thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S `1 is convergent
q in A ;
V in I ;
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 >= X ;
T is continuous ;
d <= a ;
p <= r ;
t < s ;
p <= t ;
t < s ;
let r ;
D <= E ;
assume e > 0 ;
assume 0 < g ;
p in X ;
x in X ;
Y `1 in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z `1 ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B `1 = b ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b ;
assume b in X ;
assume k <> 1 ;
f = \prod l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated 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 <= x ;
A `1 in B ;
assume i = 1 ;
let x be element ;
x `1 = x `1 ;
let X be BCK-algebra ;
assume S is non empty ;
a in [: REAL , REAL :] ;
let p be set ;
let A be set ;
let G be _Graph , a , b be Nat ;
let G be _Graph , a , b be Nat ;
let a be UNKNOWN of L ;
let x be element ;
let x be element ;
let C be Category , a , b be element ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
y be Real ;
X c= f . a
let y be element ;
let x be element ;
i in 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 <= [: NAT , NAT :] ;
let y be element ;
r2 < r2 ;
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 `1 c= B `1 ;
set s = \mathclose { -1 } ;
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 ;
the function arccot is differentiable of Z ;
the function exp is differentiable of x , Z ;
j < i2 ;
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 <> b2 ;
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 <= r2 ;
let e be Real , x be Real ;
not r in G . l
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is non trivial ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , x be element ;
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 discrete ;
let i be Nat ;
R2 is total ;
cluster downarrow x -> simplex-like ;
X <> { x } ;
x in { x } ;
q , b // M , b ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
indices M1 >= 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 , W be Subset of V ;
assume x in Form M ;
k < s . a ;
not t in { p } ;
let Y be functional set , x be element ;
M , L are_isomorphic ;
a <= g . i ;
f . x = b ;
f . x = c ;
assume L is lower-bounded & L is lower-bounded ;
rng f = Y ;
G2 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 ;
x `1 = a * y `1 ;
rng D c= A ;
assume x in K ;
1 <= i0 ;
1 <= i0 ;
p9 c= \pi ;
1 <= i0 ;
1 <= i0 ;
w in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is Lipschitzian ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
FX1 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
sup D in S ;
x \ll sup D ;
b1 >= Z ;
assume w = 0. V ;
assume x in A . i ;
g in BoundedFunctions X ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
mi c= f ;
x3 is increasing ;
let d2 be element ;
- b divides b ;
F c= \tau ( F ) ;
G1 is non-decreasing ;
G1 is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , x be element ;
assume P [ n ] ;
assume union S is independent & card S is finite ;
V is Subspace of V ;
assume P [ k ] ;
rng f c= NAT ;
assume inf X in L ;
y in rng f ;
let s , I , J ;
b `1 c= b9 ;
assume not x in [: Q + 1 , Q :] ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in BA ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y ;
i2 < i1 + 1 ;
a * h in a * H ;
p , q ] in Y ;
cluster sqrt I -> non empty ;
q1 in A1 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 & A1 c= A2 ;
\hbox { \boldmath $ p $ } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M ;
let k , m ;
a , b // b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 ;
g is continuous Function of x0 , REAL ;
assume O is symmetric ;
let x , y ;
let j be Nat ;
[ y , x ] in R ;
let x , y ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 halts_on s , P ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
Y is 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 , A be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , y be element ;
let b be Element of X , x be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D ;
P [ k + 1 ] ;
m in dom ( n -tuples_on NAT ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> with_functional ;
let R be non empty multiplicative RelStr , 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 ( the_arity_of o ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `1 ;
let M be non empty maid id ;
N be non empty multiplicative M ;
let R be transitive RelStr ;
let n , k be Nat ;
let P , Q be symmetric RelStr ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be Vector of V ;
reconsider d = x as FinSequence of D ;
assume I is non halting ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v2 ;
x <= c2 . x ;
x in F ` ;
cluster S --> T -> nonempty ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be odd Nat ;
assume F1 <> F2 ;
c in Intersect ( R ) ;
dom p1 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 ;
set i1 = i + 1 ;
assume a1 = b1 ;
dom ( g1 * g2 ) = A ;
i < len M + 1 ;
assume not - \infty in rng G ;
N c= dom ( f1 (#) f2 ) ;
x in dom sec ;
assume [ x , y ] in R ;
set d = sqrt ( x , y ) ;
1 <= len g1 ;
len s2 > 1 ;
z in dom ( f1 ^ f2 ) ;
1 in dom D2 ;
( p `2 ) ^2 = 0 ;
j2 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. px .| = 1 ;
let s be SortSymbol of S ;
\frac ( i , i ) = i ;
X1 c= dom f ;
h . x in h . a ;
let G be \times D2 ;
cluster m * n -> square ;
let k2 be Nat ;
i -' 1 > m ;
R is transitive in field R & R is transitive ;
set F = <* u , w *> ;
pP c= P3 ;
I is_closed_on t , Q ;
assume [ S , x ] is quantifiable ;
i <= len ( f2 ^ g2 ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> real-valued ;
x in dom ( f1 ^ f2 ) ;
assume [ X , p ] in C ;
BX c= X1 & X c= X2 ;
n2 <= n2 + n2 ;
A /\ ( { P } ) c= A ` ;
cluster -> x -valued for Function ;
let Q be Subset-Family of S , S ;
assume n in dom g2 ;
let a be Element of R ;
t `2 in dom f2 ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , x be Element of X ;
i . y in rng i ;
[: dom f , dom g :] c= dom f ;
f . x in rng f ;
NAT <= sqrt ( r ^2 + 2 ^2 ) ;
s2 in r0 & s2 in r ;
let z , z1 , z2 be number ;
n <= N . m ;
LIN q , p , s ;
f . x = \twoheaddownarrow x /\ B ;
set L = [ S \to T ] ;
let x be non negative extended real number ;
m in M ;
f in union rng ( F1 ^ F2 ) ;
let K be add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , n be Element of NAT ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x ;
n1 < n1 + 1 + 1 ;
n1 < n1 + 1 + 1 ;
cluster <* T . X *> -> M ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( S29 | X ) ;
b = sup dom f & sup dom f = sup dom g ;
x in Seg len q ;
reconsider X = [: D , D :] as set ;
[ a , c ] in E ;
assume n in dom h2 ;
w + 1 = [: a1 , b1 :] ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 + 1 ;
i in NAT ;
Support u = Support p \/ { x } ;
assume X is complete for m being Nat ;
assume that f = g and p = q ;
n1 <= n1 + 1 + 1 ;
let x be Element of REAL ;
assume x in rng s2 ;
x0 < x0 + 1 + 1 ;
len ( L * ( i , j ) ) = len W ;
P c= Seg len A & P c= Seg len A ;
dom q = Seg n ;
j <= width M *' ;
let IT be real-valued FinSequence ;
let k be Element of NAT ;
\int P + d < + \infty ;
let n be Element of NAT ;
assume z in atat0 ( 0 ) ;
i in dom f ;
n -' 1 = n ;
len n2 = n ;
\ ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , A , B be Subset of B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E |^ omega ;
let B1 be basis of x , y ;
3 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume \mathopen { \downarrow x } = \mathopen { \downarrow x } ;
assume b , c // b , c ;
LIN q , c , c9 ;
x in rng ( f2 | F29 ) ;
set d8 = n + j ;
let D1 be non empty set , D2 be Subset of L ;
let K be add-associative right_zeroed right_complementable non empty addLoopStr , n be Element of NAT ;
assume f = f & h = g ;
R1 - R2 is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K ` is open ;
assume that a , b are_maximal with C ;
a , b , c is_collinear ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f , M ;
cluster as q -cOne -> non empty ;
not u in { \hbox { \boldmath $ g } } ;
the support of f c= B
reconsider z = x as Vector of V ;
cluster the MSAlgebra of L -> non empty ;
r (#) H is point;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U2 be strict non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
[ x , [#] T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
reconsider ry = ry as Element of \mathclose { x } ;
let x , y be Element of X ;
A , I , I , J , J , K , L ;
[ y , z ] in O ;
Shift ( goto i , 2 ) = 1 ;
rng Sgm A = A ;
q |- |[ y , y1 ]| ;
for n holds X [ n ] ;
x in { a } & x in { d } ;
for n holds P [ n ] ;
set p = [ x , y ] ;
LIN o , a , o ;
p . 2 = Z |^ Y ;
( D2 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: Al ( ) , NAT ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Subset of L ;
set g = f1 + f2 , h = f2 + g2 , i = f3 , j = k2 , i = k2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . i1 ;
x `1 , y `2 ] in [: A2 , A1 :] ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multiplicative for FinSequence of REAL ;
x in support ( support ( t ) ) ;
assume a in [: the carrier of G , the carrier of G :] ;
i `2 <= ( y `2 ) `2 ;
assume p divides b1 + b2 ;
M2 <= sup ( M1 * M2 ) ;
assume x in W \pi ( X ) ;
j in dom ( z | ( len z ) ) ;
let x be Element of [: D , D :] ;
IC s3 = goto ( l + 1 ) ;
a = {} or a = { x } ;
set PA = Vertices G , PA = Vertices G , c = the Element of G ;
seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hh c= h & h c= h ;
]. a , b .[ c= Z ;
X1 , X2 are_separated implies X1 , X2 are_separated
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k ;
cluster -> binary for Relation of NAT ;
ex v st C = v + W ;
let G1 be non empty addLoopStr , x be Element of G1 ;
assume V is Abelian add-associative right_zeroed right_complementable distributive non empty doubleLoopStr ;
X1 \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
sup B is upper ;
let L be non empty reflexive transitive RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= H ;
dom G `2 = a ;
sqrt ( 1 - 4 ) >= - sqrt ( 1 - 4 ) ;
G . x0 in rng G ;
let x be Element of F1 ( ) , x be Element of F2 ( ) ;
D [ 0 , 0 ] ;
z in dom id B & z in dom id B ;
y in the carrier of N ;
g in the carrier of H ;
rng ( fk1 ^ <* n *> ) c= [: NAT , NAT :] ;
j + 1 + 1 in dom s1 ;
A , B , C is_collinear ;
C is non empty Subset of REAL ;
f . z1 in dom h ;
P . k1 in rng P ;
M = { A } +* {} ;
let p be FinSequence of REAL ;
f . n1 in rng f ;
M . F in [: the carrier of K , the carrier of K :] ;
\varnothing [. a , b .] = E-bound A ;
assume that the distance of V is total and Q is total ;
let a be Element of ^ ( V , C ) ;
let s be Element of ( P ) . s ;
