_in x & for x st x in Z holds f2 . x <> 0 ;
reconsider q2 = ( q - x ) / ( q - x ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + j1 & i + 1 <= j + 1 ;
assume f in the carrier of [: X , Omega Y :] ;
F . a = H / ( ( x , y ) / ( x , y ) ) ;
( ( {} T ) at ( C , u ) ) . ( ( {} T ) . ( u , u ) ) = TRUE ;
dist ( ( a * seq ) . n , h ) < r / 2 ;
1 in the carrier of [. 0 , 1 .] & 1 / 2 < 1 / 2 ;
( p2 `1 - x1 ) - x1 > - g & g - x1 < - g ;
|. r1 - `2 .| = |. a1 .| * |. thesis .| ;
reconsider S-14 = 8 , S-14 = 8 as Element of Seg 8 ;
( A \/ B ) |^ b c= A |^ b \/ B |^ b
D0W .|^ ( n ) = D0W .succ ( n + 1 ) ;
i1 = ma + n & i2 = K + n & j2 = K + n ;
f . a [= f . ( f . O1 "\/" a ) ;
pred f = v & g = u & f + g = v + u ;
I . n = Integral ( M , F . n ) ;
chi ( T1 , S ) . s = 1 & chi ( T1 , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . b3 , k1 = s . b3 as Element of NAT ;
( Comput ( P , s , 4 ) ) . GBP = 0 ;
L~ M1 meets L~ ( R * ( len M1 , j ) ) ;
set h = the continuous Function of X , R , g be Function of X , R ;
set A = { L . ( k9 . n ) : not contradiction } ;
for H st H is atomic holds P7 [ H ] ;
set b-> ( S5 , i ) --> ( i , j ) , S5 = ( S , i ) --> ( j , i ) ;
Hom ( a , b ) c= Hom ( a opp , b opp ) ;
( 1 - s ) / ( n + 1 ) < ( 1 - s ) / ( n + 1 ) ;
( l ) `1 = [ dom l , cod l ] `1 .= [ dom l , cod l ] `2 .= cod l ;
y +* ( i , y /. i ) in dom g & y . i in dom g ;
let p be Element of [: [: Al , Al :] , D :] ;
X /\ X1 c= dom ( f1 - f2 ) /\ dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) & p2 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in ( L2 /\ K0 ) \/ ( ( K /\ R ) /\ R ) ;
- 1 <= ( ( f2 ) . O ) `2 & - 1 <= ( ( f2 ) . O ) `2 ;
let f , g be Function of I[01] , TOP-REAL 2 , a , b , c , d be Real ;
k1 - k2 = k1 - k2 - k2 .= k1 - k2 - k2 ;
rng seq c= ]. x0 , x0 + r .[ & rng seq c= ]. x0 , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ & x0 < g2 + r / 2 ;
sgn ( p `1 , K ) = - ( 1_ K ) .= - ( 1_ K ) ;
consider u being Nat such that b = p |^ ( y * u ) ;
ex A being as as as as as \smallfrown of W & ex A being Ordinal of W st a = Sum A
Cl ( union H ) = union ( ( Cl H ) \ { {} } ) ;
len t = len t1 + len t2 & len t1 + len t2 = len t1 + len t2 ;
v-29 = v + w |-- A + A8 .= v + A8 ;
cv <> DataLoc ( t3 . GBP , 3 ) , cv = DataLoc ( t3 . GBP , 3 ) ;
g . s = sup ( d " { s } ) .= s . s ;
( \dot y ) . s = s . ( y . s ) ;
{ s : s < t } in NAT implies t = {} & t = {} ;
s ` \ s = s ` \ 0. X .= ( 0. X ) \ ( 0. X ) ;
defpred P [ Nat ] means B + $1 in A & A + $1 in B + $1 ;
( 329 + 1 ) ! = 329 ! * ( 329 + 1 ) ;
( U succ A ) = ( ( A * ( A * B ) ) ) * ( A * ( B * ( A * ( A * B ) ) ) ) ;
reconsider y = y , z = z 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 , q = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) \mathop { \it Boolean } , g = ( S , U ) \mathop { \it Boolean } ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , a , b , c be Real ;
( ( 1 + i ) * 'not' A ) . [ n + i , 'not' A ] <> 1 ;
ex r being Real st x = r & a <= r & r <= b ;
let R1 , R2 be Element of ( n + 1 ) -tuples_on REAL , a be Element of REAL ;
reconsider l = 0. ( V ) , m = 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 'or' ( b 'or' c ) = 'not' ( ( a 'or' b ) 'or' c ) ;
||. x9 - g1 .|| < r2 - g1 & ||. g - g1 .|| < r ;
b9 , a9 // b9 , c9 & b9 , c9 // b9 , c9 & a , b // c , c9 ;
1 <= k2 -' k1 & k2 + 1 = k2 & k2 + 1 = k1 + 1 + 1 ;
( ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 >= 0 ;
( q `2 / |. q .| - sn ) < 0 ;
E-max C in right_cell ( RCage ( C , 1 ) , 1 ) /\ L~ Cage ( C , 1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( lim F ) = Re ( lim G ) .= Re ( lim G ) ;
LIN b , a , c or LIN b , c , a ;
p `2 , a `2 // a `2 , b or p `2 , a `2 // b `2 , a `2 ;
g . n = a * Sum ( f | n ) .= f . n ;
consider f being Subset of X such that e = f and f is x0 ;
F | ( N2 ~ ) = CircleMap * ( F-4 " ) .= ( CircleMap * FK1 ) " ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r0 ) c= Ball ( m , s ) & Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of (0). V = { 0. V } .= { 0. V } ;
rng ( ( cos * 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 as ( 0 , {} ) string of S2 , C = { {} } as Element of D ;
reconsider x9 = seq . n , y9 = seq . n as sequence of REAL n ;
assume that E-max C meets L~ Cage ( C , n ) and L~ Cage ( C , n ) meets L~ go ;
- ( ( - 1 ) / 2 ) < F . n - ( - 1 ) / 2 ;
set d1 = being Subset of dist ( x1 , z1 ) , d2 = dist ( x1 , z2 ) , d2 = dist ( x2 , z1 ) ;
2 |^ ( 100 -' 1 ) = 2 |^ ( 100 ) - 1 ;
dom ( vG2 ) = Seg len ( d6 ) .= Seg len ( d6 ) ;
set x1 = - k2 + |. k2 .| + 4 * ( k + 1 ) ;
assume for n being Element of X holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= T-32 . i and T-32 . ( i + 1 ) <= 1 and T-32 . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the carrier of ( Carrier ( LT ) + L2 ) c= I2 ;
'not' Ex ( x , p ) => All ( x , 'not' p ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the normal normal w.r.t. of {} ;
Z c= dom ( ( sin (#) f1 ) `| Z ) & Z c= dom ( ( cos (#) f1 ) `| Z ) ;
|. 0. TOP-REAL 2 - ( q `2 / |. q .| - sn ) .| < r ;
ConsecutiveSet2 ( B , succ d ) c= ConsecutiveSet2 ( A , succ d ) & \mathbb L ( A , succ d ) c= L ;
E = dom L8 & L is measurable & L is measurable & L . ( k + 1 ) = L . ( k + 1 ) ;
C |^ ( A + B ) = C |^ B * C |^ A ;
the carrier of W2 c= the carrier of V & the carrier of W1 c= the carrier of V ;
I . IC Comput ( P , s , m ) = P . IC Comput ( P , s , m ) .= halt SCM+FSA ;
pred x > 0 means : Def1 : ( 1 / 2 ) ^2 = x ^2 / 2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .] and p in A ;
b , c are_connected & - C , - C - ( b , c ) + - ( b , c ) + - ( b , c ) + - ( b , c ) + - ( b , c ) + - ( b , d ) + - ( b , c ) ) + - (
assume f = id the carrier of O1 & f is Function of the carrier of O1 , the carrier of O2 ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( ( the carrier of V ) \ { 0. V } ) ;
reconsider g = f " as Function of U2 , U1 , U2 ;
A1 in the Points of G_ ( k , X ) & A2 in the Points of G ;
|. - x .| = - ( - x ) .= - ( - x ) .= - ( - x ) .= - ( - x ) ;
set S = ) +* 1GateCircStr ( x , y , c ) ;
Fib ( n ) * ( 5 * Fib ( n ) ) >= 4 * ;
vseq /. ( k + 1 ) = vseq . ( k + 1 ) .= vseq /. ( k + 1 ) ;
0 mod i = - ( i * ( 0 qua Nat ) ) .= - ( i * ( 0 qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & len M1 = n & width M1 = n ;
Line ( S\mathopen { i } , j ) = S\mathopen { i } .= S . j ;
h . ( x1 , y1 ) = [ y1 , x1 ] & h . ( y1 , y2 ) = [ y1 , y2 ] ;
|. f .| (#) ( Re ( |. f .| (#) h ) ) is nonnegative ;
assume x = ( a1 ^ <* x1 *> ) ^ b1 & y = ( a1 ^ <* x1 *> ) ^ b1 ;
Mi is_closed_on IExec ( I , P , s ) . a , P & Mi is_halting_on s , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , q , c & LIN c , q , b ;
fsuch . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) .= x1 + y1 + z1 ;
flim . a = flim . a & flim a in InputVertices S & flim a in InputVertices S ;
( p `1 ) ^2 + ( p `2 ) ^2 <= ( ( E-max C ) `1 ) ^2 + ( p `2 ) ^2 ;
set R8 = Cage ( C , n ) \circlearrowleft E8 , E8 = Cage ( C , n ) ;
( p `1 ) ^2 >= ( E-max C ) `1 / 2 * ( ( E-max C ) `1 ) ^2 ;
consider p such that p = p-20 and s1 < p and p < s2 and p < s2 ;
|. ( f /* ( s * F ) ) . l - GM .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 + 1 ) = width N ;
f1 /* s1 is convergent & f2 /* s1 is convergent & lim ( f1 /* s1 ) = x0 ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 <> 0 implies f | ( len f + 1 ) = f | ( len f + 1 )
dom ( Proj ( i , n ) * s ) = REAL m .= REAL m ;
n = k * ( 2 * t ) + ( n mod ( 2 * k ) ) ;
dom B = 2 -tuples_on the carrier of V \ { {} } .= { {} } ;
consider r such that r _|_ a and r _|_ x and r _|_ y ;
reconsider B1 = the carrier of Y1 , B2 = the carrier of Y2 as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 / 2 .] & 1 / 2 < 1 / 2 ;
for L being complete LATTICE holds <* <* <* \mathbb L *> , L *> *> , L are_isomorphic
[ gi , gj ] in Ii \ Ij ;
set S2 = 1GateCircStr ( x , y , c ) , S2 = 1GateCircStr ( y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and for r st r < x0 holds f1 . r < f2 . x0 ;
reconsider y = ( a " ) / ( F . ( len F ) ) , z = a " * ( F . ( len F ) ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = d1 & s . 2 = d2 ;
( min ( g , 1 ) - f ) . c <= h . c - f . c ;
set G2 = the as \HM { of G , v } -\langle v , w *> , G = the \langle of G , v *> ;
reconsider g = f as PartFunc of REAL , REAL-NS n , REAL-NS n ;
|. s1 . m - p .| / |. s1 . m - p .| < d / |. s1 .| ;
for x being element st x in for u being element st u in consider t holds x in ( q . u ) holds x in t
P = the carrier of ( TOP-REAL n ) | P .= [#] ( ( TOP-REAL n ) | P ) ;