let PI be non empty reflexive transitive RelStr ;
n in NAT ;
the support of g c= B ;
I = halt SCM R .= halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K ;
l <= IC ( F . j ) ;
assume x in \mathopen { \rbrack s , t .[ ;
( x - t ) in ]. - t , t .[ ;
x in then then x in then then then T ( T ) . x ;
let h be Morphism of c , a ;
Y c= { \bf Y } & Y in { \bf L } ;
A2 \/ A1 c= L1 \/ L2 ;
assume LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 , x3 is_collinear ;
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| ;
[ x `1 , x `2 ] in X ;
for n being Nat holds 0 <= x . n
[ a , b ] = [. a , b .] ;
cluster open -> closed for Subset of T ;
x = h . ( z1 , z2 ) ;
q1 , q2 , q1 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 ] ;
R , Q are_equipotent implies R , Q , R is_collinear
set d = sqrt ( 1 / n ) ;
rng ( g2 * g1 ) c= dom g2 ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V , W be Subset of V ;
I c= Program ( SCM+FSA ) ;
assume x in rng ( R * S ) ;
let b be Element of the lattice of T ;
dist ( e , z ) - r > ro ;
u1 + v1 in W2 & v1 + v2 in W1 ;
assume not the support L misses rng G ;
let L be lower-bounded RelStr ;
assume [ x , y ] in [: the carrier of G , { y } :] ;
dom ( A * e ) = NAT ;
a , b // G * ( i , j ) ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI * PI ;
o , a9 // o , y ;
{ v } c= the carrier of l ;
let x be bound of A ;
assume x in dom ( ( uncurry f ) . x ) ;
rng F c= ( product f ) .: X ;
assume D2 . k in rng D ;
f " . p1 = 0 ;
set x = the Element of X ;
dom Ser ( G ) = NAT ;
n in NAT ;
assume LIN c , a , a1 ;
cluster finite -> finite ;
reconsider d = c as Element of L1 ;
( v2 .--> I ) . I <= 1 ;
assume x in the carrier of f ;
conv @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @
reconsider B = b as Element of the carrier of T ;
J , v |= P ! l ;
cluster J . i -> non empty for TopSpace ;
sup ( Y \/ Y ) in the carrier of T ;
W1 is_\! with W1 & W2 is_\! \! | W2 implies W1 c= W2
assume x in the carrier of R ;
dom -16 = Seg n & dom -16 = Seg n ;
s2 misses s2 & s2 misses s2 implies s2 , s2 is_collinear
assume ( a 'imp' b ) . z = TRUE ;
assume that X is open and f = X --> d ;
assume [ a , y ] in Indices ( f | X ) ;
assume that that that that that that stop I c= J and J c= K and J c= K ;
Im ( seq - seq ) = 0 ;
( ( the function sin ) * ( sin * ( cos * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin * ( sin
the function sin is differentiable of Z & ( for x st x in Z holds cos . x > 0 ) implies ( ( for x st x in Z holds cos * ( x + a ) ) ^2 = 1 / (
t6 . n = t6 . n ;
dom ( cos * F ) c= dom F ;
seq1 . x = seq2 . x ;
y in W .vertices() \/ W .vertices() ;
k9 <= len ( v | ( len ( v | ( len v -' 1 ) ) ) ) ;
x * a \equiv y * a ;
proj2 .: S c= proj2 .: ( P /\ Q ) ;
h . p3 = g2 . I ;
G1 = U /. 1 .= U /. 1 .= U ;
f . r1 in rng f ;
i + 1 + 1 <= len One ;
rng F = rng ( F | ( len F ) ) ;
cluster partial non empty for doubleLoopStr ;
[ x , y ] in A [: { a } , { a } :] ;
x1 . o in L2 . o ;
the support the support of m c= B ;
not [ y , x ] in id ( X ) ;
1 + p .. f <= i + len f ;
seq ^\ ( k + 1 ) is lower ;
len ( F ^ <* Gij *> ^ ( F ^ <* Gij *> ^ <* \rangle ) ) = len ( F ^ <* Gij *> ^ <* Gij *> ) ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) . IC s = s ;
k <= k + 1 ;
reconsider c = {} _ { T } as Element of L ;
let Y be with_Biff Y is with_inof T ;
cluster there exists a Function of L , L st is monotone & for x being Element of L st x in X holds x is closed ;
f . j1 in K . ( j + 1 ) ;
cluster J => y -> total ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 ;
x1 = x or x1 = y ;
attr a <> {} means : Def5 : \frac { a } = 1 ;
assume that cf a c= b and b in a ;
s1 . n in rng s1 & s2 . n in rng s2 ;
{ o , b2 } on C2 & { o , b2 } on C2 ;
LIN o , b , b9 ;
reconsider m = x as Element of Funcs ( V , C ) ;
let f be non constant FinSequence of D ;
let F2 be non empty topological space ;
assume that h is being_homeomorphism and y = h . x ;
[ f . 1 , w ] in [: F , F :] ;
reconsider p2 = x as Subset of m ;
A , B , C is_collinear ;
cluster non empty -> non empty for satisfying_of X ;
rng c misses rng ( e .--> e1 ) ;
z is Element of gr { x } ;
not b in dom ( a .--> p1 ) ;
assume that k >= 2 and P [ k ] ;
Z c= dom ( cot * cot ) ;
the component of Q c= UBD ( A ) & Q c= UBD ( A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) ;
attr f = u * f , a * u means : Def5 : a * f = a * u ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
let x be Element of V . s ;
a , b // a , b ;
assume that S = S2 and p = S2 ;
gcd ( n1 , n2 ) = 1 / ( n1 + n2 ) ;
set oo = ( 2 * PI ) * PI , po = 2 * PI ;
seq . n < |. r1 .| ;
assume that seq is increasing and r < 0 ;
f . y1 <= a ;
ex c being Nat st P [ c ] ;
set g = { n } --> ( n + 1 ) ;
k = a or k = b or k = c ;
a9 , b9 // b9 , c9 ;
assume that Y = { 1 } and s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W4 = W . 1 .= W . 1 ;
cluster -> -> -> -> -> -> -> trivial ;
reconsider u = u as Element of Bags X ;
A in B implies A , B are_\kern1pt ;
x in { [ 2 * n + 3 ] } ;
1 >= sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) ;
f1 is_subc_of f2 & f2 is_s_of g2 implies f1 , f2 , g2 , g2 is_collinear
( f . q ) `2 <= ( q `2 ) ^2 ;
h is in the carrier of Cage ( C , n ) ;
( b - a ) * ( b - a ) <= ( p - a ) * ( b - a ) ;
let f , g be symmetric Function of X , Y ;
S /. k <> 0. K ;
x in dom ( max ( f , g ) ) ;
p2 in [: N ( ) , N ( ) :] ;
len ( the_right_argument_of H ) < len ( H ) ;
F [ A , F . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def5 : A c= C & A c= C ;
assume that r1 <> 0 or r2 <> 0 ;
rng q1 c= rng ( the Sorts of A1 ) ;
A1 , L , L is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in \bf L ( p , S ) ;
then S is atomic ;
Cl [#] ( T | [#] T ) = [#] T ;
f2 | ( A2 \ A2 ) = f2 | ( A2 \ A1 ) ;
0. M in the carrier of W ;
v , v // M , v ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V ;
let X be Subset of S ;
consider H1 such that H = 'not' H1 ;
d1 c= d1 * d1 + d2 * d2 ;
0 * a = 0. R .= a * a .= a ;
A |^ 2 = A |^ ( 2 + 1 ) ;
set pv = ( Wv ) . n , pv = ( m , n ) --> v , pv = ( m , n ) --> v , pv = ( m , n ) --> v , w
r = 0. \langle ( TOP-REAL n ) * , ( TOP-REAL n ) * \Vert \rangle ;
( f . p3 ) `2 >= 0 ;
len W = len ( W *' ) + len ( W *' ) ;
f /* ( s * G ) is divergent to \hbox { - \infty , - \infty , + \infty .[ ;
consider l being Nat such that m = F . l ;
tOne .. ( W1 , W2 ) does not contradiction ;
reconsider X1 = X1 as SubSpace of X ;
consider w such that w in F and not x in w ;
let a , b , c be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id ( L ) . x ;
\sigma ( T ) \/ \omega is Subset-Family of T ;
for x being element st x in X holds x in Y ;
cluster [ x1 , x2 ] -> pair ;
sup { a } /\ \mathopen { \downarrow } t is Subset of T ;
let X be non empty set , n be Element of NAT ;
rng f = Terminals IExec ( S , X ) ;
let p be Element of B , B ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N1 ;
0. X <= b |^ ( m * n ) ;
assume that i in I and R1 . i = R . i ;
i = j1 & ( p1 + p2 ) `1 = ( p1 + p2 ) `1 ;
assume gR in the carrier of g ;
let A1 , A2 be Point of S ;
x in h " ( P ) /\ [#] ( T | P ) ;
1 in Seg 2 & 1 in Seg 3 ;
reconsider XT = X as non empty Subset of T ;
x in ( the Arrows of B ) . i ;
cluster ET . n -> ( the Target of G ) -valued ;
n1 <= n2 + len ( g | n1 ) ;
( i + 1 ) + 1 = i + 1 ;
assume v in the carrier of G2 ;
y = Re ( y . i ) + ( Im ( y . i ) ) ;
attr ( pred - 1 ) * ( 1 , p ) = 1 ;
x2 is_differentiable_in a & ]. a , b .[ c= ]. a , b .[ ;
rng ( M2 * M2 ) c= rng ( M2 * M2 ) ;
for p be Real st p in Z holds p >= a
\bf \bf X \bf Y * ( f , g ) = ( f , g ) * ( f , g ) ;
( seq ^\ k ) . m <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \! \mathop { - 1 } ) . 2 = d ;
A \ ( B \ C ) = ( A \ B ) \ C
h \equiv ( g mod P ) mod ( P , T ) ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V , a , b be VECTOR of V ;
for V being Vector of V holds V is the carrier of V
reconsider -7 = i as Element of NAT ;
dom f c= [: the carrier of C1 , the carrier of C1 :] ;
x in ( the distance of B ) . n ;
len h2 in Seg len ( f2 ^ g2 ) ;
p1 c= the topology of T ;
]. r , s .] c= [. r , s .] ;
let B2 be Subset-Family of T ;
G * ( B * A ) = id ( the carrier of G1 ) ;
assume p , u , v is_collinear & p <> q ;
[ z , z ] in union rng ( F * G ) ;
'not' ( b . x ) 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S + 1 ;
LIN a1 , a2 , a1 & LIN a2 , a3 , a3 ;
f " ( f .: ( x ) ) = { x } ;
dom ( w2 , v2 ) = dom ( the Element of the carrier of V ) ;
assume that 1 <= i and i <= n and j <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I1 * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q9 . x in rng ( q ^ <* x *> ^ ( q ^ <* x *> ^ ( q ^ ) ) ) ;
L-43 misses L~ pion1 \/ L~ co ;
consider c being element such that [ a , c ] in G ;
assume N19 = o & o = o & o = o ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F ^ ) " { -1 } ;
P . ( k2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , g . j .] ;
attr 0 <= x & x ^2 <= x ;
p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster TrivialcaaaaaaaaaaaaaaaT ( S ) -> non empty ;
let x be Element of S , T ;
<^ F , F . ( a , b ) ^> is one-to-one ;
|. i - j .| <= - ( 2 |^ ( n + 1 ) ) ;
the carrier of [: I[01] , I[01] :] = [: the carrier of I[01] , the carrier of I[01] :] ;
th * ( n + 1 ) > 0 * PI ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b2 , b2 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 4 ;
x joins X , Y ;
set v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 ) , v2 = v2 /. ( i + 1 )
x = r . n .= ( r * ( n + 1 ) ) . 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 ( d * ( A * B ) ) = [: A , B :] ;
0 < sqrt ( p `1 + p `2 ) + sqrt ( p `2 + p `2 ) ;
e . ( m + 1 ) <= e . m ;
B \ominus X c= B "\/" Y
- \infty < \int ( g | B ) | E ;
cluster O \tt F -> with_.. F ;
let U1 , U2 be non-empty MSAlgebra over S , A be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y // x , y & x , y // y , z ;
reconsider p0 = p . x as Subset of V ;
x in the carrier of Lin ( A ) ;
let I , J be Program of SCM+FSA ;
assume - a is lower & a is lower ;
Int ( Cl A ) c= Cl ( Int ( A ) ) ;
assume for A being Subset of X holds Cl A = A ;
assume q in Ball ( x , r ) ;
( ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ) <= ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
Cl Q ` = [#] ( T | P ) ;
set S = the carrier of T ;
set I1 = \sum ( f |^ n ) , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = f |^ n , I2 = g |^ n , I2 = g |^ n , I2 = g |^ n , I2
len p -' n = len p - n ;
A is Subset of Funcs ( x , y ) ;
reconsider n2 = nI as Element of NAT ;
1 <= j + 1 & j + 1 <= len seq & seq ^ <* j *> is one-to-one ;