assume that p00 in LSeg ( p1 , p2 ) /\ LSeg ( p00 , p2 ) and p2 in LSeg ( p1 , p2 ) /\ LSeg ( p11 , p2 ) ;
( 0. X \ x ) |^ ( m * ( k + 1 ) ) = 0. X ;
let g be Element of Hom ( cod f , dom f ) ;
2 * a * b + ( 2 * c * d ) <= 2 * C1 * C2 ;
let f , g , h be Point of the complex normed space of X , Y ;
set h = Hom ( a , g ) , f = id a ;
then ( idseq n ) | ( Seg m ) = idseq m & m <= n ;
H * ( g " * a ) in the right of H & H * ( g " * a ) in the right of H ;
x in dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) & x in dom ( ( - 1 / 2 ) (#) ( sin * cos ) ) ;
cell ( G , i1 , j2 -' 1 ) misses C & cell ( G , i1 , j2 -' 1 ) misses C ;
LE q2 , p4 , P , p1 , p2 & LE q2 , p2 , P , p1 , p2 ;
attr B is BDD component means : Def1 : B c= BDD A ;
deffunc D ( set , set ) = union rng $2 & $2 = union rng $2 ;
n + - n < len ( p\frac + n - 1 ) + n - 1 ;
attr a <> 0. K means : Def1 : the_rank_of M = the_rank_of ( a * M ) & the_rank_of M = the_rank_of ( a * M ) ;
consider j such that j in dom \mathbb Z and I = len PI + j and x = len PI + 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 CC1 = Comput ( P2 , s2 , i + 1 ) , CC2 = Comput ( P2 , s2 , i + 1 ) , CC2 = Comput ( P2 , s2 , i + 1 ) , CC2 = Comput ( P2 , s2 , i + 1 ) , CC2 = Comput ( P2 , s2
set cv = 3 / 2 * ( a , b , c ) , cv = - ( b - c ) / 2 ;
conv @ W c= union ( F .: ( E " W ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( arccot ) ) ;
r3 <= s0 + ( ( |. v2 - v2 .| ) / 2 ) * ( 1 / 2 ) ;
dom ( f (#) f4 ) = dom f /\ dom f4 .= dom ( f (#) f4 ) .= dom f /\ dom f4 ;
dom ( f (#) G ) = dom ( l (#) F ) /\ Seg k .= dom ( l (#) G ) ;
rng ( s ^\ k ) c= dom f1 \ { x0 } & rng ( s ^\ k ) c= dom f2 \ { x0 } ;
reconsider g9 = gp , gq = gq as Point of TOP-REAL n1 , gq = gq as Point of TOP-REAL n1 ;
( T * h . s ) . x = T . ( h . s ) . x ;
I . ( L . ( J . x ) ) = ( I * L ) . ( J . x ) ;
y in dom <* *> & ( commute ( Frege ( A . o ) ) ) . y = ( commute ( A . o ) ) . y ;
for I being non degenerated commutative Ring holds the carrier of I is commutative non empty doubleLoopStr
set s2 = s +* Initialize ( ( intloc 0 ) .--> 1 ) , P2 = P +* Initialize ( ( intloc 0 ) .--> 1 ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 ;
lim S1 in the carrier of [. a , b .] & lim S1 = a * b ;
v . ( l-13 . i ) = ( 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 x in F1 . x ;
card Funcs ( X , 0 , x1 , x2 , x3 , x4 ) = { EF } .= 0 ;
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 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n2 , n3
mg1 . HT ( mg2 , T ) = 0. L .= 0. L ;
then H1 , H2 |^ ( n + 1 ) " & ( Cl H1 ) , ( Cl H2 ) " ;
( ( N-min L~ f ) .. f ) .. ( ( f /. len f ) + 1 ) > 1 ;
]. s , 1 .[ = ]. s , 2 .[ /\ [. 0 , 1 .] ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) & x2 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) ) ;
let f1 , f2 be continuous PartFunc of REAL , the carrier of S , x be Element of REAL ;
DigA ( t-23 , z9 ) is Element of k -tuples_on ( the carrier of K ) ;
I \mathop { \rm 22k1 } = d2k1 & I \mathop { \rm I } = k2 ;
u9 ~ { u } = { [ a , u9 ] , [ a , u9 ] } .= [ a , u9 ] ;
( w | p ) | ( w | ( w | ( w | ( w | ( w | p ) ) ) ) ) = p ;
consider u2 such that u2 in W2 and x = v + u2 and u2 in W2 and u1 in W3 ;
for y st y in rng F ex n st y = a |^ n & a < n
dom ( ( g * {} ( V \dot \to C ) ) | K ) = K ;
ex x being element st x in ( ( {} ) \/ A ) . s & x in ( {} ) ;
ex x being element st x in ( \HM { the } \HM { carrier of O1 } \/ A ) . s ;
f . x in the carrier of [. - r , r .[ & f . x = r ;
( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} & ( the carrier of X1 union X2 ) /\ ( the carrier of X2 ) <> {} ;
L1 /\ LSeg ( p00 , p2 ) c= { p11 } /\ LSeg ( p00 , p2 ) ;
( b + bLet r ) / 2 in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of GX such that z = y and P [ z ] and z in A and z in B ;
( the sequence of ( iff for e be Real st e in W holds e - x <= e ) ) implies for x be Element of X holds x - x <= e
len ( w ^ w2 ) + 1 = len w + 2 + 1 .= len w + 1 ;
assume that q in the carrier of ( TOP-REAL 2 ) | K1 and q <> 0. TOP-REAL 2 ;
f | Eq ` = g | Eq ` .= g | Eq ` .= g | Eq ` ;
reconsider i1 = x1 , i2 = x2 , j1 = y2 , j2 = y1 , j1 = y2 as Element of NAT ;
( a * A ) @ = ( a * ( A * B ) ) @ .= a * ( A * B ) ;
assume ex n0 being Element of NAT st f |^ n0 is \mathop { 0 } & f . n0 is min ;
Seg len ( ( the thesis ) * ( f2 ) ) = dom ( ( the support of ( f * f1 ) ) ) .= dom ( ( the support of ( f * f1 ) ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . n & ( Complement A1 ) . n c= ( Complement A1 ) . n ;
f1 . p = p9 & g1 . p = d & g1 . ( p + 1 ) = d & g2 . ( p + 1 ) = d ;
FinS ( F , Y ) = FinS ( F , dom ( F | Y ) ) ;
( x | y ) | z = z | ( y | x ) ;
( |. x .| |^ n ) / ( n + 1 ) <= ( r2 |^ n ) / ( n + 1 ) ;
Sum ( F-12 ) = Sum f & dom ( F-12 ) = dom g & rng ( F-12 ) = dom g ;
assume for x , y being set st x in Y & y in Y holds x /\ y in Y ;
assume that W1 is Subspace of W3 and W2 is Subspace of W3 and W1 is Subspace of W3 and W2 is Subspace of W3 ;
||. t-15 . x - t-15 . x .|| = lim ||. ( vseq . x - vseq . x ) .|| .= ||. ( vseq . x - vseq . x ) .|| ;
assume that i in dom D and f | A is lower and g | A is lower ;
( ( p `2 ) ^2 + 1 ) * ( - 1 ) <= ( - 1 ) * ( - 1 ) ;
g | Sphere ( p , r ) = id Sphere ( p , r ) .= id Sphere ( p , r ) ;
set N8 = ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
for T being non empty TopSpace holds T is countable countable implies the TopStruct of T is countable countable
width B |-> 0. K = Line ( B , i ) .= B * ( i , i ) .= B * ( i , j ) ;
attr a <> 0 means : Def1 : ( A \ B ) = ( A \ a ) Let ( B \ a ) ;
then f is_is_is_\cal 2 2 , u & pdiff1 ( f , 1 ) is_partial_differentiable_in u , 3 ;
assume that a > 0 and a <> 1 and b > 0 and c <> 1 and b <> 0 ;
w1 , w2 in Lin { w1 , w2 } & w2 in Lin { w1 , w2 } ;
p2 /. IC s = p2 . IC s .= ( IC Comput ( p2 , s2 , m ) ) + 2 .= IC Comput ( p2 , s2 , m ) ;
ind ( T-10 | b ) = ind b .= ind b .= ind b .= ind b ;
[ a , A ] in the let of the Points of an & [ a , A ] in the Indices of the P of the P of the P of the P of the Line ( A , A ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) & m in the Arrows of C ;
( ( a , CompF ( PA , G ) ) . z ) . z = FALSE ;
reconsider phi = phi , phi = phi , phi = phi , phi = phi as Element of ( D * ) * ;
len s1 - ( len s2 - 1 ) + 1 > 0 + 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 z in dom g2 and g2 . z = y ;
[#] V1 = { 0. V1 } .= the carrier of ( V + W ) .= { 0. 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 ;
h1 = f ^ ( <* p3 *> ^ <* p *> ) .= h ^ <* p *> .= h ;
c /. |[ b , c ]| `1 = c .= |[ a , c ]| `1 .= a ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Term of C , V ;
( 1 / 2 ) in the carrier of [. 1 / 2 , 1 / 2 .] ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( p1 `2 ) + D .= C * ( p1 `2 ) + D ;
R . b - b = 2 * - b .= 2 * b - b .= b ;
consider ] such that B = 1- 1 * C + ( - 1 ) * A and 0 <= 1 ;
dom g = dom ( ( the Sorts of A ) * ( the Arity of S ) ) .= the carrier of S ;
[ P . ( l ) , P . ( l ) ] in => ( T . ( k + 1 ) ) ;
set s2 = Initialize s , P2 = P +* stop I ;
reconsider M = mid ( z , i2 , i1 ) , N = L~ z as non empty Subset of TOP-REAL 2 ;
y in product ( ( the support of J ) +* ( V , { 1 } ) ) ;
1 / 2 = 1 & 0 / 2 = 0 & 1 / 2 = 1 / 2 ;
assume x in the left of g or x in the left of g or x in the right of g ;
consider M being strict non-empty MSAlgebra over A9 such that a = M and T is non-empty & M is non-empty & M is non-empty ;
for x st x in Z holds ( ( ( - 1 / 2 ) (#) f ) `| Z ) . x <> 0
len W1 + len W2 + m = 1 + len W3 + m .= len W3 + len W3 + m .= len W3 + m + 1 ;
reconsider h1 = ( vseq . n ) - t-16 as Lipschitzian Lipschitzian LinearOperator of X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is for s1 , s2 st s1 in the { of s2 } & s2 in the { the carrier of s2 } holds s1 = s2 ;
( ( for x , y being Element of X holds x , y ] ) & ( x , y ) . ( x , y ) = gcd ( x , y , x ) ;
for u being element st u in Bags n holds ( p + m ) . u = p . u + m . u
for B being Subset of u-5 st B in E holds A = B or A misses B ;
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = tree p \/ W1 , W1 = tree q , W2 = p +* q ;
x in { X where X is Ideal of L : X is Ideal of L & X is Ideal of L } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 /\ W2 c= the carrier of W1 /\ W2 ;
( for a , b holds a * id a = id a ) & ( id a ) * id b = id a ;
( dom ( X --> f ) ) . x = ( X --> dom f ) . x .= f . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( q => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= ( ( 2 |^ ( n -' m ) ) + 1 ) + 1 ;