reconsider q9 = q9 , q9 = q9 as Element of M ;
a1 in the carrier of S1 & a2 in the carrier of S2 & a3 in the carrier of S2 ;
c1 /. ( n1 + 1 ) = c1 . ( n1 + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( ( ( ( ( ( ( ( S * * ( S , T ) ) ) * ( x , y ) ) * ( x , y ) ) ) ) . x ;
consider x being element such that x in \pi \pi \pi A ;
assume r in ( dist ( o , P ) ) .: P ;
set i2 = ( E-max L~ h ) .. h ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M1 , k ) . i = M * ( i , k ) ;
reconsider m = sqrt ( x ^2 + 2 ) as Element of REAL ;
U1 , U2 , U1 , U2 is_collinear & U2 , U1 , U2 is_collinear implies U1 , U2 , U2 is_collinear
set P = Line ( a , d ) ;
len ( p1 ^ p2 ) < len ( p1 ^ p2 ) + 1 ;
T1 , T2 be with_the_topological Function of L , L ;
then x <= y & ( \ast x ) c= ( 'not' y ) . x ;
set M = n -tuples_on Seg ( m , n ) ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of ( a ) ) c= dom ( the_arity_of o ) ;
z1 " = z1 " * ( z1 * z2 ) .= z1 * ( z1 * z2 ) ;
x0 - r / 2 in L /\ dom f ;
then w is string of S & rng w /\ rng w <> {} ;
set pZ = x9 ^ <* Z *> ^ <* Z *> ^ <* Z *> ;
len w1 in Seg ( len ( w ^ <* ( len w ) *> ^ ( len w ) ) ) ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = sqrt ( a . n ) .= ( a . n ) ^2 ;
( p `1 ) ^2 <= ( G * ( len G , 1 ) `1 ) ^2 ;
rng ( g * ( len g ) ) c= L~ g \/ L~ h ;
reconsider k = i-1 * j + 1 as Nat ;
for n be Nat holds F . n is non empty ;
reconsider x9 = x9 as Vector of M ;
dom ( f | X ) = X /\ dom f ;
p , a // p , c & b , a // p , c ;
reconsider x1 = x as Element of REAL m -tuples_on REAL ;
assume i in dom ( a * p ) ;
m . \hbox { \boldmath $ g } = p . \hbox { \boldmath $ g $ } ;
a |^ ( s . m ) - ( a |^ n ) <= 1 ;
S . ( n + k ) c= S . n ;
assume B1 \/ ( B2 \/ B1 ) = ( B1 \/ B2 ) \/ ( B2 \/ B2 ) ;
X . i = { x1 , x2 } & X . i = { x1 , x2 } ;
r2 in dom ( h1 + h2 ) ;
1_minus 0 = a & bR = b ;
FQ is_closed_on t , Q & PQ is_halting_on t , Q ;
set T = IExec ( X , x0 ) , x0 = x0 ;
Int ( Int ( Int R ) ) c= Int ( R ) ;
consider y being Element of L such that c . y = x ;
rng Fwhere Fy = { F . x } ;
G1 " { c } c= B \/ S \/ S ;
then f3 is Relation of X , Y & X is Subset of X ;
set R1 = the Point of P , R2 = the Point of P , R2 = the Point of Q , R2 = the Point of Q ;
assume that n + 1 >= 1 and n + 1 <= len M ;
k2 in NAT ;
reconsider ph = u as Element of / ( n , 1 ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len ( f1 ^ f2 ) ;
reconsider T = b * N as Element of ( G * N ) * ;
len ( ( len fX2 ) + len Sgm dom Sgm dom Sgm dom X2 ) <= len ( Sgm dom Sgm dom X2 ) ;
x " in the carrier of ( ( the carrier of A1 ) \/ the carrier of A2 ) ;
[ i , j ] in Indices ( A * ) & [ i , j ] in Indices ( A * ) ;
for m being Nat holds Re ( F . m ) is simple function of S ;
f . x = a . i .= a . i .= a . i ;
let f be PartFunc of REAL , REAL ;
rng f = the carrier of ( ( TOP-REAL 2 ) | A ) ;
assume s1 = sqrt ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * 1 ) ) ) ) ) ) ) ) ;
attr a > 1 & b > 0 implies a |^ b > 0 ;
let A , B , C be Subset of Carrier ( I ) ;
reconsider X1 = X , Y = Y as Subset of X ;
let f be PartFunc of REAL , REAL ;
r * ( v1 , I ) . ( X , I ) . ( X , I ) . ( X , I ) . ( X , I ) . ( X , I ) . ( X , I ) . ( X , I ) . ( X ,
assume that V is Subgroup of X and X is Subset of Y ;
t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3 , t-3
Q [ d1 \/ { v } ] ;
g .. ( z .. z ) = z .. z + 1 .= z .. z + 1 ;
|. [ x , v ] - [ x , v ] .| = vI ;
- f . w = - ( L * w ) ;
z -' y <= x iff z <= x + y
sqrt ( ( 7 + 1 ) to_power ( 1 + 1 ) ) > 0 ;
assume X is BCK-algebra implies X is BCK-algebra of 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 ,
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( ( ( tan * tan ) ) `| Z ) = ( ( tan * tan ) ) `| Z ) ;
i2 = ( ( f | ( len f -' 1 ) ) ^ ( f | ( len f -' 1 ) ) ) ^ ( f | ( len f -' 1 ) ) ;
X1 = X1 \/ X2 \/ ( X1 \ X2 ) ;
[. a , b .] = [. 1_ G , b .] ;
let V , W be non empty VectSpStr over K ;
dom g2 = the carrier of I[01] & dom g2 = the carrier of I[01] ;
dom ( f2 | the carrier of I[01] ) = the carrier of I[01] ;
( proj2 | X ) .: X = proj2 .: ( X ) ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 < a1 . n & a1 . n < x0 ;
|. ( f /* s ) . k - ( f /* seq ) . k .| < r ;
len Line ( A , i ) = width A & width A = width B ;
SIf ^ S = ( S , T ) .: ( S , T ) ;
reconsider f = v + u as Function of X , the carrier of Y ;
intloc 0 in dom ( Initialized p ) \/ dom DataPart Initialized p ;
i1 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2 & i2 <> i2
arcsin + r * J + r * J = sqrt ( 2 * J + r * J ) + r * J ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = sqrt ( q `1 - x `2 ) as Element of REAL ;
( 0 qua Nat ) + 1 <= i + 1 ;
assume f in the carrier of ( X --> [#] Y ) ;
F . a = H . ( x , y ) ;
true ( T , u ) = TRUE ;
dist ( a * seq . n , h ) < r ;
1 in the carrier of [. 0 , 1 .] ;
( ( p2 - ( p2 - ( p2 - ( - 1 ) ) ) ) * ( ( - 1 ) * ( ( - 1 ) * ( p2 - 1 ) ) ) ) > - 1 ;
|. r1 - q1 .| = |. r1 - q1 .| * |. q1 - q2 .| ;
reconsider S-14 = 8 as Element of Seg ( len S ) ;
( A \/ B ) |^ ( b + c ) c= A |^ ( b + c ) ;
DW { W : W { \rm .first ( ) } = DW } + 1 ;
i1 = [: NAT + n , n :] & i2 = [: the carrier of K , the carrier of K :] ;
f . a [= f . ( O , o1 ) "\/" ( f . ( O , o1 ) ) ;
attr f = v & g = u + v ;
I . n = \int ( F . n , M ) | E ;
( \raise .4ex \hbox { $ \chi $ } , T ) . s = 1 ;
a = VERUM ( A ) or a = {} & a = {} ;
reconsider k2 = s . k2 as Element of NAT ;
( Comput ( P , s , 4 ) ) . DataLoc ( 0 , 2 ) = 0 ;
L~ M1 meets L~ ( M1 * ( i , j ) ) ;
set h = the continuous Function of X , R ;
set A = { L . ( k + 1 ) } ;
for H st H is atomic holds P [ H ]
set bOne = ( S . ( i + 1 ) ) . ( i + 1 ) ;
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 / n + 1 / n ) < sqrt ( 1 / n ) ;
( l - 1 ) * ( l - 1 ) = [ l , l * ( l - 1 ) ] ;
y +* ( i , y ) in dom g ;
let p be Element of QC-WFF ( Al ) ;
X /\ X1 c= dom ( f1 - f2 ) ;
p2 in rng ( f /^ ( len f -' 1 ) ) ;
1 <= indx ( D2 , D1 , j ) + 1 ;
assume x in W2 /\ ( W1 + W2 ) ;
- 1 <= ( f2 . O ) `2 ;
f , g be Function of I[01] , TOP-REAL 2 ;
k1 -' k2 = ( k + 1 ) - k2 + k2 ;
rng ( seq + c ) c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ \/ ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - ( - 1 , 1 ) ;
consider u being Nat such that b = p |^ y * u ;
attr a = e or a = Sum A ;
Cl ( union ( Cl ( Cl Cl Cl Cl Cl Cl Cl Cl ( Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl Cl
len t = len ( ( len t ) + 1 ) + 1 ;
p-29 = v + w & w = v + w ;
DataLoc ( ( Initialize ( t2 . GBP ) ) , 3 ) <> IC SCMPDS ;
g . s = sup ( d " { s } ) ;
( \dot y ) . s = s . y ;
{ s : 0 < s & s < t } c= [: Q , Q :]
s ` \ s = s ` \ ( s ` ) .= s ` \ ( s ` ) ` ;
defpred P [ Nat ] means B + 1 in A ;
( 319 + 1 ) ! = 319 * ( 319 + 1 ) ;
U . ( succ A ) = T . ( A , B ) ;
reconsider y = y as Element of ( len y ) -tuples_on the carrier of K ;
consider i2 being Integer such that y = p * i2 + i2 * i2 ;
reconsider p = Y | Seg k as FinSequence of ( the carrier of K ) * ;
set f = ( S , U ) -TruthEval z ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL 2 ;
SAT ( M . ( n + i , A ) , i ) <> 1 ;
ex r be Real st x = r & a <= r & r <= b & a <= b & r <= b & r <= b ;
R1 , R2 be Element of ( n -tuples_on the carrier of K ) , a , b be Element of ( n -tuples_on the carrier of K ) , c , d be Element of ( n -tuples_on the carrier of K ) , d be Element of ( n -tuples_on the carrier of K ) ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e + |. w .| .| + |. w + |. w + - w .| .| ;
consider y being Element of S such that z <= y and y in X ;
a to_power ( b to_power c ) = 'not' ( a to_power ( b to_power c ) ) ;
||. ( x9 - y9 ) - ( g - y ) .|| < r2 + r2 ;
b9 , a9 // b9 , c9 ;
1 <= k2 & k2 + 1 <= k2 & k2 + 1 <= k2 + 1 ;
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) >= 0 ;
sqrt ( ( q `1 ) ^2 - ( q `2 ) ^2 ) < 0 ;
E-max ( C ) in LSeg ( ( R /. 1 ) , ( R /. 1 ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( ( lim G ) | D ) ;
LIN b , a , c or LIN b , a , c ;
p `1 , a // a , b or p `1 = b & p `2 = c ;
g . n = a * Sum ( f1 ^ f2 ) .= f . n * a ;
consider f being Subset of X such that e = f and f is bijective ;
F | ( N2 , S ) = ( F * ( N2 , S ) ) | ( the carrier of S ) ;
q in LSeg ( q , v ) \/ LSeg ( p , q ) ;
Ball ( m , x0 ) c= Ball ( m , x0 ) ;
the carrier of ( ( 0. ( V ) ) --> 0. ( V ) ) = { 0. V } ;
rng ( ( the function cos ) * ( - 1 ) ) = [. - 1 , 1 .] ;
assume Re ( seq ) is summable & Im ( seq ) is summable ;
||. vseq . n - seq . m .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t2 as 0 string of S2 ;
reconsider x9 = seq as sequence of REAL n , x9 = seq . n , y9 = seq . n as Element of REAL n ;
assume that Index ( E-max C , E-max C ) meets L~ Cage ( C , n ) and Index ( E-max C , E-max C ) in L~ Cage ( C , n ) ;
- ( Cl Cl ( Cl Cl F ) ) < F . n - ( Cl F ) . n ;
set d1 = dist ( x1 , y1 ) , z1 = dist ( y1 , y2 ) , z2 = dist ( y2 , z2 ) , z2 = dist ( y2 , z2 ) , z2 = dist ( y2 , z2 ) , z2 = dist ( y2 , z2 ) , z2 = dist ( y2 , z2 )
2 |^ sqrt ( sqrt ( 1 + sqrt 5 ) ^2 ) = 2 ^2 + ( sqrt 5 ) ^2 ;
dom ( ( the Sorts of U2 ) * ( the Arity of U2 ) ) = [: dom ( the Arity of U2 ) , dom ( the Arity of U2 ) :] ;
set x1 = - ( k + 1 ) + ( k + 1 ) ;
assume for n being Element of NAT holds 0. <= F . n & 0. <= F . n ;
assume that 0 <= ST . i and ( T . i ) `2 <= 1 ;
for A being Subset of X holds c . ( c . A ) = c . A
the support ( ( Carrier ( L2 ) ) + ( L2 ) ) c= { I } ;
'not' ( All ( x , p ) => ( 'not' ( x , p ) ) ) is valid ;
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the carrier of G ;
Z c= dom ( ( ( ( ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) ) ;
|. 0. TOP-REAL 2 - ( q `1 / q `2 - q `2 ) .| < r / 2 ;
ConsecutiveSet2 ( B , C ) c= ConsecutiveSet2 ( A ) & \bf d in C ;
E = dom ( L + ( L + R ) ) & ( L + R ) . x = ( L + ( R + R ) ) . x ;
C |^ ( A + B ) = C |^ ( B + C ) ;
the carrier of W2 c= the carrier of W1 + the carrier of W2 ;
I . IC Comput ( P , s , 2 + 1 ) = P . IC Comput ( P , s , 2 ) ;
attr x > 0 means : Def5 : \frac { 1 } * x = x |^ ( 1 / x ) ;
LSeg ( f ^ g , i ) = LSeg ( f , i ) ;
consider p being Point of T such that C = [. p , q .] ;
b , c are_connected & a , b are_connected implies b , c are_connected
assume f = id ( the carrier of I[01] ) ;
consider v such that v <> 0. V and f . v = L * v ;
let l be Linear_Combination of {} ( the carrier of V ) ;
reconsider g = f " as Function of ( the carrier of U1 ) , the carrier of U2 ;
A1 in the carrier of ( G | X ) ;
|. - x .| = - x .= - x ;
set S = 1GateCircStr ( x , y , c ) ;
Fib ( n ) * ( 5 * n ) >= 4 * n * n ;
f3 /. ( k + 1 ) = f3 . ( k + 1 ) .= f3 . ( k + 1 ) ;
0 mod i = sqrt ( i * ( 0 qua Nat ) ) ;
Indices ( M1 * M2 ) = [: Seg n , Seg n :] & len ( M1 * M2 ) = [: Seg n , Seg n :] ;
Line ( SX2 , j ) = [: Seg n , Seg n :] ;
h . x1 = [ y1 , y2 ] ;
|. f .| (#) ( |. ( |. f .| ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
assume x = ( a1 ^ <* b1 *> ) ^ ( b1 ^ ( b1 ^ b2 ) ) ;
PI is_closed_on s , P & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on P , Q & I is_halting_on s , P & I is_halting_on s , P & I is_halting_on P
DataLoc ( t1 . a , 4 ) = intpos ( 0 + 4 ) .= intpos ( 0 + 4 ) ;
x + y < - x + y & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , b , d ;
ft9 . 1 = f . 0 .= a ;
x + ( y + z ) = x1 + ( y1 + y2 ) ;
fa1 . a = f2 . a & v in InputVertices S & v in InputVertices S & v in InputVertices S & v in InputVertices S ;
( p `1 ) ^2 <= ( E-max L~ Cage ( C , n ) ) ^2 ;
set Rseq = Cage ( C , n ) .. Cage ( C , n ) ;
( p `1 ) ^2 >= ( ( E-max C ) ^2 + ( E-max C ) ^2 ) ;
consider p such that p = pw and s1 < p and s1 < s2 and s2 < s2 ;
|. ( f /* ( s * F ) - G ) . l - ( f /* ( s * F ) - G ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , i , q ) ;
len ( Line ( N , k + 1 ) ) = width N & len ( Line ( N , k ) ) = width N ;
f1 /* ( s1 + s2 ) is convergent & f2 /* ( s1 + s2 ) is convergent ;
f . x1 = x1 & f . x2 = y1 & f . y2 = y2 ;
len f <= len f + 1 & len f + 1 = len f + 1 & len f + 1 + 1 <= len f ;
dom ( Proj ( i , n ) * s ) = [: the carrier of TOP-REAL n , the carrier of TOP-REAL n :] ;
n = k * ( 2 * t ) + ( k mod 2 ) ;
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 Y as Subset of X ;
1 in the carrier of [. 1 / 2 , 1 / 2 .] ;
let L being complete LATTICE , L being RelStr , n , m being Element of NAT st n <= m holds L . m = isomorphic ( n , L ) & L . n = isomorphic ( m , L ) ;
[ gi , gj ] in [: I \ I , I :] ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume that f1 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 and f2 is_differentiable_in x0 ;
reconsider y = ( a ` ) / ( F . i ) as Element of L ;
dom s = { 1 , 2 , 3 } & s . 1 = <* 2 , 3 *> & s . 2 = <* 3 , 1 *> ;
( min ( g , h ) ) . c <= h . c ;
set G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G , G2 = the Vertex of G ;
reconsider g = f as PartFunc of REAL , REAL ;
|. s1 . m - p .| < d / 2 ;
for x being element st x in u holds x in u iff x in v & x in u & x in v ;
P = the carrier of ( TOP-REAL n ) | P ;
assume 10 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) ;
( 0. X ) \ x = 0. X ;
g be Element of Hom ( cod f , cod g ) ;
2 * a * b + ( 2 * c ) * ( 2 * b + ( 2 * c ) * ( 2 * c ) ) <= 2 * ( 2 * ( 2 * c ) ) ;
f , g be Point of X , h be Point of X , i , j be Nat st f = h & g is continuous & h = i holds f is continuous
set h = Hom ( a , g ) ;
then idseq ( n ) | Seg m = idseq m | Seg m ;
H * ( g " * a ) in the carrier of H ;
x in dom ( ( ( #Z n ) * ( sin ) ) `| Z ) ;
cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , p2 , P , p1 , p2 & LE p1 , p2 , P & LE p1 , p2 , P & LE p1 , p2 , P & p1 , p2 , P & p1 , p2 , P & p1 , p2 , P & p1 , p2 , P & P , p1 , P , p1 , p2 &
attr B is Subset of A means : Def5 : B c= BDD A & B c= BDD B ;
deffunc D ( set , set ) = union rng $2 & $2 in rng $2 & $2 = union rng $2 ;
n + - ( n + 1 ) < len ( ( p ^ q ) ^ ( n + 1 ) ) ;
attr a <> 0. K means : Def5 : the_rank_of ( a * M ) = the_rank_of ( a * M ) ;
consider j such that j in dom F12 and I = Seg n + j ;
consider x1 such that z in x1 and x1 in ( P * ) . x1 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
set CB = Comput ( P2 , s2 , i + 1 ) , P2 = P2 +* I , s2 = P2 +* I , P2 = P2 +* I , s2 = P2 +* I , P2 = P2 +* I , s2 = P2 +* I , P2 = P3 = P3 ;
set \cal \cal v = 3 -tuples_on { a , b } , h = 3 -tuples_on the carrier of G , h = 3 -tuples_on the carrier of G , g = 3 -tuples_on the carrier of G , h = 3 -tuples_on the carrier of G , g = 3 -tuples_on the carrier of G , h = 4 -tuples_on the carrier of
conv @ W c= union ( F .: ( E \ { W } ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( - 1 ) ) ;
s3 <= s2 + ( s2 - s1 ) * ( s2 - s1 ) ;
dom ( f * ( f2 * f3 ) ) = dom f /\ dom ( f2 * f3 ) ;
dom ( f * G ) = dom ( l * F ) /\ Seg k ;
rng ( s ^\ k ) c= dom f1 \ ( f1 /* ( x0 + k ) ) ;
reconsider gp = gp as Point of TOP-REAL 2 ;
( T * h ) . x = T . ( h . x ) ;
I . J . J = ( I * J ) . J ;
y in dom \mathopen { \rm qua } \HM { means : for o being element st o in dom the \! \! : = ( the Arity of A ) . o } ;
for I being non degenerated integral of I , L being non empty doubleLoopStr holds I is commutative iff I is commutative
set s2 = s +* ( intloc 0 , 1 ) , P2 = P +* Start-At ( 0 , 1 ) , P2 = P +* Start-At ( 0 , SCM+FSA ) , P2 = P +* Start-At ( 0 , SCM+FSA ) , P2 = P +* Start-At ( 0 , SCM+FSA ) , P2 = P +* Start-At ( 0 , SCM+FSA ) ;
P1 /. IC s1 = P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 . IC s1 .= P1 ;
lim S1 in the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | the carrier of ( TOP-REAL 2 ) | K1 ;
v . ( Carrier ( v ) ) . i = ( v *' ) . i ;
consider n being element such that x in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F ( x ) and F1 . x <> 0 ;
\frac { 0 , 0 , x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 *> = { E , x5 , x5 , x5 } ;
j + ( 2 * ( k2 + 1 ) ) + ( 2 * ( k2 + 1 ) ) > j + ( 2 * ( k2 + 1 ) ) ;
{ s , t } on ( a3 , a2 ) & { s , t } on ( a3 , a2 ) & { s , t } on ( a3 , a2 ) & { s , t } on ( a3 , a2 ) & { s , t } on ( a3 , a2 ) & { s , t } on ( a3
n1 > len ( ( p2 ^ ( p2 ^ ( p1 ^ ( p2 ^ ( p2 ^ ( p2 ^ ( p1 ^ ( p2 ^ ( p2 ^ ( p1 ^ ( p1 ^ ( p1 ^ ( p2 ^ ( p1 ^ ( p2 ^ ( p1 ^ ( p1 ^ ( p1 ^ ( p1 ^ ( p1 ^ ( p1 ^ (
g1 . ( HT ( g2 , T ) ) = 0. L ;
then H1 , H2 are_are \kern1pt : where H1 , H2 , Z , H , H is \kern1pt } ;
( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) + ( E-max L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) > 1 ;
]. s , 1 .[ = ]. s , 1 .[ /\ ]. 0 , 1 .[ ;
x1 in [#] ( ( TOP-REAL 2 ) | K1 ) ;
let f1 , f2 be continuous PartFunc of REAL , REAL ;
indx ( t\times t\times \times \times L~ z , \times L~ z ) is Element of k -tuples_on BOOLEAN ;
I = S11111111111I & I = ( k2 , k1 ) --> k2 ;
u in { [ a , u ] } \/ { [ a , b ] } ;
( w | p ) | ( w | p ) = p | ( w | p ) ;
consider v2 such that v2 in W2 and x = v + v2 and v2 in W1 and x = v + v2 ;
for y st y in rng F ex n st y = a |^ n & n <= len F ;
dom ( g * ( ( g * ( id V ) --> C ) ) = K ;
ex x being element st x in ( the Sorts of U1 ) . s \/ ( the Sorts of U1 ) . s ;
ex x being element st x in ( w \/ ( A \/ B ) ) . s & x in ( w \/ B ) . s ;
f . x in the carrier of [. - r , r .] ;
( the carrier of X1 union X2 ) /\ the carrier of X2 <> {} implies X1 /\ X2 <> {} & X1 /\ X2 <> {} implies X1 /\ X2 <> {} implies X1 /\ X2 = {}
L1 /\ LSeg ( p1 , p2 ) c= { p1 , p2 } ;
sqrt ( b + ( b-2 ) / 2 ) < r / 2 ;
sup { x } in { x } & x "\/" y in { x } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G1 such that z = y and P [ z , y ] ;
( the \overline of ( the carrier of ( M , 1 ) ) . i <= e ;
len ( w ^ ( w ^ ( w ^ w ) ) ) + 1 = len w + ( len w + 1 ) ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | E ` ` = g | E ` .= g | E ` ` .= g | E ` ;
reconsider i1 = x1 , i2 = x2 as Element of NAT ;
( a * A ) |^ ( n + 1 ) = ( a * A ) |^ ( n + 1 ) ;
assume ex x0 be Element of NAT st f |^ x0 is Ax0 & f |^ x0 = f . x0 ;
Seg ( len ( ( f2 ^ ) ^ <* f2 *> ) ) = dom ( ( f2 ^ <* f1 ^ <* f2 *> ) ^ <* f1 ^ f2 *> ) ;
( Complement ( A * B ) ) . m c= ( Complement ( A * B ) ) . m ;
f1 . p = p1 & ( f2 . p ) `2 = p2 `2 ;
FinS ( F , Y ) = FinS ( F , Y ) ^ ( F | Y ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( |. x .| ^2 + |. x .| ^2 ) <= sqrt ( ( |. x .| ^2 + |. x .| ) ^2 ) ;
Sum ( f | X ) = Sum f & dom ( g | X ) = dom g ;
assume for x , y st x in Y holds x in Y & y in Y ;
assume that W1 is Subspace of W2 and W2 is Subspace of W1 and W2 is Subspace of W2 ;
||. vseq . x - vseq . x .|| = lim ( ||. vseq . x - vseq . x .|| ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) <= sqrt ( 1 + ( p `2 ) ^2 ) ;
g | divset ( p , r ) = id ( TOP-REAL 2 ) | divset ( p , r ) ;
set Nmin = ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
let T being non empty TopSpace , T be non empty TopSpace ;
width B |-> ( 0. K ) = width ( B @ ) .= width ( B @ ) .= width ( B @ ) .= width ( B @ ) .= width ( B @ ) ;
attr a <> 0 means : Def5 : ( A \ B ) c= ( A \ B ) \ ( A \ B ) ;
then f is_partial u0 , u & pdiff1 ( f , 1 ) is_differentiable_in u ;
assume that a > 0 and a <> 1 and b <> 0 and c <> 0 and a <> 0 and b <> 0 and c <> 0 and d <> 0 and d <> 0 and d <> 0 and c <> 0 and d <> 0 and b <> 0 and c <> 0 and d <> 0 and d <> 0 and d <> 0 and d
w1 , w2 // ( the carrier of G ) , ( the carrier of G ) --> ( the carrier of G ) ;
p2 /. IC Comput ( p2 , s2 , i + 1 ) = p2 . IC Comput ( p2 , s2 , i + 1 ) .= i ;
ind ( T | b ) | b = ind ( T | b ) | b .= ind ( T | b ) | b ;
[ a , A ] in Indices ( Line ( A , i ) ) & [ a , A ) = [ a , A * ( i , j ) ] ;
m in ( the Arrows of C ) . ( o1 , o2 ) ;
( 'not' a , CompF ( PA , G ) ) . z = TRUE ;
reconsider \varphi = \varphi , \varphi = \varphi , \varphi = l , \varphi = l , \varphi = l , \varphi = l , l = l , \varphi = l , l = l , l = l , t = l = l , l = ( l , t ) = ( l , t ) `1 ;
len s1 " * ( len s2 ) + ( len s2 ) * ( len s2 ) + 1 > 0 ;
\delta ( f , D ) . ( sup A ) - f . ( sup A ) < r ;
[ f21 , f21 ] in the InternalRel of A & [ ( the InternalRel of A ) \/ the carrier of B ) = the carrier of A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = the carrier of ( TOP-REAL 2 ) | K1 & K1 = ( TOP-REAL 2 ) | K1 ;
consider z being element such that z in dom g2 and p = g2 . z ;
[#] ( V1 ) = { 0. V } .= the carrier of ( W1 + W2 ) \/ the carrier of W2 ;
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 x0 - r < s and ||. x1 - x0 .|| < s ;
h1 = f ^ <* p3 *> ^ ( <* p3 *> ^ ( f ^ <* p3 *> ^ ( f ^ h ) ) .= h ^ ( f ^ h ) .= h ^ ( f ^ h ) ;
c /. [ b , c ] = c /. ( a , b ) .= c /. ( a , b ) ;
reconsider t1 = p1 , t2 = p2 as Element of the carrier' of C ;
sqrt ( 1 / 2 ) in the carrier of ( TOP-REAL 2 ) | K1 ;
ex W being Subset of X st p in W & h .: W c= V ;
( h . p1 ) `2 = C * ( ( h . p2 ) `2 ) + D ;
R . b = 2 * PI .= 2 * PI .= PI * PI ;
consider \lambda such that B = 1- C * ( 1- C ) and 0 <= \lambda and 0 <= 1- C ;
dom g = dom ( the Sorts of A ) & dom ( the Sorts of A ) = the carrier' of S ;
[ P . ( l , .. P ) , P . ( l , .. P ) ] in Indices ( T * ( l , len P ) ) ;
set s2 = Initialize s , P2 = P +* I , P2 = P +* I , P3 = P +* I , P4 = P +* I , P4 = P +* I , P4 = P +* stop I , P4 = P +* stop I , P4 = P2 +* stop I , P4 = P2 ;
reconsider M = mid ( z , i2 , i2 ) as Matrix of ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) , ( len z ) -tuples_on ( L~ z ) ) ;
y in product ( the support of J ) \/ { V } ;
1 / |[ 0 , 1 ]| = 1 & |[ 0 , 1 ]| = |[ 0 , 1 ]| ;