( reproj ( 1 , z0 ) ) . x in dom ( f1 (#) f2 ) /\ dom ( f2 (#) f3 ) ;
assume that b1 . r = { c1 } and b2 . r = { c2 } and c1 . r = c1 . r ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & d on P & a on P & b on P & c on P & d on P & d on P & a on P & b on P & b on P & c on P & d on P & d on P & a , b
reconsider gf = g * f `2 , hf = h * g `2 as strict Element of X ;
consider v1 being Element of T such that Q = ( downarrow v1 ) ` and v1 in ( downarrow v2 ) ` and v1 in ( downarrow v2 ) ` ;
n in { i where i is Nat : i < n0 + 1 & i < n + 1 } ;
( F * ( i , j ) ) `2 >= ( F * ( m , k ) ) `2 ;
assume K1 = { p : p `1 >= sn & p `2 >= 0 & p `1 >= 0 } ;
ConsecutiveSet ( A , succ O1 ) = ( ConsecutiveSet ( A , O1 ) ) ^ ( A , O1 ) ;
set I1 = Macro ( a , intloc 0 ) , I2 = AddTo ( a , intloc 0 ) , I2 = goto 2 , I3 = goto 3 ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1 ;
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) & the carrier of L1 c= the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a & x9 |^ 3 = b |^ 3 ;
reconsider ee = ee , ff = ff , ff = f , ff = f as Element of D ;
ex O being set st O in S & C1 c= O & M . O = 0. ( Cl F ) ;
consider n being Nat such that for m being Nat st n <= m holds S . m in U1 and S . n in U2 ;
f (#) g * reproj ( i , x ) is_differentiable_in ( proj ( i , m ) ) . x ;
defpred P [ Nat ] means A + succ $1 = succ $1 & A + succ $1 = succ $1 + A & A = succ $1 ;
the left st - g = the left of ( - g ) & the carrier of ( - g ) = the carrier of ( - g ) ;
reconsider pp = x , pp = y , pp = z , pp = w as Point of TOP-REAL 2 ;
consider g3 such that g3 = y and x <= ex g2 st g2 <= x0 & g2 <= x0 & g2 <= g2 & g2 <= x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ] & X [ n , r ]
len ( x2 ^ y2 ) = len x2 + len y2 .= len ( x2 ^ y2 ) + len y2 .= len ( x2 ^ y2 ) + len y2 ;
for x being element st x in X holds x in the set of the set of \HM { 0 } & x is Element of NAT implies x is Element of NAT
LSeg ( p11 , p2 ) /\ LSeg ( p00 , p2 ) = {} /\ LSeg ( p11 , p2 ) .= {} ;
func \rm such that for X being set holds X in \mathop { \rm [ id X , id X ] } implies X is thesis & X is thesis implies X is thesis
len ( ( CR ( CC ) ) * ( CR /. 1 ) ) <= len ( CC ) + len ( CC ) ;
attr K is a1 means : Def1 : a <> 0. K & v . ( a |^ i ) = i * v . a ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] and o in rng p ;
for x st x in X ex y st x c= y & y in X & y is \Omega of f . x
IC Comput ( P-6 , smeans : Let : IC Comput ( P-6 , smeans , k ) in dom san & IC Comput ( PE , sE , k ) in dom I ;
attr q < s means : Def1 : r < s & s < q & q < s ;
consider c being Element of Class f such that Y = ( F . c ) . ( [ c , 3 ] , [ c , 3 ] ) ;
the ResultSort of S2 = id the carrier of S2 & the ResultSort of S2 = id 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 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z 2 ) * ( ( #Z
r-7 in Int cell ( GoB f , i , GoB 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 & i - len f <= len f - len f + len f - 1 ;
for n ex x st x in N & x in N1 & h . n = - x0 & h . n > 0
set s0 = ( \mathop { a , I , p , s ) . i , p = ( \mathop { a , I } , p , s ) . i , q = ( \mathop { a , I } , p , s ) . i , s = ( \mathop { a , I } , p ) . i , p = ( \mathop { a , I } , p ) . i , q =
( p . k ) . 0 = 1 or ( p . k ) . 0 = - 1 or p . k = 1 ;
u + Sum ( L-18 ) in ( U \ { u } ) \/ { u + Sum ( L-18 ) } ;
consider x9 being set such that x in x9 and x9 in V1 and x9 in V1 and x9 = [ x9 , x9 ] ;
( p ^ ( q | k ) ) . m = ( q | k ) . ( ( len p ) - len p ) ;
g + h = gg + hh & for n holds Nat holds Nat ( g + h , X , X ) . n = g + h . n ;
L1 is distributive & L2 is distributive implies for L being distributive LATTICE holds L1 ~ is distributive & L2 ~ is distributive & for x being Element of L holds x in the carrier of L implies x is distributive
pred x in rng f means : Def1 : y in rng ( f . x ) & f . x = f . y ;
assume that 1 < p and p + 1 / q = 1 and 0 <= a and a <= b and b <= q ;
F* ( f , <* the carrier of L *> ) = rpoly ( 1 , the carrier of L ) *' t .= ( 0. L ) *' t ;
for X being set , A being Subset of X holds A ` = {} implies A = {} & A = {} & A = {} & A = {} & A = {} & A = {} implies A = {}
( ( ( ( ( ( ( Y ) ) | X ) ) | Y ) ) ) `1 <= ( ( ( ( ( ( ( Y ) ) | X ) ) | Y ) ) `1 ) `1 ;
for c being Element 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 .= s . GBP .= s . GBP .= s . GBP ;
for a , b being Real holds [ a , b ] in ( y >= 0 ) implies b >= 0 & a >= 0 & b >= 0
for x , y being Element of X holds x ` \ y = ( x \ y ) ` & y \ x = 0. X ;
mode BCK-algebra of i , j , m , n , m , n , m , k be Element of NAT ;
set x2 = |( ( Re y ) , ( Im x ) )| ;
[ y , x ] in dom u5 & u5 . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A & A c= A implies A = B
0 <= delta ( S2 . n ) & |. delta ( S2 . n ) .| < ( e / 2 ) * ( e / 2 ) ;
( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) <= ( - ( q `1 / |. q .| - cn ) ) / ( 1 + cn ) ;
set A = ( 2 / b-a ) ;
for x , y being set st x in R" holds x , y are__Let x , y implies x , y are__Let y
deffunc FG2 ( Nat ) = b . $1 * ( M * G ) . $1 * ( M * G ) . $1 ;
for s being element holds s in consider f 'or' g iff s in means : Let s being Element of S holds s in means : Let s ;
for S being non empty non void non empty ManySortedSign st S is connected 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 and n1 < r + s ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( A ) is Subspace of Lin ( B ) ;
set n-15 = n-13 '&' ( M . ( x , y ) qua Element of BOOLEAN ) , n-15 = M . ( x , y ) , n-15 = M . ( x , y ) ;
f " V in such that f " V in D and f " V in D & f " V in D ( the carrier of X , p ) ;
rng ( ( a ^\ c ) +* ( 1 , b ) ) c= { a , c , b } ;
consider y being Was Vertex of G1 such that y `1 = y and dom y `2 = WWg ;
dom ( 1 / f ) /\ ]. - 1 , x0 .[ c= ]. - 1 , x0 .[ & f . x = - 1 / ( g . x ) ;
as Element of as Element of as Element of as of as of as of as of as Element of as ( Seg i , j , n , - r ) ;
v ^ ( nE |-> 0 ) in Lin ( rng ( ( B | c1 ) | c2 ) ) & v ^ ( B | c2 ) = v ^ ( B | c2 ) ;
ex a , k1 , k2 st i = a := k1 & i = a := k2 & i = a := k2 & i = a := k2 & i = a := k2 ;
t . NAT = ( NAT .--> succ i1 ) . NAT .= succ i1 .= succ i1 .= succ i1 .= succ i1 .= succ i1 .= succ i1 ;
assume F is bbfamily & rng p = Seg ( n + 1 ) & dom p = Seg ( n + 1 ) & for k st k in Seg ( n + 1 ) holds p . k = F ( k + 1 ) ;
not LIN b , b9 , a & not LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , c & LIN a , a9 , b & LIN a , a9 , c ;
( L1 \HM { L } \HM { , L2 } ) \& O c= ( L1 \HM { L } \HM { , L2 } ) Let ( L2 \HM { L } )
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( contradiction ) = b * ( -w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( |. $1 .| ) ;
u = cos . ( x , y ) * x + ( cos . ( x , y ) ) * y .= v + ( cos . ( x , y ) ) * y .= v ;
dist ( ( seq . n ) + x , g ) + dist ( x , g ) <= dist ( ( seq . n ) , g ) + 0 ;
P [ p , |. p .| ^ {} , {} , id ( the Sorts of A ) . ( id the Sorts of A ) ] ;
consider X being Subset of CQC-WFF ( Al ) such that X c= Y and X is non empty and X is non empty and X is non empty and X is non empty and X is non empty ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= l1 & l1 <= h & l1 <= g & l1 <= h } ;
vol ( ( G . n ) vol ) <= vol ( ( G . n ) vol ) ) * vol ( ( G . n ) vol ) ;
f . y = x .= x * ( ( power L ) . ( y , 0 ) ) .= x * ( power L ) . ( y , 0 ) ;
NIC ( goto i1 , ( succ i1 ) + ( succ i1 ) ) = { i1 , succ i1 } .= { succ i1 , succ i1 } .= { succ i1 , succ i1 } ;
LSeg ( p00 , p2 ) /\ LSeg ( p1 , p11 ) = { p1 } /\ LSeg ( p11 , p2 ) /\ LSeg ( p00 , p2 ) ;
Product ( ( the support of I-15 ) +* ( i , { 1 } ) ) in Z & Z . i = Z ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s1 , n ) +* Following ( s2 , n ) .= Following ( s2 , n ) +* Following ( s2 , n ) ;
W-bound Qs2 <= ( q1 `1 ) / 2 & q1 `1 <= ( q1 `1 ) / 2 & q1 `1 <= ( q1 `1 ) / 2 ;
f /. i2 <> f /. ( ( i1 + len g -' 1 ) + len g -' 1 ) & f /. ( i1 + 1 ) = f /. ( i1 + 1 ) ;
M , v / ( x. 3 , m ) / ( x. 4 , m ) / ( x. 4 , m ) |= H ;
len ( ( P ^ ) ) in dom ( ( P ^ ) ) & len ( ( P ^ ) ) = len ( ( P ^ ) ) + len ( ( P ^ ) ) ;
A |^ ( n , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , n ) ;
( R |^ n ) \ { q : |. q .| < a } c= { q1 : |. q1 .| >= a } ;
consider n1 being element such that n1 in dom p1 and y1 = p1 . n1 and y = p1 . n1 and n1 in dom p1 and p1 . n1 = p2 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q & Z <> X holds not Z in X ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA .= halt SCM+FSA .= halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng |. ( id the carrier of V ) . v .| & ||. v .|| = ||. v .|| . v