assume x in the left of g or x in the carrier' of g ;
consider M being strict Subgroup of A such that a = M and T is SubSpace of M ;
for x st x in Z holds ( ( ( ( ( ( ( ( ( ( ( ( ( ^ ) ) ^ ) ) + ( ( ( ( ( ( ( ( ( ( ( ^ ) ) ^ ) ) ) ) ) ) ) ) ) ) ) `| Z ) ) = x ;
len ( W1 + W2 ) + len ( W2 + W1 ) = 1 + ( len W1 + ( len W1 + len W2 ) ) ;
reconsider h1 = v2 . n - v1 as Lipschitzian LinearOperator of X , Y ;
( Y. -' len ( p ^ q ) ) + 1 in dom ( p ^ q ) ;
assume that s2 is conjunctive and F is conjunctive and F is conjunctive & F is conjunctive & not F is conjunctive & not F is conjunctive & not G is subformula & G is subformula of ( the carrier of G ) \ the carrier of G ;
( ( ( \mathclose { \bf 2 } ) * ( x , y ) ) ) `1 = ( ( \mathclose { \bf 2 } ) * ( x , y ) ) `1 ;
for u being element st u in Bags n holds ( p *' ) . u = p . u
for B being Subset of PA st B in E holds A = B or A misses B or A misses B or A misses B ;
ex a being Point of X st a in A & A /\ { y } = { a } ;
set W2 = <* p *> \/ ( <* p *> ^ <* p *> ) ;
x in { X where X is Subset of L : X is Subset of L : X in F } ;
the carrier of W1 /\ the carrier of W2 c= the carrier of W1 + the carrier of W2 ;
[ a1 + b * ( a + b ) , id ( a + b ) * ( a + b ) ] = [ a1 , b1 + b1 * ( a + b ) ] ;
( dom ( X --> f ) ) . x = ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => r ) in TAUT ( A ) ;
set \pi = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
set \pi = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) ) ;
- 1 + 1 <= sqrt ( 2 |^ ( n -' m ) ) + 1 ;
( reproj ( 1 , z ) ) . x0 in dom ( ( f1 * f2 ) * ( reproj ( 1 , z ) ) ) ;
assume that b1 . r = { c1 } and c1 . r = { c1 } ;
ex P st a1 on P & a2 on P & b on P & c on P & d on P & b on P & c on P & d on P & d , b is_collinear & d , c is_collinear & d , d is_collinear & d , b is_collinear & d , c is_collinear & d , d is_collinear & d , c is_collinear & d , d is_collinear
reconsider gf = g * f , g = h * f as strict Element of X ;
consider v1 being Element of T such that Q = ( \mathopen { \uparrow } v1 ) ` ;
n in { i where i is Nat : i < n + 1 } ;
( F /. ( i , j ) ) `2 >= ( F /. ( m + k ) ) `2 ;
assume K = { p : |. p .| - |. p .| >= 0 } ;
ConsecutiveSet2 ( A , succ succ succ succ O ) = ( sequence of A ) ^ ( succ O ) ;
set I1 = Stop SCM+FSA , i2 = Stop SCM+FSA , i2 = Stop SCM+FSA , i2 = Stop SCM+FSA , i2 = Stop SCM+FSA , i1 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , i2 = halt SCM+FSA , j2 = halt
for i being Nat st 1 < i & i < len z holds z /. i <> z /. 1 & z /. ( i + 1 ) <> z /. 1 ;
X c= ( the carrier of L1 ) \/ the carrier of L2 ;
consider x9 being Element of GF ( p ) such that x9 |^ i = a and x9 in the carrier of GF ( p ) ;
reconsider f3 = d1 , f3 = d1 as Element of D * ;
ex O being set st O in S & ( C c= O ) & ( M c= O ) & ( M = O ) & ( M = O implies M = {} implies M = {} ) & M = {} implies M = {} )
consider n being Nat such that for m being Nat st n <= m holds S . m in U ;
f * g is_differentiable_in x & reproj ( i , x ) . i = ( proj ( i , m ) * g . x ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ $1 + 1 & A c= succ $1 & A c= succ $1 implies A c= succ $1 + 1 ;
the left of ( - g ) = the .[ & - ( g - f ) = the .[ ;
reconsider pOne = x , pOne = y as Point of TOP-REAL 2 ;
consider g2 such that g2 = y and x <= g2 and g2 <= x0 and g2 <= x0 and g2 <= g2 and g2 < x0 ;
for n being Element of NAT ex r being Element of REAL st X [ n , r ]
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len y2 + len y2 .= len y2 + len y2 ;
for x being element st x in X holds x in the Element of the carrier of K & x in the carrier of K implies x in the carrier of K
LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) = { p1 , p2 } ;
redefine func v IExec ( X , Y ) -> set equals : the carrier of X ;
len ( ( the connectives of C ) * ( len the connectives of C ) ) <= len ( the connectives of C ) & len ( the connectives of C ) + len the connectives of C = len the connectives of C ;
attr K is has J means : Def5 : for a , b st a <> b & b <> c holds v . ( a , b ) = i * ( a , b ) ;
consider o being OperSymbol of S such that t . {} = [ o , the carrier of S ] -tree p ;
for x st x in X ex y st x c= y & y in X & y in Y ;
IC Comput ( P-6 , 2 , k + 1 ) in dom ( P21 +* I , 2 ) ;
attr q < s & r < s & ]. p , q .[ c= ]. p , s .] ;
consider c being Element of Class ( F . c , F . ( c , 1 ) ) such that Y = F . ( F . ( c , 1 ) , F . ( c , 1 ) ) ;
the Arity of S2 = id the carrier' of S2 & the carrier' of S2 = the carrier' of S2 & the carrier' of S2 = the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] , y2 = [ <* z , x *> , f2 ] , f2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( ( ( 1 / 2 ) * ( ( ( #Z ) ) * ( ( #Z 2 ) * ( ( #Z ) ) * ( ( #Z ) ) * ( ( #Z 2 ) * ( ( #Z ) ) * ( ( #Z 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
-7 in Int cell ( GoB f , i , j ) \ cell ( GoB f , i , j ) ;
( q `2 ) ^2 >= ( ( ( q `2 ) ^2 + ( q `2 ) ^2 ) ;
set Y = { a "/\" a ` where a is Element of L : a in X } ;
i -' len f + len ( f /^ ( len f -' 1 ) ) <= len f + len f - len f ;
for n holds ex x st x in N & x in dom ( f | X ) & h . x = - x ;
set s3 = ( a , I ) --> ( a , I ) , s3 = ( a , I ) --> ( a , I ) ;
( p . k ) . 0 = 1 or ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 & ( p . k ) . 0 = 1 & ( p . k = 1 ) & ( p . k = 2 ) & ( p . k = 3 )
u + Sum ( Lw ) in ( U \ { u } ) \/ { u } ;
consider x9 being set such that x in x9 and x9 in the carrier of G and x9 in the carrier of G and x9 in the carrier of G ;
( p ^ q ) . m = ( q | k ) . m + ( p | k ) . m ;
g + h = g1 + g2 & bounded ( g + h ) = g + h + ( h + g ) ;
L1 is distributive & L2 is distributive implies for L being distributive non empty doubleLoopStr , u being Element of L st u in the carrier of L holds ( L "\/" ( L ) ) . u = L . u "\/" ( L ) . u
attr x in rng f & y in rng ( f /^ ( n ) ) & x in rng f implies f | ( n + 1 ) = f /^ ( n + 1 ) ;
assume that 1 < p and sqrt ( 1 + p ) + sqrt ( 1 + p ) = 1 and 0 <= p and p `1 = p `1 and p `2 = p `2 and p `2 = p `2 ;
Fh * ( f , smin ) = ( 1 , t ) *' + ( 1 , t ) *' ( 1 , t ) .= ( 1 , t ) *' + ( 1 , t ) *' + ( 1 , t ) *' ( 1 , t ) *' ( 1 , t ) ;
let X being set , A being Subset of X , B being Subset of X st A = B holds A is open iff B is open
( ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ) `2 <= ( E-max L~ Cage ( C , n ) ) `2 ;
let c being Element of the bound of A , a , b being Element of the bound of A holds c <> a & b <> a & a <> b & c <> b & b <> a & a <> b & c <> a & b <> a & c <> a & a <> b & b <> b & c <> a & a <> b & c <> b & b <> a & c <> a & a <>
s1 . DataLoc ( s2 . GBP , 2 ) = ( s2 . DataLoc ( s2 , 2 ) , 2 ) .= ( s2 . DataLoc ( s2 , 2 ) , 2 ) .= ( s2 . DataLoc ( s2 , 2 ) , 2 ) .= ( s2 . DataLoc ( s2 , 2 ) , 2 ) ;
let a , b be Real , y be Real ;
for x , y being Element of X holds x \ y = ( x \ y ) \ x ;
mode f0 of i , j , m , n , k , m , n , k , m , n , m , k be Nat ;
set x2 = ( Re ( y ) ) | ( |. y .| ) ;
[ y , x ] in dom ( u | ( dom ( v | ( dom u ) ) ) ) & ( u | ( dom v ) ) . ( y , x ) = g . ( y , x ) ;
]. lower_bound divset ( D , k ) - lower_bound divset ( D , k ) + lower_bound divset ( D , k ) - lower_bound divset ( D , k ) <= lower_bound divset ( D , k ) - lower_bound divset ( D , k ) ;
0 <= h2 . ( S . n ) & |. h2 . n - h2 . n .| < e / 2 ;
( - ( - ( - ( q `1 / |. q .| - q `1 ) / |. q .| - ( q `1 / |. q .| - q `1 ) / |. q .| - ( q `1 / |. q .| - q `2 ) / |. q .| - ( q `1 / q `1 ) / |. q .| - ( q `1 / q `1 / |. q .| - q `2 ) ) ) ^2 ) <= (
set A = sqrt ( 2 * PI ) ;
for x , y being set st x in { R } & y in { R } holds x in { R }
deffunc F2 ( Nat ) = b . ( M * ( i , j ) ) * ( M * ( i , j ) ) ;
for s being element holds s in |= ( f \/ g ) iff s in |= ( f \/ g ) & s in |= ( f \/ g ) ;
let S being non empty non void non empty non void ManySortedSign ;
max ( ( |. z .| ) ^2 + ( |. z .| ) ^2 + ( |. z .| ) ^2 ) >= 0 ;
consider n1 being Nat such that for k being Nat holds seq . k - seq . k < r + r ;
Lin ( A /\ B ) is Subspace of Lin ( A ) & Lin ( B ) is Subspace of Lin ( A ) /\ Lin ( B )
set n-15 = n -tuples_on ( Seg ( n + 1 ) ) ;
f " ( V ) in [: X , Y :] & f " ( V ) in [: X , Y :] & f " ( V , X ) in [: X , Y :] ;
rng ( a +* c ) c= { a , b } \/ { c } ;
consider y being Vertex of G1 such that y = y and dom y = dom the WG of G and dom y = dom the WG of G and dom y = dom the WG of G ;
dom ( ( 1 / 2 ) (#) ( f ^ ) ) /\ ]. - 1 , 1 .[ c= ]. - 1 , 1 .[ ;
i is Matrix of n , n , K & i is Element of Seg n , n , K ;
v ^ ( ( n |-> 0 ) |-> ( n + 1 ) ) in dom ( ( ( ( n |-> 0 ) --> ( n + 1 ) ) --> ( n + 1 ) ) ;
ex a , k1 , k2 st i = a & k1 = b & k2 = c & k2 = d & k2 = d ;
t . NAT = ( NAT .--> ( i -' 1 ) ) . NAT .= ( i .--> ( i -' 1 ) ) . NAT .= i ;
assume that F is bbfamily of X and rng p = { n } and rng p = { n } and dom p = { n } ;
not LIN b , b9 , b9 & not LIN b , b9 , b9 & not b , b9 // b , b9 & b , b9 // b9 , c9
( L1 .--> L2 ) . O c= ( L1 "/\" L2 ) . O & ( L2 "\/" L2 ) . O = ( L1 "\/" L2 ) . O ;
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 * ( u , w ) = b * ( w , y ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. Sum ( $1 , D ) - Sum ( $1 , D ) .| <= Sum ( $1 , D ) - Sum ( $1 , D ) ;
u = sin ^ ( x , y ) * ( x , y ) .= v + ( x , y ) * ( x , y ) .= v + ( x , y ) * ( x , y ) .= v + y * ( x , y ) .= v + x ;
dist ( seq . n + x , x ) <= dist ( seq . n + x ) + ( g . n ) ;
P [ p , |. p .| , id ( the carrier of A ) ] ;
consider X being Subset of [: Al ( ) , NAT ( ) , NAT ) , NAT :] such that X c= Y and X is non empty ;
|. b .| * |. ( eval ( f , z ) ) .| >= |. b .| * |. ( eval ( f , z ) ) .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) + 1 ;
l in { l where l is Real : l <= l & l <= 1 } ;
Partial_Sums ( G . n ) <= Partial_Sums ( G . n ) + Partial_Sums ( G . n ) ;
f . y = x * 0. L .= x * 0. L .= x * 0. L .= x * 0. L .= x * 0. L ;
NIC ( ( i1 , i1 ) , ( i1 + 1 ) ) = { i1 , i2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 , p2 } ;
product ( the support of ( I +* ( i , 1 ) ) ) . ( i + 1 ) in [: the carrier of ( I +* ( i , 1 ) ) , the carrier of ( I +* ( i , 1 ) ) :] ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s , n ) | ( the carrier of S2 ) ;
W - ( W - ( 1 / 2 ) ) <= ( ( 1 / 2 ) * ( 1 / 2 ) ) / ( 1 + 2 * ( 1 / 2 ) ) ;
f /. i2 <> f /. ( i1 + len g -' 1 ) ;
M , ( ( ( ( the _ of M ) | ( ( the carrier of M ) | ( the carrier of M ) ) \/ ( the carrier of M ) ) ) |= H ;
len ( ( P ^ Q ) ^ ( P ^ Q ) ) in dom ( P ^ Q ) ;
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( m , n ) c= A |^ ( m , n ) ;
\ { q : |. q .| - |. q .| < a } c= { q1 : |. q .| - |. q .| } ;
consider n1 being element such that n1 in dom p1 and y = p1 . n1 and p1 . n1 = p1 . n1 ;
consider X being set such that X in Q and for Z being set st Z in Q holds X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCMPDS ;
for v being Vector of l holds ||. v .|| = ||. ( the _ of l ) . v .||
for \varphi st \varphi in X holds phi in X & not contradiction holds not contradiction
rng ( Sgm dom ( f | ( dom f ) ) ) c= dom ( f | ( dom f \ dom g ) ) ;
ex c being FinSequence of D st len c = k & for k being Nat st k in dom c holds P [ k , c . k ] ;
the_arity_of ( a , b ) = <* <* ( a , b ) , ( b , c ) *> *> , <* <* a , b *> *> *> *> *> *> *> *> *> , <* <* a , b *> *> *> *> *> *> *> *> is <* <* <* a , b *> , <* a , b *> *> *> , <* a , b *> *> , <* b , c *> *> *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f . x - f . x .| and f1 is continuous and f2 is continuous and continuous ;
a1 = b1 & a2 = b2 & a3 = b2 & a2 = b2 & a3 = b1 & a2 = b2 & a3 = b2 & a2 = b2 & b2 = b3 & b3 = b3 & b3 = b3 & b2 = b3 & b3 = b3 & not b2 = b3 & b1 = b3 & not b2 = b3 & b1 = b3 & not b2 = b3 & not b2 = b3 & b1 = b3 & b2 = b3 & not b2 = b3 & b1 =
D2 . indx ( D2 , D1 , n1 ) = D1 . indx ( D2 , D1 , n1 ) + 1 ;
f . ( |. r .| ) = |. r .| .= |. r .| .= |. r .| .= |. r .| .= |. r .| .= |. r .| ;
consider n being Nat such that for m being Nat st n <= m holds C . m = C ( m ) ;
consider d being Real such that for a , b being Real st a in X & b in Y holds a <= b ;
||. L - ( h - c ) .|| <= ||. ( h - c ) * ( h - c ) .|| ;
attr F is commutative means : Def5 : for b being Element of X holds F . ( b , f . b ) = f . b ;
p = 1- ( 0 + 1 ) * ( 0 + 1 ) .= 1 * ( 0 + 1 ) .= ( 0 + 1 ) * ( 0 + 1 ) * ( 0 + 1 ) .= ( 0 + 1 ) * ( 0 + 1 ) .= ( 0 + 1 ) * ( 0 + 1 ) .= ( 0 + 1 ) * ( 0 + 1 ) * ( 0 + 1 ) .= ( 0 + 1 ) * ( 0 + 1
consider z1 such that b `1 , z1 , z2 is_collinear and o <> z1 and o <> z1 and o <> z1 and o <> z1 and o <> z1 and o <> z1 and o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o <> z1 & o
consider i such that Arg ( ( ( Rotate ( s , r ) ) . i ) = s + ( Arg ( s ) ) ;
consider g such that g is one-to-one and dom g = Seg ( len f ) and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom f and rng g = dom
assume that A = P2 \/ P2 and P = P1 and P = P2 and Q c= P2 and P c= P1 and Q c= P2 and P c= P1 and Q c= P2 and Q c= P2 and P c= P1 and Q c= P2 and Q c= P2 and P c= P1 and Q c= P2 and P c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q
attr F is associative means : only : F .: ( F .: ( f , g ) ) = F .: ( f .: ( f , g ) ) ;
ex x being Element of NAT st m = x & x in { i } & m < i & i < j + 1 } ;
consider k2 being Nat such that k2 in dom ( P2 | ( Seg k2 ) ) and l = ( P2 | k2 ) . k2 ;
seq = r * seq implies for n holds seq . n = r * seq . n & seq . n = r * seq . n
F1 . [ f . a , a ] = [ f . a , f . b ] ;
{ p } "\/" D2 = { p } "\/" ( { y } "\/" D ) ;
consider z being element such that z in dom ( ( F . 0 ) | ( dom F ) ) and ( ( F . 0 ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds f . x = f . y
cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( i + 1 , j ) `1 } ;
consider e being element such that e in dom ( T | E ) and ( T | E ) . e = v ;
( F ' * b1 ) . x = ( ( ( Len ) . b2 ) * ( B12 ) ) . x ;
- 1 = ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) *
attr for x being set st x in dom f /\ dom g holds g . x <= f . x ;
len ( f1 . j ) = len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) + len ( f2 ^ g2 ) .= len ( f1 ^ f2 ) + len g2 .= len ( f1 ^ f2 ) + len g2 .= len ( f1 ^ f2 ) + len g2 ;
All ( 'not' a , A , G ) '&' All ( 'not' a , A ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not' a , B ) '&' All ( 'not'
LSeg ( E . k2 , F . k2 ) c= Cl ( ( L~ Cage ( C , m ) ) | ( L~ Cage ( C , m ) ) ;
x \ ( a |^ m ) = x \ ( a |^ ( k + 1 ) ) .= ( x |^ ( k + 1 ) ) \ a .= x |^ ( k + 1 ) ;
k in func func func func func func func func func func func func func func func func func func func func I -> Element of ( the carrier of S ) --> ( the carrier of S ) , the carrier of S ) equals ( the carrier of S ) --> ( the carrier of S ) --> ( the carrier of S ) ;
for s being State of A1 , n being Nat holds Following ( s , n + 1 ) is stable of Following ( s , n + 1 ) & Following ( s , n + 1 ) is stable ;
for x st x in Z holds ( ( ( ( 1 / 2 ) (#) ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ) ) ) `| Z ) . x <> 0 & ( ( ( 1 / 2 ) * ( 1 / 2 ) ) ) . x <> 0 ;
support ( ( support ( m ) ) \/ support ( m ) ) c= support ( m ) \/ support ( m ) \/ support ( m ) \/ support ( n ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ;
- ( a * sqrt ( 1 + b ^2 ) ) <= - ( - sqrt ( 1 + sqrt ( 1 + sqrt ( 1 + a ) ^2 ) ) ;
\varphi /. ( succ ( a , b ) ) = g . a & \varphi . ( a , b ) = f . b ;
assume that i in dom ( F ^ <* p *> ^ <* p *> ) and j in dom ( F ^ <* p *> ^ <* p *> ) ;
{ 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 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5
the Sorts of U1 /\ ( the Sorts of U1 ) c= the Sorts of U1 "\/" ( the Sorts of U2 ) ;
( - ( 2 * a ) * ( b + a ) ) + ( - 2 * a ) * ( b + a ) * ( b + a ) ) > 0 ;
consider W00 such that for z being element holds z in W iff z in W & not [ z , y ] ;
assume that ( the Arity of S ) . o = <* a *> and ( the Arity of S ) . o = r ;
Z = dom ( ( ( ( ( ( ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( 1 * ( arctan * ( arctan * ( 1 / ( 1 / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
integral ( f , S ) is convergent & integral ( f , S ) = integral ( f , S ) + integral ( g , S ) ;
( for a holds ( f . a ) => ( g . ( a , b ) ) ) => ( f . ( a , b ) ) => ( f . ( a , b ) ) ) in the carrier of \alpha ;
len ( M2 * ( M2 * ( M2 * ( M2 * ( M1 , n ) ) ) ) = n & width ( M2 * ( M1 * ( M1 * ( M1 , n ) ) ) = n ;
attr X1 \/ X2 is open means : where X1 , X2 , X1 , X2 , X1 , X2 , X2 , X1 , X2 , X1 , X2 , X2 , X1 , X2 , X1 , X2 , X2 , X1 , X2 , X2 , X1 , X2 , X1 , X2 is_collinear : X1 , X2 , X1 , X2 is_collinear & X1 , X2 , X1 , X2 is_collinear & X1 , X2 , X1 , X2 is_collinear & X1 , X2 , X2 is_collinear & X1 , X2 , X1 , X2
let L being antisymmetric transitive RelStr , X be Subset of L , Y being Subset of L , X being Subset of L st X = { "/\" ( X , L ) } holds X "/\" Y = X "/\" Y
reconsider f29 = ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c ) ) . ( ( F . ( b , c
consider w being FinSequence of M such that the carrier of M = { <* s *> *> ^ w ;
g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= g . ( a |^ 0 ) .= 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 ) ;
ex L being Subset of X st L = L & for x being Element of X st x in L holds L . x = 0. X ;
( the carrier of C1 ) /\ ( the carrier of C2 ) c= the carrier of C1 & the carrier of C1 c= the carrier of C2 & the carrier of C1 c= the carrier of C2 & the carrier of C2 c= the carrier of C1 & the carrier of C2 c= the carrier of C2 & the carrier of C1 = the carrier of C2 ;
reconsider oo = o `1 as Element of TS ( D ) * , the carrier of G ;
1 * ( x1 + x2 ) + ( 0 * x2 + 0 ) = x1 + ( 0 * x2 ) + ( 0 * x2 ) .= x1 + ( 0 * x2 ) + ( 0 * x2 ) * x2 .= x1 + ( 0 * x2 ) + 0 * x2 + 0 * x2 .= x1 + ( 0 * x2 ) + 0 * x2 ;
E " . 1 = ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= ( E qua Function ) . 1 ;
reconsider u1 = the carrier of U1 as Subset of ( the carrier of U1 ) + the carrier of U2 ;
( x "/\" z ) "\/" ( x "/\" z ) <= ( x "/\" z ) "\/" ( x "/\" z ) ;
|. f . ( s1 . ( l + 1 ) ) - f . ( s1 . ( l + 1 ) ) .| < sqrt ( 1 / ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
LSeg ( ( W-min C ) * ( i , j ) , ( W-min C ) * ( i + 1 , j ) ) is horizontal ;
( f | Z ) /. x0 - ( f | Z ) /. x0 = L * ( ( f | Z ) /. x0 - R /. x0 ) + R * ( f | Z ) /. x0 - R * ( f | Z ) /. x0 ;
g . c * ( 1- ( g . c ) ) <= h * ( - ( g . c ) ) + ( - g . c ) * f ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
attr ColVec2Mx ( f , g ) in Indices ( A , B ) & width ( A , B ) = width A & width ( B , C ) = width B ;
len ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - ( ( - ( - ( - ( - ( ( - ( - ( - ( - ( ( - ( ( - ( 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( - ( - ( - ( - ( - ( - ( 1 ) ) ) ) ) ) ) & ( len ( - (
let n being Nat , i be Element of NAT , n be Element of NAT , i be Element of NAT , j be Element of NAT , n be Element of NAT , i be Element of NAT , i be Element of NAT ;
pdiff1 ( f1 , 2 ) is partially differentiable of x0 , i & pdiff1 ( f2 , 2 ) . ( i + 1 ) = ( f1 + f2 ) . ( i + 1 ) ;
attr a <> 0 & b <> 0 & a <> 0 & b <> 0 & a <> 0 & b <> 0 & c <> 0 implies b * c + c * d + d * d + d * d + d * c + d * d + d * d + d * c + d * d + d * d + c * d + d * c + d * d + d * d + d * c + d * d + c * d + d + d * c + d * d + d * c +
for c being set st c in [. a , b .] holds not c in Intersection ( the InternalRel of G , a )
assume that V1 is linearly closed and ( for v st v in V1 holds ( ( for x st x in V1 holds ( x in V1 ) iff x in V1 ) & ( x in V1 ) & ( x in V1 ) & ( x in V1 ) & ( x in V1 ) & ( x in V1 ) & ( x in V1 ) & ( x in V1 ) & y in V1 ) ;
z * ( x1 + y1 ) * ( x1 + y1 ) + ( x1 + y1 ) * ( x2 + y2 ) * ( x1 + y2 ) + ( z + y2 ) * ( x2 + y2 ) * ( x1 + y2 ) + ( z + y2 ) * ( x2 + y2 ) * ( z + y2 ) * ( z + y2 ) ) + ( z + y2 ) * ( z + y2 ) * ( z + y2 ) ) * ( z + y2 + y2 + z + y2 ) * ( z + y2 +
rng ( ( P1 * ( i , j ) ) " ) = Seg ( len ( P1 * ( i , j ) ) ) .= Seg len ( P1 * ( i , j ) ) .= Seg len ( ( P1 * ( i , j ) ) " ) ;
consider s2 being Real such that s2 is convergent and b = ( lim s2 ) / 2 and s2 is convergent and for n st n <= m holds s2 . n = lim s2 ;
h2 " . n = h2 " . n & 0 < h2 . n & 0 < h2 . n & h . n = h2 . n ;
( Partial_Sums ( ||. seq .|| ) ) . m = ( Partial_Sums ( ||. seq .|| ) ) . m .= ( Partial_Sums ( ||. seq .|| ) ) . m .= ( Partial_Sums ( ||. seq .|| ) ) . m ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P1 , s1 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( Comput ( P2 , s2 , 1 ) ) . b .= ( P2 , s2 ) . b .= ( P2 , s2 ) . b ;
- v = ( - v ) * ( - v ) & - v = - ( - v ) * ( - v ) & - v = - ( - v ) * ( - v ) * ( - v ) * ( - v ) = - ( - v ) * ( - v ) * ( - v ) * ( - v ) ;
sup ( [: k , D :] .: ( k , D ) ) = sup ( k , D ) .= sup ( k , D ) .= ( k , D ) .: ( k , D ) .= ( k , D ) .: ( k , D ) ;
( A |^ k ) .. ( A |^ ( k + l ) ) = ( A |^ ( k + l ) ) * ( A |^ ( k + l ) ) ;
let R being add-associative right_zeroed right_complementable well-unital non empty addLoopStr , I , J , K , L be Subset of R , I , J , J , L be Subset of R , I , J be Subset of R , K , J be Subset of R , L , J be Subset of I , L be Subset of R , J , M being Subset of R , L , N being Subset of R , M being Subset of R , L , N being Subset of R , M being Subset of R , M being Subset of R ,
( f . p ) `1 = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) ;