for phi holds phi in X implies phi in X & not phi in X & phi in X & phi is not ( phi in X implies phi is not R )
rng ( ( Sgm dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f | dom ( f
ex c being FinSequence of D ( ) st len c = k & P [ c ] & a = c & c = d ;
( the_arity_of a , b , c ) = <* Hom ( b , c ) , Hom ( c , d ) *> .= <* Hom ( a , b ) , Hom ( c , d ) *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f1 | X is continuous ;
a1 = b1 & a2 = b2 or a1 = b1 & b1 = b2 & b1 = b2 & b2 = b3 & b1 = b3 & b2 = b3 & b3 = b3 ;
D2 . ( indx ( D2 , D1 , n1 ) + 1 ) = D1 . ( n1 + 1 ) & D1 . ( n1 + 1 ) = D2 . ( n1 + 1 ) ;
f . ( ||. r .|| ) = ||. [ r , r ] .|| /. 1 .= <* r *> . 1 .= r .= x ;
consider n being Nat such that for m being Nat st n <= m holds C-25 . m = C-25 . m and C-25 . n = C-25 . m ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= d & b <= b holds d <= b ;
||. L /. h - K * ( K + |. h .| ) + K * ( K + |. h .| ) <= p0 + K * ( K + |. h .| ) ;
attr F is commutative means : Def1 : for b being Element of X holds F \hbox { b } = f . b ;
p = - ( - ( p `1 / |. p .| - cn ) ) .= 1 * ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) .= ( 1 - cn ) * ( ( p `1 / |. p .| - cn ) ) .= ( 1 - cn ) * ( p `2 / |. p .| - cn ) ;
consider z1 such that b , x3 , x1 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8
consider i such that Arg ( ( Rotate ( s ) ) . q ) = s + Arg ( q ) + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card f and rng g = { x } and rng g = { x } and g . x = y ;
assume A = P2 \/ Q2 & Q <> {} & Q <> {} & Q <> {} & Q is closed & P = Q & Q is closed & Q is closed & P = Q & Q is closed & Q is closed & P = Q & Q is closed & Q is closed & Q is closed & P \/ Q = Q \/ Q ;
attr F is associative means : Def1 : F .: ( F .: ( f , g ) , h ) = F .: ( f , F ) ;
ex x being Element of NAT st m = x `1 & x in z `1 & x < i or m = i & i < n or m = n & n < i & n < m ;
consider k2 being Nat such that k2 in dom P-2 and l = P-2 . k2 and ( for k st k in dom P-2 holds P-2 . k = ( P-2 . ( k2 + 1 ) ) * ( k + 1 ) ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n & seq is convergent & lim seq = r * seq . n
F1 . [ ( id a ) , [ a , a ] ] = [ f * ( id a ) , f * ( id a ) ] ;
{ p } "\/" D2 = { p "\/" y where y is Element of L : y in D2 & p in D1 } ;
consider z being element such that z in dom ( ( dom F ) . z ) and ( ( dom F ) . z ) . y = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds x = y
cell ( G , i ) = { |[ r , s ]| : r <= G * ( 0 + 1 , 1 ) `1 } ;
consider e being element such that e in dom ( T | E1 ) and ( T | E1 ) . e = v and ( T | E1 ) . e = v ;
( F `1 * b1 ) . x = ( Mx2Tran ( J , BT , \leq len ( b1 . x ) ) ) . ( ( b1 /. j ) ) .= b1 /. j ;
- 1 / ( - 1 ) = mm (#) D .= mm (#) D .= ( - 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 <= g . x ;
len ( f1 . j ) = len ( f2 /. j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) .= len ( f2 . j ) ;
All ( 'not' 'not' a , A , G ) '<' Ex ( 'not' All ( 'not' a , A , G ) , A , G ) ;
LSeg ( E . k , F . ( k + 1 ) ) c= Cl RightComp Cage ( C , k + 1 ) \/ { F . ( k + 1 ) } ;
x \ a |^ m = x \ ( a |^ k * a ) .= ( x \ ( a |^ k ) ) \ a .= ( x \ a ) \ a ;
k -ininininininin1 = ( commute ( I-5 ) ) . k .= ( commute ( I-5 ) ) . k .= ( commute ( I-5 ) ) . k .= ( commute ( I-5 ) ) . k ;
for s being State of A holds Following ( s , n ) . 0 + ( n + 2 ) * n + 2 * n * n + 1 * n * n + 2 * n * n + 2 * n + 1 * n * n + 2 * n + 1 * n * n + 2 * n + 1 * n * n + 2 * n + 1 * n * n + 2 * n * n + 1 * n * n + 2 * n * n + 1
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x <> 0 & ( f1 - f2 ) . x <> 0 ) implies f1 - f2 is_differentiable_on Z & ( f1 - f2 ) . x = 1
support ( ^ ( being bag of n ) ) \/ support ( ( support ( m ) ) ) c= support ( ( m ) ) \/ support ( ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , ( the carrier' of B ) * the Arity of C , ( the carrier' of C ) * the Arity of C ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( b * sqrt ( 1 + a ^2 ) ) ;
phi /. ( succ b1 ) = g . a & phi /. ( succ b1 ) = f . ( g . a ) & phi . ( succ b1 ) = f . ( g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( ( F ^ <* p *> ) | ( dom <* p *> ) ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } = { x1 } \/ { x2 , x3 , x4 , x5 , x5 } \/ { x5 , x5 , x5 , x5 } ;
the Sorts of ( U1 /\ U2 ) "\/" ( ( U1 "\/" U2 ) "\/" ( U2 "\/" W3 ) ) c= the Sorts of ( U1 "\/" U2 ) "\/" ( ( U1 "\/" U2 ) "\/" ( U1 "\/" W3 ) ) ;
( - ( 2 * a * ( b - a ) ) ) ^2 + b ^2 - ( 2 * a * ( b - a ) ) ^2 > 0 ;
consider W00 such that for z being element holds z in W00 iff z in N ~ N & P [ z ] and W [ z ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = a ;
Z = dom ( ( exp_R * ( arccot ) ) `| Z ) /\ dom ( ( exp_R * ( f1 + #Z 2 ) ) ^2 ) ;
lim ( f , SS1 ) is convergent & lim ( lim ( f , SS1 ) ) = integral ( f , SS1 ) ;
( X . a9 ) => ( g . x9 ) => ( x9 => x9 ) in J . ( a9 , b9 ) => ( g . x9 ) ;
len ( M2 * M3 ) = n & width ( M3 * M3 ) = n & width ( M3 * M3 ) = n & width ( M3 * M3 ) = n & width ( M3 * M3 ) = n ;
attr X1 union X2 is open means : Def1 : X1 , X2 are_separated & X2 , X1 union X2 are_separated & X1 , X2 union X2 are_separated & X1 , X2 union X2 are_separated ;
for L being upper-bounded antisymmetric antisymmetric RelStr , X being non empty Subset of L holds X "\/" { Top L } = { Top L }
reconsider f-1-129 = F1 . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . ( b . b ) ) )
consider w being FinSequence of I such that the InitS of M = the InitS of M & the InitS of M = q ^ w ^ w ^ w ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= 1_ H .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f ex z being Element of L st f . i = rpoly ( 1 , z ) & z in D & f . i = rpoly ( 1 , z ) ;
ex L being Subset of X st Carrier ( L ) = L & for K being Subset of X st K in C holds L /\ K <> {} ;
( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C1 & ( the carrier' of C1 ) /\ ( the carrier' of C2 ) c= the carrier' of C2 ;
reconsider o-21 = o `2 , op = p `2 , oq = p `2 , oq = p `2 as Element of TS ( ( the Sorts of A ) * ( the_arity_of o ) ) ;
1 * x1 + ( 0 * x2 ) + ( 0 * x3 ) = x1 + <* \underbrace { 0 , 0 } , 0 , 0 , 1 *> .= x1 + <* \underbrace { 0 , 0 } , 0 , 0 *> ;
Ei " . 1 = ( Ei qua Function ) " . 1 .= ( ( 1 - 1 ) * ( 1 - 1 ) ) " . 1 .= ( 1 - 1 ) * ( 1 - 1 ) " .= ( 1 - 1 ) * ( 1 - 1 ) " .= ( 1 - 1 ) * ( 1 - 1 ) " .= ( 1 - 1 ) * ( 1 - 1 ) ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) , v1 = the carrier of U2 /\ ( U1 "\/" U2 ) as non empty Subset of U0 ;
( ( x "/\" z ) "\/" ( x "/\" y ) ) "\/" ( z "/\" y ) <= ( x "/\" ( z "\/" y ) ) "\/" ( z "/\" y ) ;
|. f . ( s1 . ( l1 + 1 ) ) - f . ( s1 . ( l1 + 1 ) ) .| < ( 1 - M ) * ( 1 - M ) ;
LSeg ( ( Cage ( C , n ) ) * ( i , 1 ) , ( Cage ( C , n ) ) * ( i + 1 ) ) is vertical ;
( f | Z ) /. x - ( f | Z ) /. x0 = L /. ( x- x0 ) + R /. ( x- x0 ) ;
g . c * ( - g . c ) + f . c <= h . c * ( - f . c ) + f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) .= f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx f in the carrier of ( the carrier of A ) and ColVec2Mx b = ( the carrier of A ) \ ( the carrier of B ) and len f = width A and width f = width A and width f = width B ;
len ( - M3 ) = len M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 & width ( - M3 ) = width M1 ;
for n , i being Nat st i + 1 < n holds [ i , i + 1 ] in the InternalRel of [: ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of ( TOP-REAL n ) | ( the carrier of S ) ) ) ) ;
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 implies pdiff1 ( f2 , 2 ) is_partial_differentiable_in u0 , 2
attr a <> 0 & b <> 0 & Arg a = Arg b & Arg b = Arg a & Arg a = Arg b & Arg b = Arg a & Arg b = Arg b ;
for c being set st not c in [. a , b .] holds not c in Intersection ( the } \HM { set } , the carrier of a , the carrier of b )
assume that V1 is linearly closed and V2 is closed and V = { v + u : v in V1 & u in V1 & u in V1 & v in V1 } ;
z * x1 + ( 1 - z ) * x2 in M & z * ( y1 + ( 1 - z ) ) * x2 in N implies z * ( y1 + ( 1 - z ) ) * ( y2 + ( 1 - z ) ) in N
rng ( ( PS1 qua Function ) " * SS1 ) = Seg card dS1 .= Seg card dS1 .= Seg card dS1 .= Seg card dS1 .= Seg card dS1 .= Seg card dS1 .= Seg card dS1 .= Seg card dS1 ;
consider s2 being rational Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n = b . n and s2 . n = ( lim s2 ) / 2 ;
h2 " . n = h2 . n " & 0 < - ( ( 1 / 2 ) |^ n ) & 0 < ( ( 1 / 2 ) |^ n ) / 2 ;
( Partial_Sums ( ||. seq1 .|| ) ) . m = ||. seq1 .|| . m .= ||. seq1 .|| . m .= ||. seq1 .|| . m .= ||. seq1 .|| . m .= ||. seq1 .|| . m .= ||. seq1 .|| . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b .= Comput ( P2 , s2 , 1 ) . b ;
- v = - 1_ G & - w = - ( - 1_ G ) * v & - ( - w ) * v = - ( - 1_ G ) * w ;
sup ( ( k .: D ) .: D ) = sup ( ( k .: D ) .: D ) .= k . ( ( k .: D ) . D ) .= k . ( ( k . D ) . D ) ;
A |^ ( k , l ) \mathbin ( n , .. A ) = ( A |^ ( n , .. A ) ) ^^ ( A |^ ( k , .. A ) ) .= A |^ ( k , .. A ) ;
for R being add-associative right_zeroed right_complementable non empty addLoopStr , I , J , K being Subset of R holds I + ( J + K ) = ( I + J ) + K
( f . p ) `1 = ( p `1 ) ^2 + ( p `2 ) ^2 .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
for a , b being non zero Nat st a , b are_relative_prime holds ( a * b ) mod ( a * b ) = \mathop { \rm ^ ( a * b ) } + \mathop { \rm ^ ( a * b ) }
consider A9 being countable Nat such that r is countable & A9 is Element of CQC-WFF ( Al ) and A9 is Element of CQC-WFF ( Al ) and A9 is ( len A ) -element ;
for X being non empty addLoopStr , M being Subset of X , x , y being Point of X st x in M & y in M holds x + y in M + M
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , y1 ] , [ y1 , y2 ] } ;
h . ( f . O ) = |[ A * ( f . O ) + C * ( f . O ) + D , C * ( f . O ) + D ]| ;
( Gauge ( C , n ) ) * ( k , i ) in L~ Upper_Seq ( C , n ) /\ L~ Upper_Seq ( C , n ) ;
cluster m , n are_relative_prime for Nat , p be prime Nat , n be Nat , m be Nat , p be prime Nat , k be Nat , n be Nat , m be Nat , m be Nat , n be Nat , m be Nat st n < m & m < n & n < m holds n divides m & m divides n implies n divides m
( f (#) F ) . x1 = f . ( F . x1 ) & ( f (#) F ) . x2 = f . ( F . x2 ) & ( f (#) F ) . x2 = f . ( F . x2 ) ;
for L being LATTICE , a , b , c being Element of L st a \ b <= c & b \ c <= c holds a \ b <= c
consider b being element such that b in dom ( H / ( x. 0 ) ) and z = ( H / ( x. 1 ) ) . b and b = ( H / ( x. 0 ) ) . b ;
assume that x in dom ( F (#) g ) and y in dom ( F (#) g ) and ( F (#) g ) . x = ( F (#) g ) . y ;
assume ex e being element st e Joins W . 1 , W . 5 , G or e Joins W . 3 , W . 5 , G ;
( \cal x0 ) . ( 2 * n ) = ( x0 (#) ( f | n ) ) . ( 2 * n ) .= ( ( x0 (#) f ) | n ) . ( 2 * n ) ;
j + 1 = ( len h11 + 2 ) + 1 .= i + 1 - 2 + 1 .= i + 1 - 2 - 1 .= i + 1 - 1 ;
( *' S ) . f = *' S . ( ( *' f ) . ( f . ( f . x ) ) ) .= S . ( ( *' f ) . ( f . x ) ) .= S . ( f . x ) ;
consider H such that H is one-to-one and rng H = the carrier of L2 and Sum ( L2 * H ) = Sum ( L2 ) and Sum ( L2 ) = Sum ( L2 ) ;
attr R is >= >= 1 means : Def1 : for p , q st p in R & q <> q holds ex P st P is special & p in P & q in P & p <> q ;
dom product ( product ( X --> f ) ) = meet ( ( X --> f ) . ( dom f ) ) .= meet ( X --> f ) .= dom f .= dom f ;
upper_bound ( proj2 .: ( Upper_Arc C /\ Lower_Arc C ) ) <= upper_bound ( proj2 .: ( Lower_Arc C /\ Lower_Arc C ) ) & upper_bound ( proj2 .: ( Lower_Arc C /\ Lower_Arc C ) ) <= upper_bound ( proj2 .: ( Lower_Arc C /\ Lower_Arc C ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - x0 .| < r
i * fN - fN = i * fN - ( i * fN ) .= i * ( ( f - f ) . ( y - y0 ) ) .= i * ( f . ( y - y0 ) ) ;
consider f being Function such that dom f = 2 -tuples_on X ( ) & for Y being set st Y in 2 -tuples_on X ( ) holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] Y and g2 in union C and g = [ g1 , g2 ] and g1 in union C and g2 in C and g2 in D and g1 in D ;
func d |-count n -> Nat means : Def1 : d |^ ( n + 1 ) divides n & ( d |^ ( n + 1 ) divides n implies d |^ ( n + 1 ) divides n ) & ( a |^ ( n + 1 ) ) divides n ;
f\in f . [ 0 , t ] .= f . [ 0 , t ] .= ( - P ) . ( 2 * x ) .= ( - P ) . ( 2 * x ) .= a ;
t = h . D or t = h . B or t = h . C or t = h . E or t = h . F or t = h . J or t = h . M or t = h . N or t = h . N or t = h . N or t = h . M ;
consider m1 be Nat such that for n st n >= m1 holds dist ( ( seq . n ) , ( seq . n ) ) < 1 / ( ( seq . n ) + ( seq . n ) ) ;
( q `1 ) ^2 / ( |. q .| ) ^2 <= ( ( q `1 ) ) ^2 / ( |. q .| ) ^2 / ( |. q .| ) ^2 ;
h0 . ( i + 1 + 1 + 1 ) = h21 . ( i + 1 + 1 -' len h11 + 2 ) .= h11 . ( i + 1 + 1 -' 1 ) ;
consider o being Element of the carrier' of S , x2 being Element of { the carrier' of S } such that a = [ o , x2 ] and o in { [ o , x2 ] } ;
for L being RelStr , a , b being Element of L holds a <= { b } iff a >= b & b >= a & a >= b & b >= a & b >= b & a >= b implies a >= b
||. h1 . n .|| = ||. h1 . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h . n .|| .= ||. h .|| . n ;
( ( - 1 ) (#) ( ( #Z 2 ) * ( f ^ ) ) ) . x = f . x - ( ( #Z 2 ) . ( f . x ) ) .= ( - 1 ) * ( f . x ) .= ( - 1 ) * ( f . x ) .= ( - 1 ) * ( f . x ) ;
attr r = F .: ( p , q ) means : Def1 : len r = len p & for i st i in dom r holds r . i = min ( p . i , q . i ) ;
( r / 2 ) ^2 + ( r / 2 ) ^2 + ( r / 2 ) ^2 <= ( r / 2 ) ^2 + ( r / 2 ) ^2 ;
for i being Nat , M being Matrix of n , K st i in Seg n holds Det ( M @ ) = Sum ( ( Det ( M @ ) ) ) ) & Det ( M @ ) = Sum ( ( Det ( M @ ) ) )
then a <> 0. R & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v & a " * ( a * v ) = 1 * v ;
p . ( j - 1 ) * ( q /* ( i + 1 ) - j ) = Sum ( p . ( j -' 1 ) * ( q . ( i + 1 ) - j ) ) ;
deffunc F ( Nat ) = L . 1 + ( ( R /* h ) ^\ n ) * ( ( R /* h ) ^\ n ) " .= ( R /* h ) . ( h . $1 ) ;
assume that the carrier of H2 = f .: the carrier of H1 and the carrier of H2 = f .: the carrier of H2 and the carrier of H1 = the carrier of H2 and the carrier of H2 = the carrier of H1 and the carrier of H2 = the carrier of H2 ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ) . o .= ( the Sorts of Free ( S , X ) ) . o ;
H1 = n + 1 - h .= n + 1 - h .= n + 1 - h .= n + 1 - h .= n + 1 - h .= n + 1 - h ;
( 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 implies O = O
F1 .: ( dom F1 /\ dom F2 ) = F1 .: ( dom F1 /\ dom F2 ) .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } .= { f /. ( n + 2 ) } ;
attr b <> 0 & d <> 0 & b <> d & ( not a = b & d = d ) & ( not a = b & d = d implies a = ( - b ) / ( - d ) ) ;
dom ( ( f +* g ) | D ) = dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) /\ D .= dom ( f +* g ) ;
for i be set st i in dom g ex u , v being Element of L st g /. i = u * a & u in v & v in u & u in v + W
g `2 * P * g " = g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g `2 * ( g " * P ) .= g `2 * ( g " * P ) ;
consider i , s1 such that f . i = s1 and not ( ex s st s = f . i & not ( ex s st s <> i & s <> i & s <> i ) & not ( ex s st s <> i & s <> i & s <> i ) & not ( s <> i & s <> i & s <> i ) ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] ] in R & [ s2 , t2 ] in R & [ s2 , t2 ] in R implies [ s2 , t2 ] in R
then H is negative & H is non negative & H is non negative & H is non empty implies H is not non -g-gfor f being Function of H , H st f is not negative -gfor H holds f is not an -gfor f being Function of H , H holds f is not implies f is not implies f is not implies f is not One -gfor H
attr f1 is total means : Def1 : f1 is total & f2 is total & ( for c st c in dom f1 holds f1 . c = f1 . c ) implies f1 - f2 is total & ( f1 - f2 ) (#) ( f2 - f3 ) = f1 . c - f2 . c * ( f2 - f3 ) ;
z1 in W2 " ( W2 " ( { y } ) ) or z1 = z2 & not ( ex z st z in W2 & z in W1 & z in W2 & z in W2 ) implies ( z in W1 implies z in W2 )
p = 1 * p .= a " * a * p .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) .= a " * ( b " * q ) ;
for r be Real for K be Real st for n be Nat holds seq1 . n <= K holds upper_bound rng ( seq1 ^\ K ) <= ( seq1 ^\ K ) . n
Cage ( C , n ) meets L~ go \/ L~ pion1 or Cage ( C , n ) meets L~ Cage ( C , n ) or not E-max L~ Cage ( C , n ) meets L~ Cage ( C , n ) or E-max L~ Cage ( C , n ) meets L~ Cage ( C , n ) or E-max L~ Cage ( C , n ) meets L~ Cage ( C , n ) ;
||. f . ( g . ( k + 1 ) ) - g . ( k + 1 ) .|| <= ||. g . 1 - g . 0 .|| * ( K * ( K * ( K / ( k + 1 ) ) ) ) ;
assume h = ( ( B .--> B ' ) +* ( C .--> D ) +* ( E .--> E ' ) +* ( F .--> J ' ) +* ( J .--> M ) +* ( J .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( M .--> N ' ) ) +* ( N .--> N ' ) +* ( M .--> N ' ) +* ( N .--> N ' ) +* ( N .--> N ' ) +*
|. ( ( upper_volume ( H . n , T ) ) `| A ) . k - ( ( upper_volume ( H . n , T ) ) `| A ) . k .| <= e * ( b-a - ( lim ( H . n ) ) ) ;
( ( the Sorts of A ) . ( v ) ) . e = [ the | ( the carrier of I1 ) , the carrier of I1 , the carrier of I1 ] -tree q .= ( the | ( the carrier of I1 ) ) . e ;
{ x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x1 , x2 , x1 , x2 , x3 } = { x1 , x1 , x2 , x3 } ;
assume that A = [. 0 , 2 * PI / 2 .] and integral ( ( #Z n ) * ( sin ) , A ) = 0 and integral ( ( #Z n ) * ( cos ) , A ) = 0 ;
p `2 is Permutation of dom f1 & p `2 * ( Sgm Y ) " = ( ( Sgm Y ) * p ) * ( Sgm Y ) " .= ( Sgm Y ) * ( Sgm X ) " ;