let a , b , c , d , f , g , h , g , h , i , i , f , g , h , i , g , h , i , f , g , h , i , g , h be Element of INT.Ring ( a , b , f , g ) ;
consider A5 being Nat such that r is countable & rng r c= the carrier of ( Al ( ) ) ) & ( for n being Nat holds r is [: the carrier of Al ( ) , the carrier of Al ( ) :] ) & ( ex S being Subset of NAT st S = ( the carrier of Al ( ) ) ) & ( ex r being Element of NAT st r is [: the carrier of Al ( ) , NAT :] ) ;
for X being non empty addLoopStr for M being Subset of X , x , y being Point of X st x + y in M holds x + y in M + M
{ [ x1 , y1 ] } , [ x1 , y2 ] } c= { x1 , x2 } ;
h . O = [ A * ( f . O ) + B * ( f . O ) + C * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D * ( f . O ) + D + D * ( f
( Gauge ( C , n ) * ( k , i ) ) * ( k , i ) in L~ Cage ( C , n ) ;
cluster m gcd n -> non zero ;
( f * F ) . x1 = f . x1 & ( f * F ) . x2 = f . x2 ;
let L being LATTICE , a , b , c , d being Element of L st a \ b = c & b \ d = c holds a \ b <= c
consider b being element such that b in dom ( H _ { ( { x } \leftarrow y ) } ) and z = ( H _ { ( { x } \leftarrow y ) } ) . b ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , W & W . 1 = W . ( 3 + 1 ) & W . ( 3 + 1 ) in G ;
( ( ( Shift ( f , h ) * ( 2 * n ) ) ) . x = ( ( Shift ( f , h ) ) * ( 2 * n ) ) . x ;
j + 1 = j + ( len h1 + 2 ) .= i + ( len h1 + 2 ) .= i + ( len h2 + 2 ) .= i + ( len h2 + 2 ) ;
S *' ( S *' ) = S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) .= S *' ( S *' ) ;
consider H such that H is one-to-one and rng ( L (#) ( L (#) ( L (#) ( L (#) ( ( L (#) ( ( L (#) ( ( L (#) ( ( L ) ) ) * ( ( ( L ) ) * ( ( ( L ) ) * ( ( ( ( L ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) )
attr R is ld sorder means : Def5 : for p , q st p in R & q <> q & p <> q & p <> q & p <> q & q <> r & p <> r & r <> s & s in P & p <> s & p <> s & q <> s & p <> s & p <> s & p <> s & p <> s & s <> q & p <> q & p <> s & p <> s & q <> s & p <> s & p
dom ( ( product ( X --> f ) ) | ( X --> f ) ) = meet ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom f /\ ( X --> f ) .= dom f /\ ( X --> f ) .= dom f /\ ( X --> f ) ;
sup ( ( proj2 .: ( LSeg ( w , e ) ) /\ LSeg ( w , e ) ) ) <= sup ( ( .: ( LSeg ( w , e ) ) /\ LSeg ( w , e ) ) ) ;
for r be Real st 0 < r ex m be Nat st for n be Nat st n <= m holds |. S . n - ( S . m ) - ( S . n ) .| < r
i * f9 = i * ( ( ( i - j ) * ( i - j ) ) * ( i - j ) ) .= i * ( ( i - j ) * ( i - j ) ) * ( i - j ) ) .= i * ( ( i - j ) * ( i - j ) ) * ( i - j ) ;
consider f being Function such that dom f = 2 -tuples_on X and for x being element st x in 2 -tuples_on X holds f . x = F ( x ) ;
consider g1 , g2 being element such that g1 in [#] ( Y ) and g2 in ( Y | ( X ) ) and [ g1 , g2 ] in the InternalRel of Y ;
redefine func d \! ( n ) -> Nat means : Def5 : for k being Nat st k in Seg n holds it . k = a & it . k = b ;
fj . [ 0 , t . ( 0 , t ) ] = f . [ 0 , t . ( 0 , t ) ] .= a .= a ;
t = h . D or t = h . B or t = h . C & t = h . D & t = h . E & t = h . J & h = h . J & t = h . J & h = h . J & t = h . M & h = h . J & h = h . M & h = h . M & h = h . J & h = h . M & h = h . J & h = h . M & h = h . J & h =
consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( ( vseq . n ) - ( vseq . n ) ) ) < 1 / 2 ;
sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) <= sqrt ( ( q `1 ) ^2 + ( q `2 ) ^2 ) ;
h2 . ( i + 1 ) = h2 . ( i + 1 ) .= h2 . ( i + 1 ) ;
consider o being Element of the carrier' of S such that a = [ o , the carrier' of S ] and a = [ o , the carrier' of S ] ;
let L being transitive RelStr , a , b be Element of L , c be Element of L , a , b be Element of L st a <= b & b <= c holds a <= b & b <= c & c <= a & b <= c & a <= c & b <= c & c <= a & a <= b & b <= c & c <= c & a <= c & b <= c & c <= d & d <= d ;
||. h1 .|| . n = ||. h1 . n - h .|| .= ||. h1 . n - h .|| .= ||. h1 . n - h .|| ;
( ( - ( ( ( ( the function of exp ) * ( ( #Z ) ) * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) ) ) ) . x = ( ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) . x .= ( ( ( #Z n ) * ( ( #Z n ) ) ) . x ) .= ( ( ( - n ) * ( ( - n
attr r = F .: ( p , q ) , r = F .: ( p , q ) , s = F .: ( p , q ) ;
sqrt ( r / 2 ) + sqrt ( 2 * ( r / 2 ) + sqrt 2 ) <= sqrt ( 2 * ( r / 2 ) + sqrt 2 ) + sqrt 2 ;
let i being Nat , M be Matrix of n , K , n , K , a , b , M be Nat , M be Matrix of n , K , K , a , b , M , a , b , M , Nat st len ( M * ( i , n ) ) = n & ( ( M1 * ( i , n ) ) * ( i , n ) = ( ( M1 * ( i , n ) ) * ( i , n ) ) * ( i , n ) ) * ( i , n ) ) * ( i , n ) ) * ( i , n ) * ( i , n ) ) * ( i , n , n , n ) *
then a <> 0. R & a * ( a * b ) = 1 * a * b & a * b = 1 * b * b ;
p . ( j -' 1 ) * ( q /. ( i + 1 ) ) = Sum ( p | ( j + 1 ) ) * ( q /. ( j + 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( R /* ( h + c ) ) * ( R /* ( h + c ) ) ;
assume that the carrier of ( H ) = f .: the carrier of ( H ) & the carrier of ( H ) c= f .: the carrier of ( H ) ;
Args ( o , Free ( S , X ) ) = ( ( the Sorts of Free ( S , X ) ) * the Arity of S ) . o ;
H1 = n + 1 + ( 2 to_power ( n + 1 ) ) .= n + ( 2 to_power ( n + 1 ) ) .= n + ( 2 to_power ( n + 1 ) ) .= n + ( 2 to_power ( n + 1 ) ) ;
( O * ( O * ( O , 1 ) ) ) `1 = 0 & ( O * ( O , 1 ) ) `1 = 0 & ( O * ( O , 1 ) ) `1 = 0 & ( O * ( O , 1 ) ) `1 = 0 & ( O * ( O , 1 ) ) `1 = 0 & ( O * ( O , 1 ) ) `1 = 0 ;
F1 .: ( dom ( F1 | ( dom F1 ) ) ) = ( F1 | ( dom F1 ) ) | ( dom F1 /\ ( dom F1 ) ) .= { f . ( f . ( F1 . ( len F1 ) ) ) } ;
attr b <> 0 & d <> 0 & b <> 0 implies a = b & b = sqrt ( b + c ) & sqrt ( a + c ) = sqrt ( b + c ) ;
dom ( f +* g ) = dom ( f +* g ) .= dom ( f +* g ) \/ dom g .= dom f \/ dom g .= dom f \/ dom g .= dom g \/ dom g .= dom f \/ dom g .= dom f \/ dom g .= dom f \/ dom g ;
for i being set st i in dom g ex u being Element of L st u = g * u & g /. i = a * u + a * v ;
g * P = g * P * P .= g * P * P * P * Q .= g * P * P * Q .= g * P * Q * Q .= g * P * Q * Q * Q * Q * P * Q .= g * P * Q * Q .= g * P * Q * Q .= g * P * Q * Q .= g * P * Q * Q .= g * Q * Q .= P * Q * Q * Q .= P * Q * Q * Q * Q * Q * Q * Q .= P * Q * Q * Q * Q
consider i , s1 such that f . i = s1 & not ( f . i = s2 & not f . i = s1 & ( f . i = s2 ) & not f . i = s2 & f . i = s2 & f . i = s2 & f . i = s2 & f . i = s2 ;
h5 | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , s2 ] in R & [ s2 , s2 ] in R implies [ s2 , s2 ] in R & [ s2 , s2 ] in R & [ s2 , s2 ] in R & [ s2 , s2 ] in R implies [ s2 , s2 ] in R & [ s2 , s2 ] in R & [ s2 , s2 ] in R & [ s2 , s2 ] in R implies [ s2 , s2 ] in R & [ s2 , s2 ] in R implies [ s2 , s2 ] in R & [ s2 , s2 ] in R implies [ s2 , s2 ] in R & [ s2 , s2 ] in
then H is negative & H is not negative & H is not negative ;
attr f1 is total means : Def5 : 1 / ( 2 * ( 2 * ( 1 / 2 ) ) ) & ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) = ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ;
z1 in W2 ` & z2 in W2 ` & z in W1 & z in W2 & ( z in W2 ) & ( z in ( W1 + W2 ) + ( z + z ) ) & ( z in ( W1 + W2 ) + ( z + z ) ) & ( z in ( W1 + W2 ) + ( z + z ) ) & ( z in ( W1 + W2 ) + ( z + z ) ) & ( z in ( W1 + W2 ) + ( z in W2 ) & ( z in W2 ) & ( z in W2 ) & ( z in W2 ) & ( z in W2 )
p = 1 * p .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) .= a * ( b * p ) ;
for rseq be sequence of real numbers for n be Nat st for k be Nat st k in dom seq holds seq . k <= K . k holds upper_bound rng seq <= K * ( k + 1 )
the_arity_of ( E-max C ) meets ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/ ( E-max C ) \/
||. ( f . ( k + 1 ) - g ) . ( k + 1 ) .|| <= ||. ( g . ( k + 1 ) - g . ( k + 1 ) - g .|| ;
assume h = ( B .--> B ) +* ( C .--> D ) +* ( D .--> E ) ;
|. ( ( the lower of T ) . n ) - ( ( the _ of T ) . n ) - ( ( the _ of T ) . n ) .| <= e * ( ( the _ of T ) . n ) ;
( is_{ \rm qua Element of S ) . v = [ the <* *> *> , the carrier of S ] -tree ( the Arity of S ) . v .= [ <* the carrier of S *> , the carrier of S ] -tree ( the Arity of S ) . v ;
{ 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 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/ x5 \/
assume that A = [. 0 , PI / 2 .] and \int ( ( ( ( ( #Z n ) * ( ( #Z n ) * ( #Z n ) ) ) ) `| Z ) = 0 ;
p `1 is Permutation of dom ( ( Sgm Y ) * ( i , j ) ) & p `2 = ( Sgm Y ) * ( i , j ) `2 ;
for x , y st x in A holds |. ( ( 1 / 2 ) (#) ( f + g ) ) . x - ( 1 / 2 ) * ( f + g ) . y ) .| <= 1 * |. ( f + g ) . x - ( 1 / 2 ) * ( f + g ) . y .|
( ( p2 `1 ) * ( ( p2 `1 ) * ( - ( - ( ( - ( q `1 ) ) ) ) ) ) ) = |. ( ( - ( q `1 ) ) ) ) * ( ( - ( q `1 ) ) ) ) * ( - ( q `2 ) * ( - ( q `2 ) ) ) ;
let f being PartFunc of the carrier of C , the carrier of C , the carrier of C ;
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( 'not' ( a , CompF ( B , G ) ) ) . x = TRUE ;
consider FF being FinSequence of REAL such that dom ( F | n1 ) = n1 & for k being Nat st k in dom F holds Q [ k , F . k ] ;
ex u , u1 st u <> u1 & u1 <> u2 & u , u1 // u1 , v1 & u1 , v1 // u1 , v1 & u2 , u1 // u1 , v1 & u1 , v1 // v1 , v2 implies u , u1 // u1 , v1 or u , v // v1 , v2 or u , v1 // u1 , v2 or u , u1 // v1 , v2 or u , v1 // v1 , v2 or u , v2 // v1 , v2 or u , v1 // v2 , v2 , v1 , v2 or u , v2 , u1 , u1 , v2 , v1 , v2 // v1 , v2 or u , v2
let G being Group , A , B being Subset of G , N being normal of G , B being normal Subgroup of N st N = N * ( N , B ) holds ( N , B ) * ( N , B ) = N * ( N , B ) * ( N , B ) ;
for s be Real st s in dom F holds F . s = \int ( ( R , f ) . ( s + 1 ) ) dx ( M , ( f , g ) . x )
width ( ( ( ( ( ( ( ( ( 1 ) ) * ( len ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( ( ( ( ( ( ( ( ( ( ( ( 1 ) ) * ( len ( ( ( ( ( ( ( 1 ) ) ) ) ) ) ) ) ) ) ) ) .= len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 ) ) * ( len ( (
f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ & f | ]. - PI / 2 , PI / 2 .[ = f | ]. - PI / 2 , PI / 2 .[ ;
assume that X is closed and a in X and for x , y st x in X & y in { n } holds a in X & x in { n } ;
Z = dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ;
redefine func still_not-bound_in ( l ) -> Subset of V equals { l . k : 1 <= l & l <= len l & l . k = l . k } ;
let L be non empty reflexive transitive RelStr , M be Subset of L , N be net of L , M , x being Element of L st x in N & M is N & x in M holds x in M & y in M & x in M & y in M & x in M & y in M & x in M & y in M & x in M & y in M & x in M & y in M
for s being Element of NAT holds ( ( for v being Element of NAT holds ( ( the _ of ( the carrier of G ) ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) . ( v , ( the carrier of G ) )
then z = W-min ( L~ z ) .. z + ( W-min ( L~ z ) ) .. z + ( W-min ( L~ z ) ) .. z + 1 ) .. z < ( W-min L~ z ) .. z + ( W-min ( L~ z ) ) .. z + ( W-min ( L~ z ) ) .. z ;