for x , y st x in A holds |. ( 1 / f ) . x - ( 1 / f ) . y .| <= 1 * |. ( f . x - 1 ) .|
( p2 `2 ) ^2 = |. q2 .| * ( ( ( q2 `2 ) - sn ) / ( 1 + sn ) ) ^2 .= ( ( q2 `2 ) - sn ) / ( 1 + sn ) ;
for f being PartFunc of the carrier of CNS , REAL st dom f is compact & f is compact holds rng f is compact & rng f c= dom f & f is compact implies f is compact
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( Ex ( a , A , G ) ) . x = TRUE ;
consider FM such that dom FM = n1 and for k be Nat st k in n1 holds Q [ k , FM . k ] and for k be Nat st k in n1 holds Q [ k , FM . k ] ;
ex u , u1 st u <> u1 & u , u1 , v / ( a , v ) / ( a , u1 ) // v , v1 & u , v1 / ( a , v ) // v , v1 & u1 / ( a , v1 ) , v1 / ( a , v ) // v , v1 & u1 / ( a , v ) // v , v1 ;
for G being Group , A , B being non empty Subgroup of G , N being normal Subgroup of G holds ( N , A ) ` * ( N , B ) ` = N ` A ` * N `
for s be Real st s in dom F holds F . s = integral ( R / ( f + g ) ) - integral ( integral ( f , ( f + g ) (#) e ) , x )
width AutMt ( f1 , b1 , b2 ) = len b2 .= len ( ( b1 - b2 ) * ( b1 - b2 ) ) .= len b1 - len b2 .= len b1 - len b2 .= len b1 - len b2 .= len b1 - len b2 .= len b1 - len b2 .= len b1 - len b2 ;
f | ]. - PI / 2 , PI / 2 .[ = f & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ implies f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[
assume that X is closed and a in X and a c= X and y in X and not x in { [ n , x ] } \/ { [ n , y ] } \/ { [ n , x ] } ;
Z = dom ( ( - 1 / 2 ) (#) ( ( arctan ) * ( arctan ) ) ) /\ dom ( ( - 1 / 2 ) (#) ( ( arctan ) * ( arctan ) ) ) ;
func TAUT ( V ) -> Subset of V means : Def1 : for k st 1 <= k & k <= len l holds it . k = V . k & for k st 1 <= k & k <= len l holds it . k = V . k ;
for L being non empty TopSpace , N being net of L , M being net of L st c is convergent holds c is convergent & for c being Element of N st c in N holds c . c is convergent & c . c = c . ( N . ( N . ( N . i ) ) )
for s being Element of NAT holds ( ( for v being Element of NAT holds v + ( id C\rm seq ( X ) ) . s = ( ( id C\rm seq ( X ) ) . s ) . v ) . s + ( ( id C\rm seq ( X ) ) . s ) . s ) . v = ( ( id Cseq ( X ) ) . s ) . s
then z /. 1 = ( ( N-min L~ z ) .. z ) .. z & ( ( N-min L~ z ) .. z ) .. z < ( ( E-max L~ z ) .. z ) .. z ;
len ( p ^ <* ( 0 qua Real ) *> ) = len p + len <* ( 0 qua Real ) * ( 0 qua Real ) *> .= len p + 1 .= len p + 1 .= 1 ;
assume that Z c= dom ( - ( ln * f ) ) and f = ( - ( ln * f ) ) . x and for x st x in Z holds f . x = x and f . x = a and f . x = x and f . x = a ;
for R being add-associative right_zeroed right_complementable commutative associative commutative distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R holds ( I + J ) *' ( I + J ) c= I /\ J
consider f being Function of [: B1 , B2 :] , B1 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) and f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len x .= Seg len ( x (#) z ) .= dom ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= Seg len ( x (#) ( y (#) z ) ) .= dom ( x (#) ( y (#) z ) ) ;
for S being Functor of C , B for c being Object of C holds card S . ( id c ) = id ( ( the Arrows of C ) . ( id c ) )
ex a st a = a2 & a in f6 /\ f5 & card ( \mathop { \rm f , g } ) = card ( \mathop { f , g } ) & rng ( f * ( a , a ) ) = card ( \mathop { f , g } ) ;
a in Free ( ( H / ( x. 4 , x. k ) ) '&' ( H2 / ( x. k , x. k ) ) ) & a in Free ( ( x. k , x. k ) '&' ( H2 / ( x. k , x. k ) ) ) ;
for C1 , C2 being f1 , C2 being stable Function of C1 , C2 st @ f = g holds for f being Function of C1 , C2 st f is stable holds f = g iff f = g
( W-min L~ go \/ L~ co ) `1 = W-bound L~ go \/ E-bound L~ co .= ( W-min L~ go \/ L~ co ) `1 .= W-bound L~ pion1 \/ E-bound L~ co .= ( W-min L~ co ) `1 .= E-bound L~ co ;
consider u , y0 , y0 , z0 such that f = <* x0 , y0 *> and f is_partial differentiable on u , 1 and SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . y0 = SVF1 ( 3 , pdiff1 ( f , 1 ) , u ) . y0 ;
then ( t . {} ) `1 in Vars & ex x being Element of Vars st x = ( t . {} ) `1 & t . {} = x & t . {} = y & t . {} = x & t . {} = y & t . {} = y ;
Valid ( p '&' p , J ) . v = Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v '&' Valid ( p , J ) . v .= Valid ( p , J ) . v ;
assume for x , y being Element of S st x <= y for a , b being Element of T st a = f . x & b = f . y holds a >= b & b >= a & a >= b & b >= c holds a >= c ;
func Class R -> Subset-Family of R means : Def1 : for A being Subset of R holds A in it iff ex a being Element of R st a in A & a in A & it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( \HM { the } \HM { vertices } \HM { of G ) ) . $1 c= G ^2 & ( the carrier' of G ) . $1 c= G ^2 & ( the carrier' of G ) . $1 = G ^2 ;
assume that dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W2 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 and dim ( W1 ) = 0 ;
mamae ( m . t ) = ( m . t ) . {} .= ( [ m . t , the carrier of C ] `1 ) . {} .= [ m . {} , the carrier of C ] `1 .= m . {} ;
d11 = x9 ^ d11 .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= f . ( y9 , d22 ) .= ( f | d22 ) . ( y9 , d22 ) .= ( f | d22 ) . ( y9 , d22 ) .= ( f | d22 ) . ( y2 , d22 ) .= d22 ;
consider g such that x = g and dom g = dom f0 and for x being element st x in dom f0 holds g . x in f0 and g . x = f . x ;
x + 0. F_Complex = x + len x .= ( x + 0. F_Complex ) .= ( x + 0. F_Complex ) + ( x + 0. F_Complex ) .= ( x + 0. F_Complex ) + 0. F_Complex .= x + 0. F_Complex .= x ;
( k - ( k + 1 ) ) + 1 in dom ( f | ( ( k -' 1 ) + 1 ) ) & ( k + 1 ) + ( k + 1 ) = len f + ( k + 1 ) - ( k + 1 ) ;
assume that P1 is_an_arc_of p1 , p2 and P2 is_an_arc_of p1 , p2 and P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P1 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } and P1 = { p2 , p1 } and P2 = { p1 , p2 } ;
reconsider a1 = a , b1 = b , c1 = c , c1 = d , c2 = c , c1 = d , c2 = d , c1 = e , c2 = c , c1 = d , c2 = e , c1 = d , c2 = e , c1 = d , c2 = e , c1 = d , c2 = e , c1 = d , c2 = e , c2 = d , c1 = d , c2 = e , c1 = d , c2 = d , c1 = e , c2 = d , c2 = d , c1 = d , c2 = e , c2 = e , c2 = d , c1 = e , c2 = e , c2 = e , c2 = d , c1 = e , c2 = d , c2 = d , c2 = d , c2 =
reconsider thesis , } = G1 . ( t , [. b , a .] ) * ( f . a ) as Morphism of ( G1 * F1 ) . ( a , b ) , ( G2 * F2 ) . ( b , a ) ;
LSeg ( f , i + i1 -' 1 ) = LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 ) ) .= LSeg ( f /. ( i + i1 -' 1 ) , f /. ( i + 1 -' 1 ) ) ;
Integral ( M . m , P . m ) | dom ( P . n ) <= Integral ( M . n , P . m ) | dom ( P . n ) ;
assume that dom f1 = dom f2 and for x , y being element st x in dom f1 & y in dom f2 holds f1 . ( x , y ) = f2 . ( x , y ) and f2 . ( y , x ) = f2 . ( y , x ) ;
consider v such that v = y and dist ( u , v ) < min ( ( G * ( i , 1 ) ) `1 , ( G * ( i + 1 , 1 ) `1 ) - G * ( i + 1 , 1 ) `1 ) ;
for G being Group , H being Subgroup of G , a being Element of H st a = b holds for i being Integer holds a |^ i = b |^ i & a |^ i = b |^ i holds a = b
consider B being Function of Seg ( S + L ) , the carrier of V1 such that for x being element st x in Seg ( S + L ) holds P [ x , B . x ] and P [ B , B . x ] ;
reconsider K1 = { p9 where p9 is Point of TOP-REAL 2 : P [ p9 ] & p9 `2 >= 0 } , K1 = { p where p is Point of TOP-REAL 2 : P [ p ] } as Subset of TOP-REAL 2 ;
( ( ( N - S ) / 2 ) * ( ( N - S ) / 2 ) ) / 2 <= ( ( N - S ) / 2 ) / 2 * ( ( N - S ) / 2 ) ;
for x being Element of X , n be Nat st x in E holds |. ( Re F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x & |. ( Im F ) . n .| <= P . x
len ( @ ( @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
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 ) ) / ( x. 0 , m2 ) / ( x. 4 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 , m2 ) / ( x. 0 ) / ( x. 0
consider r being Element of M such that M , v / ( x. 3 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) |= / ( x. 0 , m3 ) / ( x. 0 , m3 ) / ( x. 4 , m3 ) / ( x. 0 , m3 ) / ( x. 0 ,
func w1 \ w2 -> Element of Union ( G , R8 ) means : Def1 : for w1 , w2 being Element of Union ( G , R8 ) holds it . ( w1 , w2 ) = ( ( ( ( the HN8 of G ) * the Arity of G ) * the Arity of G ) * the Arity of G ) . ( w1 , w2 ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s2 . b2 .= Exec ( n2 , s2 ) . b2 .= s . b2 .= s . b2 .= s . b2 .= s . b2 .= s . b2 ;
for n , k being Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ) . ( n + k ) - Partial_Sums ( |. seq .| ) . n ;
set F = S -\mathop { {} } ;
( Partial_Sums ( seq ) ) . ( n + 1 ) + Partial_Sums ( seq ) . ( n + 1 ) >= ( Partial_Sums ( seq ) ) . ( n + 1 ) + ( Partial_Sums ( seq ) ) . ( 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 ) ;
func the closed Subset of \HM { a , b , c , d , e , f , g , h h , i , i , g , i ) -> Subset of TOP-REAL 2 equals LSeg ( a , b , c , d ) \/ LSeg ( b , c , i ) ;
a * b ^2 + ( a * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( c * a ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 + ( b * c ) ^2 ) >= 6 * a * a * b * c + c * b * c + ( b * c ) ^2 ;
v / ( x1 , m1 ) / ( 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 ) ) ;
consider consider consider consider consider consider consider \mathop ( Q ^ <* x *> , M1 *> ) such that len \mathop { ^ <* x *> , M ^ ( ( len M ) --> FALSE ) = len ( ( M ^ <* x *> ) +* ( ( len M ) --> FALSE ) ) and len ( M ^ <* x *> ) = len ( M ^ <* x *> ) ;
Sum ( FM ) = r |^ n1 * Sum ( CM ) .= C . n1 * ( CM . n1 ) .= C . n1 * ( CM . n1 ) .= C . n1 * ( CM . n1 ) .= C . n1 * ( CM . n1 ) .= C . n1 * ( CM . n1 ) .= C . n1 ;
( ( GoB f ) * ( len GoB f , 2 ) ) `1 = ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 .= ( ( GoB f ) * ( len GoB f , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( ( a * ( $1 + 1 ) ) * ( $1 + 1 ) ) + b * ( $1 + 1 ) * ( $1 + 1 ) ;
( the_arity_of g ) . ( ( the Arity of S ) . g ) = ( the Arity of S ) . ( ( the Arity of S ) . g ) .= ( the Arity of S ) . ( ( the Arity of S ) . g ) .= ( the Arity of S ) . g .= g ;
( X \times Y ) c= X |^ Z & card ( X * Y ) = card X * card Y implies X = Y * Z
for a , b being Element of S , s being Element of NAT st s = n & a = F . n & b = F . n & b = F . ( n + 1 ) holds b = N . ( n + 1 ) \ G . ( n + 1 )
E , f |= All ( x. 2 , ( x. 2 ) => ( x. 2 ) ) => ( ( x. 2 ) '&' ( x. 2 ) ) => ( ( x. 2 ) '&' ( x. 2 ) ) '&' ( x. 2 ) ) ;
ex R2 being 1-sorted st R2 = ( p | n-11 ) . i & ( the support of p ) . i = the carrier of R2 & ( the support of p ) . i = the carrier of R2 & ( the support of p ) . i = the carrier of R2 ;
[. a , b + 1 / ( k + 1 ) .[ is Element of the _ of the _ of a & ( the partial F of f ) . k is Element of the carrier of a & ( the partial F of f ) . k is Element of the carrier of a & ( the partial F of f ) . k is Element of the carrier of a ;
Comput ( P , s , 2 + 1 ) = Exec ( P . 2 , Comput ( P , s , 2 ) ) .= Exec ( a3 , Comput ( P , s , 2 ) ) .= Exec ( a3 , s ) ;
card ( h1 ) . k = ( power ( F_Complex ) ) . ( ( - 1_ F_Complex ) . k ) * Sum u .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - 1_ F_Complex ) . ( - 1_ F_Complex ) ) * u .= ( ( - ( - 1_ F_Complex ) ) *' ) . k ;
( f - g ) /. c = f /. c * ( g /. c ) " .= f /. c * ( ( 1 - g ) * ( 1 - g ) ) .= ( f (#) ( 1 - g ) ) /. c * ( ( 1 - g ) * ( 1 - g ) ) .= ( f (#) ( 1 - g ) ) /. c ;
len Cv - len ( ( the carrier of ( C | ( len Cv ) ) ) ) = len Cv - len ( ( the carrier of ( C | ( len Cv ) ) ) ) .= len ( ( the carrier of ( C | ( len w ) ) ) ) .= len ( ( the carrier of ( C | ( len w ) ) ) ) .= len ( ( the carrier of ( C | ( len w ) ) ) ) ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) .= dom ( r (#) f ) .= dom ( r (#) f ) .= dom ( r (#) f ) .= dom ( r (#) f ) ;
defpred P [ Nat ] means for n holds 2 * Fib ( n + $1 ) = Fib ( n ) * Fib ( n + $1 ) + ( 5 * Fib ( n + $1 ) ) * Fib ( n + $1 ) ;
consider f being Function of INT , INT such that f = f and f is onto and f is onto and for n st n < k holds f " { f . n } = { n } and f " { f . n } = f " { n } ;
consider c9 being Function of S , BOOLEAN such that c9 = chi ( A \/ B , S ) and E7 = Prob ( A , S ) and E7 = Prob ( A , S ) and E7 = Prob ( A , S ) ;
consider y being Element of Y ( ) such that a = "\/" ( { F ( x , y ) where x is Element of X ( ) : P [ x ] } , y ) and Q [ y , x ] ;
assume that A c= Z and f = ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( - 1 ) * ( ( id 1 ) * ( ( id 1 ) * ( ( id Z ) * ( id Z ) ) ) ) ) ) ) ) ) ) ) `| Z ) ) = f ;
G * ( i , j2 ) `2 = G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * ( 1 , j2 ) `2 .= G * (
dom Shift ( Seq q2 , len Seq q2 ) = { j + len Seq Seq q2 where j is Nat : j in dom Seq Seq q1 & len Seq q2 = len Seq q2 + len Seq q2 } .= len Seq q2 + len Seq q2 + len Seq q2 .= len Seq q2 + len Seq q2 + len Seq q2 .= len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 .= len Seq q2 + len Seq q2 + len Seq q2 + len Seq q2 ;
consider G1 , G2 , G2 being Element of V such that G1 <= G2 and f is Morphism of G1 and g is Morphism of G2 , G2 and g * f is Morphism of G2 , G2 and g * g is Morphism of G2 , G2 and g * g is Morphism of G1 , G2 & g * g is Morphism of G2 , 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 & {} <> a holds for v holds union ( L , v ) |= ( L , v ) iff for u holds u , v |= ( L , u ) iff u , v |= ( L , u ) ) ;
consider i1 , j1 such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) ;
consider i , n such that n <> 0 and sqrt p = ( i - n ) * ( i - n ) and for n1 being Nat st n1 <> 0 & n1 < n holds ( i - n ) * ( i - n ) = ( i - n ) * ( i - n ) ;
assume that not 0 in Z and Z c= dom ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) (#) ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 ) / 2 ) * ( ( - 1 / 2 ) * ( ( - 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) ) ) and for x st x ) ) `| Z ) holds ( ( ( 1 / 2 ) * ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) `| Z ) . x ) = - 1 / 2 ) ) ;
cell ( G1 , i1 -' 1 , i2 -' 1 ) \ ( ( m -' 1 ) / 2 ) c= BDD L~ f & ( ( m - 1 ) / 2 ) \ ( ( m - 1 ) / 2 ) c= BDD L~ f ;
ex Q1 being open Subset of X st s = Q1 & ex F8 being Subset-Family of Y st F8 c= F & F8 is open & ( for x being set st x in F8 holds F8 . x is finite ) & ( ex y being set st y in F8 & y in y holds y in F8 . x ) & ( y in F8 . y ) & y in y ) ;
gcd ( A1 , r2 , s1 , s2 , Amp ) . ( ( 1 - s1 ) * ( 1 - s2 ) ) = 1 - s1 * s2 * s2 * s1 .= 1 - s1 * s2 * s2 * s2 * s2 .= s2 * s2 * s2 * s2 .= s2 * s2 * s2 .= s2 * s2 * s2 .= s2 * s2 .= s2 * s2 ;
R8 = ( ( the the A1 of s2 ) . ( m2 + 1 ) ) . m2 .= ( ( the A1 of s2 ) . ( m2 + 1 ) ) . m2 .= [ 3 , ( the A1 of s2 ) . ( m2 + 1 ) ] .= [ 3 , ( the A1 of s2 ) . ( m2 + 1 ) ] .= [ 3 , ( the Comput of s2 ) . ( m2 + 1 ) ) ] ;
CurInstr ( P-6 , Comput ( P3 , s3 , m1 + m2 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m1 ) ) .= halt SCMPDS .= halt SCMPDS .= ( ( card I + 1 ) + 1 ) .= halt SCMPDS ;
P1 /\ P2 = ( { p1 } \/ LSeg ( p11 , p2 ) ) \/ ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ ( LSeg ( p11 , p2 ) \/ LSeg ( p11 , p2 ) ) \/ { p2 } ) ;
func the still of f -> Subset of the Sorts of A means : Def1 : ex p st p in dom f & ex a st a in dom f & for i st i in dom f holds a . i = f . i & a . i = f . i & a . i = f . i & a . i = f . i & a . i = f . i ;
for a , b being Element of F_Complex st |. a .| > 1 & f . b >= 1 holds f . ( len f ) >= 1 implies a * ( f . b ) is ] & f . ( len f ) >= 1 & f . ( len f ) >= 1 & f . ( len f ) >= 1 implies f . ( f . b ) is ]
defpred P [ Nat ] means 1 <= $1 & $1 <= len g implies for i , j st [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G & [ i , j ] in Indices G holds G * ( i , j ) = G * ( i , j ) ;
assume that C1 , C2 such that f , g are_as as \vert and for s1 , s2 being State of C1 , s2 being State of C2 holds s1 = s2 iff for f being State of C1 , s1 , s2 being State of C2 st f = s2 holds f * ( f * g ) is stable iff f * g is stable & f * g is stable ;
( ||. f .|| | X ) . c = ||. f .|| . c .= ||. f .|| /. c .= ||. f .|| /. c .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| .= ||. f /. c .|| ;
|. q .| ^2 = ( q `1 ) ^2 + ( q `2 ) ^2 & 0 + ( q `2 ) ^2 < ( q `2 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T7 st F is open & {} in F & not {} in F & A <> {} & A <> {} holds card F = card ( A \/ B ) & card F = card ( A \/ B ) & card F = card ( A \/ B ) implies card F = card ( A \/ B )
assume that len F >= 1 and len F = k + 1 and len F = len G and for k st k in dom F holds H . k = g . k and for k st k in dom F holds H . k = g . k * k + h . k * h . k ;
i |^ ( ( \mathop { n } - i ) |^ s ) = i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) - i |^ ( s + k ) .= i |^ ( s + k ) - i |^ ( s + k ) ;
consider q being oriented oriented Chain of G such that r = q and q <> {} and ( for q being Element of G st q <> {} & q <> {} holds ( q . ( len q ) ) = v1 ) and rng q c= rng ( p ) and rng q c= rng ( q . ( len q ) ) and q . ( len q ) = v2 ;
defpred P [ Element of NAT ] means $1 <= len g implies ( g ) . $1 = ( ( g , Z ) ^ I ) . ( len g + $1 ) & ( g . $1 = ( ( g , Z ) ^ I ) . ( len g + $1 ) ) . ( len g + $1 ) ;
for A , B being square Matrix of n , K holds len ( A * B ) = len A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B & width ( A * B ) = width A & width ( A * B ) = width B
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st 1 <= i & i <= len s ex a , b being Element of R st s . i = a * b & a in I & b in J & s . i = b * a & s . i = b * b ;
func |( x , y )| -> Element of COMPLEX equals |( ( Re x ) , ( Re y ) )| - ( ( Re y ) ^2 ) , ( ( Re x ) ^2 + ( ( Im y ) ^2 ) ) * ( ( Im y ) ^2 + ( ( Im y ) ^2 ) ) , ( ( Im y ) ^2 + ( ( Im y ) ^2 ) ) * ( ( Im x ) ^2 + ( ( Im y ) ^2 ) ) ) ;
consider \mathop being FinSequence of F such that g is continuous and rng g c= A and for x st x in A holds g . x = A . x and for x st x in A holds g . x = F ( x ) and g . x = F ( x ) and g . x = f ( x ) and g . x = f ( x ) ;
then n1 >= len p1 & n2 >= len p1 & n3 ( p1 , p2 , n1 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , n4 , p1 , p2 , n1 , n2 , n3 , n3 , n3 , n4 , n4 , p1 , p2 , n2 , n3 , n3 , n3 , n2 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n4 , n4 , n4 , n3 , n2 , n3 , n3 , n3 , n4 , n4 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n3 , n2 , n3 , n2 ,
( q `1 ) * a <= ( q `1 ) * a & - q `1 <= ( q `2 ) * a or q `1 >= ( q `1 ) * a & - q `1 <= ( q `2 ) * a & - q `2 <= ( q `2 ) * a ;
Fv . ( p9 . ( len p9 ) ) = Fv . ( p . ( len p9 ) ) .= ( ( v . ( len p9 ) ) . ( len p9 ) ) .= ( ( v . ( len p9 ) ) . ( len p9 ) ) . ( len p9 ) ) . ( len p9 ) .= ( ( v . ( len p9 ) ) . ( len p9 ) ) . ( len p9 ) .= ( v . ( len p9 ) ) . ( len p9 ) .= ( v . ( len p9 ) ) . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9 ) ) . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len p9 ) .= ( v . ( len
consider k1 being Nat such that k1 + k = 1 and a := k = ( <* a := intloc 0 *> ) ^ ( ( intloc 0 ) --> ( intloc 0 ) ) ^ <* halt SCM+FSA *> ) ^ <* halt SCM+FSA *> ^ ( ( intloc 0 ) .--> ( intloc 0 ) ) ^ <* halt SCM+FSA *> ) ;
consider B8 being Subset of B1 , y8 being Function of B1 , B2 such that B8 is finite and D8 is finite and [: B1 , B2 :] = { Let ( B1 , B2 ) , y1 , y2 } and B1 in B2 and y2 in B1 and y2 in B1 and y2 in B2 and y1 in B2 and y2 in B1 and y2 in B2 and B1 in B2 ;
v2 . b2 = ( curry ( F2 , g ) * ( ( curry id B ) . b2 ) ) . ( ( ( curry id B ) . b2 ) . ( ( ( curry id B ) . b2 ) ) .= ( ( curry id B ) . b1 ) . ( ( ( curry id B ) . b2 ) . b2 ) .= ( ( curry id B ) . b1 ) . ( id b2 ) .= ( ( curry id B ) . b1 ) . b2 ) . ( id b2 ) .= ( ( ( id b2 ) . b1 ) . b1 ) . b2 ) .= ( ( ( id b2 ) . b1 ) . b1 .= ( ( ( b1 . b2 ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( ( ( id b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 ) . b2 .= ( ( ( b2
dom IExec ( then then for s being State of SCMPDS , P holds s = the carrier of SCMPDS & P = dom IExec ( I , P , Initialize s ) +* Start-At ( ( card I + 2 ) , SCMPDS ) +* Start-At ( ( card I + 2 ) , SCMPDS ) ) ;
ex d-32 be Real st d-32 > 0 & for h be Real st h <> 0 & |. h .| < e holds |. h .| " * ||. ( R + L ) /. h .|| < e / ( R + R ) /. h .|| < e / ( R + R ) /. h ) ;
LSeg ( G * ( len G , 1 ) + |[ - 1 , 1 ]| , G * ( len G , 1 ) + |[ 1 , 1 ]| ) c= Int cell ( G , len G ) \/ { |[ 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 + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE p1 , q , P & LE p2 , q , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p2 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p1 , P & LE p1 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p1 , p1 , P & LE p1 , p1 , P & LE p2 , p1 , P & LE p2 , p1 , P & LE p2 ,
( ( - x ) .|. y ) = - ( ( - 1 ) * ( x .|. y ) ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) .= ( - 1 ) * ( x .|. y ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) ^2 ) = ( ( p `1 ) ^2 + ( p `2 ) ^2 ) * sqrt ( 1 + ( p `2 / p `1 ) ^2 ) .= ( p `1 ) ^2 + ( p `2 ) ^2 ;
( ( W + U ) (#) ( W + U ) ) . ( n + 1 ) = ( ( W + U ) (#) ( W + U ) ) . ( n + 1 ) .= ( W + U ) (#) ( W + U ) .= ( W + U ) (#) ( W + U ) ) . ( n + 1 ) .= W * ( W + U ) .= W * ( W + U ) ;
func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def1 : dom it = dom f & for x st x in dom it holds it . x = - h . x + h . x * ( - h . x ) ;
assume that 1 <= k and k + 1 <= len f and [ i + 1 , j ] in Indices G and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) and f /. k = G * ( i + 1 , j ) ;
assume that not y in Free H and not x in Free H and not ( x in Free H or x in Free H or x in Free H & not y in Free H or x in Free H & not y in Free H or x in Free H & y in Free H or x = y or x = y or x = y or x = y or x = y or x = y ) ;
defpred P11 [ Element of NAT , Element of NAT , Element of NAT , Element of NAT ] means ( P [ $1 ] implies $2 = ( p |^ $1 ) |^ ( 2 * $1 ) ) & ( $1 = 2 * $1 ) |^ ( 2 * $1 ) & ( $1 = 2 * $1 ) |^ ( 2 * $1 ) ) & ( $1 = 2 * $1 implies $2 = 2 * $1 ) ;
func \sigma ( C ) -> non empty Subset-Family of X means : Def1 : for A , B being Subset of X holds A c= it iff for W being Subset of X holds W c= ( W \ A ) \/ ( W \ B ) & W is open & it = ( W \ A ) \/ ( W \ B ) ;
[#] ( ( dist ( ( dist ( P ) ) ) .: Q ) = ( dist ( ( dist ( P ) ) ) .: Q ) .: Q & lower_bound ( ( dist ( P ) ) .: Q ) = lower_bound ( ( dist ( P ) ) .: Q ) ;
rng ( F | ( [: S , T :] ) ) = {} or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } or rng ( F | ( [: S , T :] ) ) = { 1 } or rng ( F | ( [: S , T :] ) ) = { 2 } ;
( f " ( rng f ) ) . i = f . i " .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i .= ( f " ( rng f ) ) . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p1 , p2 } and P1 \/ P2 = { p2 , p1 } and P1 \/ P2 = { p2 , p1 } ;
f . p2 = |[ ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 + ( p2 `2 ) ^2 * ( p2 `2 ) ^2 ;
( the \HM { AffineMap ( a , X ) ) " . x = ( the \HM { |[ a , 0 ]| , |[ a , 0 ]| , |[ b , 0 ]| ) . x .= ( ( the \HM { |[ a , 0 ]| } , |[ b , 0 ]| ) . x ) . x .= ( ( the \HM { |[ b , 0 ]| } , |[ b , 0 ]| ) . x ) .= ( ( the p1 of X ) . x ) . x ;
for T being non empty normal TopSpace , A , B being closed Subset of T st A <> {} & A is closed & B is closed & A is open & B is open holds for p being Point of T , r being Real st p in A & r in B holds p in A & p in B holds p in A & p in B implies p in A
for i st i in dom F for G1 , G2 being strict normal Subgroup of G st G1 = F . i & G2 = G . i & G2 = G . ( i + 1 ) & G1 = F . ( i + 1 ) holds G1 = G2 & G2 = G . ( i + 1 ) & G2 = G . ( i + 1 ) implies G1 = G2
for x st x in Z holds ( ( ( - 1 / 2 ) (#) ( ( arctan ) * ( arctan ) ) `| Z ) . x = ( ( - 1 / 2 ) (#) ( arctan ) ) `| Z ) . x / ( 1 + x ^2 ) - ( ( 1 / 2 ) (#) ( arctan ) ) . x / ( 1 + x ^2 )
attr f is right continuous means : Def1 : for x0 st x0 in dom f & x0 in dom f holds f . x0 - f . x0 = ( f /* x0 ) . x0 & for x st x in dom f holds f . x - f . x0 = ( f /* ( x0 - x0 ) ) . x ;
then X1 , X2 are_separated & X1 , X2 are_separated & ( X1 union X2 ) , X2 are_separated & ( X1 , X2 are_separated & X2 , X1 union X2 are_separated & X1 , X2 are_separated & X2 , X1 , X2 are_separated & X1 , X2 , X2 are_separated & X2 , X1 , X2 are_separated & X1 , X2 , X2 are_separated & X1 , X2 , X2 , X1 union X2 are_separated & X2 , X1 , X2 is_collinear & X1 , X2 , X2 is_collinear & X2 , X1 , X2 are_separated ;
ex N being Neighbourhood of x0 st N c= dom SVF1 ( 1 , f , u ) & ex L , R st for x st x in N holds SVF1 ( 1 , f , u ) . x - SVF1 ( 1 , f , u ) . x0 = L . ( x - u ) + R . ( x - u ) + R . ( x - u )
( ( p2 `1 ) * sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) * ( ( p2 `2 ) * sqrt ( 1 + ( p2 `2 / p2 `1 ) ^2 ) ) >= ( ( p2 `2 ) * sqrt ( 1 + ( p2 `1 / p2 `1 ) ^2 ) ) * ( ( p2 `2 ) * sqrt ( 1 + ( p2 `1 / p2 `1 ) ^2 ) ) ;
( ( 1 - t ) (#) ( ||. f1 .|| ) ) . x = ( ( 1 - t ) (#) ( ||. g1 .|| ) ) . x .= ( ( 1 - t ) (#) ( ||. g1 .|| ) ) . x .= ( ( 1 - t ) (#) ( ||. g1 .|| ) ) . x ;
assume that for x holds f . x = ( ( - 1 / 2 ) (#) ( sin ) - cos ) . x and for x st x in Z holds ( ( - 1 / 2 ) (#) ( sin ) - cos ) ) . x = - 1 / ( sin . x ) ^2 and for x st x in Z holds ( ( - 1 / 2 ) (#) ( sin ) ) . x = 1 / ( sin . x ) ^2 ) ;
consider X-23 being Subset of Y , Y1 being Subset of X such that Y1 is open and Y1 is open and ex Y1 being Subset of X st Y1 = Y1 & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open & Y1 is open ;
card S = card { [ d , ( a |^ 3 ) + b |^ 3 ] where d is Element of GF ( p ) , b is Element of GF ( p ) : [ d , ( a |^ 3 ) + b ] in Indices GF ( p ) } .= 0 + p |^ 3 + b |^ 3 + d |^ 3 + d |^ 3 ) .= 0 + d ;
( W-bound D - W-bound D ) * ( ( W-bound D - W-bound D ) / 2 ) = ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 ) .= ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 ) .= ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - W-bound D ) / 2 ) .= ( W-bound D - W-bound D ) / 2 * ( ( W-bound D - E-bound D ) / 2 ) ;