len ( p ^ <* 0 qua Nat *> ) = len p + len ( p ^ <* 0 qua Nat *> ) .= len p + 1 ;
assume that Z c= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) and and for x st x in Z holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
let R being add-associative right_zeroed right_complementable distributive non empty doubleLoopStr , I being Subset of R , J being Subset of R , I being Subset of R , J being Subset of R st I c= J holds ( I + J ) *' ( I + J ) c= I + J
consider f being Function of B1 , B2 such that for x being Element of B1 holds f . x = F ( x ) ;
dom ( x2 + y2 ) = Seg len ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( ( x2 + y2 ) ^ y2 ) .= dom ( ( ( x2 + y2 ) ^ y2 ) ^ y2 ) .= dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( len ( x ^ y2 ) ^ y2 ) ^ y2 ) ^ y2 ) ) ^ y2 ) ^ y2 ) ^ y2 ) ^ y2 ) ^ y2 ) ^ y2 ) ^ y2 ) ^
for S being category , B being object of C , c being object of C holds ( the Arrows of B ) . ( c , b ) = id ( the carrier' of C )
ex a st a = a2 & a in dom ( f | ( dom ( f | X ) ) ) & for a , b st a in dom ( f | X ) & b in dom ( f | X ) holds f . ( a , b ) = f . ( a , b ) ;
a in Free ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( \ \ ) ) ) ) ) ) ) | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( \ \ ) ) ) ) | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( \ \ ) ) ) ) ) ) ) ) ) ) ) | ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
let C1 , C2 be non empty set , f , g be Function of C1 , C2 holds C1 = C2 & C2 = C2 implies C1 = C2 & C2 = C2 & C1 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 implies C1 = C2 & C2 = C2 & C2 = C2 implies C1 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 & C2 = C2 &
( W-min L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) = ( W-min L~ Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
assume that u = <* x0 , y0 , z0 , z0 *> and f u is_differentiable_in x0 and f . 3 = y0 ;
then ( t . {} ) `1 = ( t . {} ) `1 & ( t . {} ) `2 = ( t . {} ) `1 ;
Valid ( p '&' J , J ) . v = Valid ( p , J ) . v .= J . v ;
assume for x , y being Element of S st x <= y & y = f . x & x >= y & y >= f . y ;
redefine func Classes R -> Subset of R means : Def5 : for a being Element of R st a = it holds it . a = a ;
defpred P [ Nat ] means ( the Target of G ) . ( the Element of G ) c= ( the Target of G ) . ( the Element of G ) ;
assume that dim ( W1 ) = 0 and dim ( W2 ) = 0 & dim ( W1 ) = 0 & dim ( W2 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W2 ) = 0 implies dim ( W1 ) = 0 & dim ( W1 ) = 0 & dim ( W1 ) = 0 ;
mam in LSeg ( m , t ) \/ LSeg ( m , t ) .= { m , t } ;
d11 = ( x9 ^ <* y *> ^ ( x ^ y ) ) ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( x ^ y ) ) .= f ^ ( y ^ ( y ^ (
consider g such that x = g and dom g = dom ( f | X ) and for x being element st x in dom ( f | X ) holds g . x = f . x ;
x + ( len x ) = x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) ;
k9 -' ( len ( f /^ ( k -' 1 ) ) + 1 ) in dom ( f /^ ( k -' 1 ) ) ;
assume that P1 is walk of p1 , p2 and P2 = { p1 , p2 } and P = { p1 , p2 } and P = { p1 , p2 } and P = { p1 , p2 } and Q = { p1 , p2 } and P = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p1 , p2 } and Q = { p1 , p2 , p3 , p3 = { p1 , p2 , p3 , p3 and Q = { p1 , p3 , p3 , p3 , p3 ,
reconsider a1 = a , b1 = b , c1 = c , c2 = d , c2 = b , c2 = c , c2 = d , c2 = d , c2 = b , c2 = d , c1 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , product = d , product = d , product c = d , c2 = d , product c = d , product 8 = d , product 8 = d , product 8 = d , product
reconsider FFFf = ( t * f ) . ( t * f ) as Morphism of ( ( G * f ) . ( t * f ) ) , ( G * f ) . ( t * f ) . ( t * f ) . ( t * f ) . ( t * f ) . ( t * f ) . ( t * f ) . ( t * f ) ;
LSeg ( f , i + 1 ) = LSeg ( f , i + 1 ) .= LSeg ( f , i ) ;
\int P . m , ( P . n ) | dom ( P . m ) | E , ( P . n ) | E ) = \int ( P . n , ( P . m ) | E ) ;
assume that dom ( f1 * f2 ) = dom ( f1 * f2 ) and for x being element st x in dom f1 holds f1 . x = f2 . x ;
consider v such that v = y and dist ( u , v ) < r and dist ( u , v ) < r / 2 ;
let G being Group , a , b , c being Element of G , a , b , c being Element of G , b , d being Element of G , i being Element of G , i being Element of dom ( a , b , c ) holds ( a , b , c , d ) . i = ( a , b , c ) . i
consider B being Function of Seg ( k + L ) , the carrier of V1 such that for x being element st x in Seg ( k + L ) holds P [ x , B . x , B . x ] ;
reconsider K = { p where p is Point of TOP-REAL 2 : p in P & p `1 <= 0 } as Subset of TOP-REAL 2 ;
sqrt ( ( Gauge ( C , n ) ) ^2 + ( Gauge ( C , n ) ) ^2 ) <= sqrt ( ( Gauge ( C , n ) ) ^2 + ( ( Gauge ( C , n ) ) ^2 ) ;
for x being Element of X , n being Nat st x in E holds |. ( Im ( F . n ) ) . x - ( Im ( F . n ) ) . x .| <= P & |. ( Im ( F . n ) ) . x - ( Im ( F . n ) ) . x .| <= P
len ( @ @ @ @ ( <* 2 *> ^ q ) ) = len ( @ @ ( 2 ^ q ) ) + len ( @ @ ( 2 ^ q ) ) .= len ( @ @ ( 2 ^ q ) ) + len ( @ @ ( 2 ^ q ) ) .= len ( @ ( 2 ^ q ) ) + len ( ( 2 ^ q ) ) + len ( 2 ^ q ) ;
v / ( ( x , m ) / ( x , m ) ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) ) = ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / ( x , m ) / (
consider r being Element of M such that M , ( ( the _ of M ) | ( ( the carrier of M ) | ( the carrier of M ) ) ) |= ( ( the _ of M ) | ( the carrier of M ) ) ;
redefine func w \ ( the Element of G ) -> Element of product ( G \ the carrier of G ) , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G , the carrier of G means : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
s2 . b2 = ( s2 | ( n + 1 ) ) . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 . b2 .= s2 .
for n be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) ) . n + ( Partial_Sums ( |. seq .| ) ) . n
set F = S \! \mathop { 0 } ;
( Partial_Sums ( seq ) ) . n + ( Partial_Sums ( seq ) ) . n >= ( Partial_Sums ( seq ) ) . n + ( Partial_Sums ( seq ) ) . n + ( Partial_Sums ( seq ) ) . n ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x - ( f | Z ) . x = L . x - R . x ;
the closed of TOP-REAL 2 = ( TOP-REAL 2 ) | LSeg ( a , b ) \/ LSeg ( b , c ) ;
a * b + b * c + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) ) >= - a * ( a * b ) + ( a * c ) * ( a * c ) + ( a * c ) + ( a * b ) * ( a * c ) + ( a * b ) * ( a * c ) * ( a * b ) * ( a * c ) + ( a * c )
v / ( x1 , m1 ) / ( x1 , m1 ) / ( x2 , m1 ) = v / ( x1 , m1 ) / ( x2 , m1 ) / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x1 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) = v / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 ) / ( x2 , m1 )
Rotate ( Q ^ <* x *> , M ^ <* x *> ) = ( M ^ <* x *> ^ ( M ^ <* x *> ^ ( M ^ <* x *> ^ ( M ^ <* x *> ^ ( M ^ <* x *> ^ ( M ^ <* x *> ^ ( M ^ ( M ^ <* x *> ) ) ) ) ) ;
Sum ( ( F |^ ( n + 1 ) ) * ( F |^ ( n + 1 ) ) ) = ( F |^ ( n + 1 ) ) * ( F |^ ( n + 1 ) ) .= ( F |^ ( n + 1 ) ) * ( F |^ ( n + 1 ) ) .= ( F |^ ( n + 1 ) ) * ( F |^ ( n + 1 ) ) .= ( F |^ ( n + 1 ) ) * ( F |^ ( n + 1 ) ;
( ( GoB f ) * ( len \alpha , 1 ) ) `1 = ( GoB f ) * ( 1 , 1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * b ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) * ( $1 + 1 ) + b * ( $1 + 1 ) = ( a + 1 ) * ( $1 + 1 ) ;
the_arity_of g = ( the Arity of S ) . ( ( the Arity of S ) . ( g . o ) ) .= ( the Arity of S ) . ( g . o ) ;
( X \times Y ) \/ ( X \/ Y ) c= X implies card ( X \/ Y ) = card ( X \/ Y ) & card ( X \/ Y ) = card ( X \/ Y ) & card ( X \/ Y ) = card ( X \/ Y ) & card ( X \/ Y ) = card ( X \/ Y ) & card ( X \/ Y ) = card ( X \/ Y ) ;
for a , b being Element of S , s being Element of NAT st s = F . n & b = F . n holds b = G ( n ) & a = G ( n ) & b = G ( n ) & b = G ( n ) & c = G ( n ) & ( n <= n implies b = G ( n ) ) & ( n <= n implies n <= n implies n <= n ) implies n <= n & n <= n & n <= n implies n <= n & n <= n & n <= n & n <= n & n <= n implies n <= n & n <= n & n <= n & n <= n & n <= n & n <= n & n <= n & n <= n
E , f |= ( ( ( ( ( x , y ) , x ) ) to_power ( ( ( x , y ) to_power ( ( x , y ) to_power ( ( x , y ) to_power ( x , y ) ) ) ) ) ) ) ) ) ;
ex R2 being RelStr st R = ( p | ( Seg ( n + 1 ) ) ) & ( the carrier of R ) \ ( the carrier of R ) = the carrier of R & ( the carrier of R ) \ ( the carrier of R ) = the carrier of R ;
[. a , b + sqrt ( 1 + ( k + 1 ) ) .] is Element of the metric of S & ( ( the partial of G ) . ( k + 1 ) ) . ( k + 1 ) is Element of the carrier of S ;
Comput ( P , s , 2 + 1 ) = Exec ( ( P , s ) . 2 ) .= Exec ( ( P , s ) . 2 ) .= Exec ( ( ( P , s ) . 2 , s ) . 2 .= Exec ( ( P , s ) . 2 , s ) . 2 .= Exec ( ( P , s ) . 2 , s ) . 2 ;
card h1 . k = ( 1_ ( K , n ) ) * ( k , i ) .= ( 1_ ( K , n ) ) * ( k , i ) .= ( ( ( ( ( ( ( ( ( ( ( n , n ) , n ) ) * ( k , i ) ) ) * ( k , i ) ) ) * ( k , i ) ) .= ( ( ( ( n , n ) --> n ) * ( k , i ) ) * ( k , i ) ) * ( k , i ) ) * ( k , i ) ) * ( k , i ) ) * ( k , i ) ) * ( k , i ) * ( k , i ) * ( k , i ) ) * (
sqrt ( ( f /. c ) * ( g /. c ) ) = f /. ( c ) * ( g /. c ) .= ( f /. c ) * ( g /. c ) * ( g /. c ) ) .= ( f /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) ) .= ( f /. c ) * ( g /. c ) * ( g /. c ) ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) * ( g /. c ) .= ( f /. c )
len ( ( the connectives of C ) * ( len the connectives of C ) ) = len ( the connectives of C ) & len ( the connectives of C ) = len ( the connectives of C ) & ( the connectives of C ) * ( the connectives of C ) = ( the connectives of C ) * ( the connectives of C ) ;
dom ( ( r (#) f ) | X ) = dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X .= dom ( r (#) f ) /\ X ;
defpred P [ Nat ] means for n st 2 * n + 2 * $1 + 2 * n ) = n * ( n + 1 ) + 2 * ( n + 1 ) * n + 2 * n + 2 * n ) ;
consider f being Function of [: { 0 } , { 1 } :] , { 0 } :] , { 1 } such that f = f and f is one-to-one and f is one-to-one and f is one-to-one and f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & g = g implies f is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & g is one-to-one & f is one-to-one & f is one-to-one & f is one-to-one & g is one-to-one & f is
consider c9 being Function of S , BOOLEAN such that c9 = ( A \/ B ) \ ( A \/ B ) and \ { c9 } = { Prob ( A , B ) , D , E , F , G , G , G , G , G , G ) ;
consider y being Element of [: Y , X :] such that a = "\/" ( { F . ( x , y ) } , L ) and y in the carrier of L and Q [ x , y ] ;
assume that A c= dom f and f = ( ( for x st x in Z holds f . x ) * ( ( \HM { the } \HM { function } ) ^ ) ) `| Z ) ;
( f /. ( i + 1 ) ) `1 = ( ( GoB f ) * ( 1 , j ) ) `1 .= ( ( GoB f ) * ( 1 , j ) ) `1 .= ( GoB f ) * ( 1 , j ) ) `1 .= ( GoB f ) * ( 1 , j ) `1 ;
dom Shift ( q1 , len ( q2 ^ q2 ) ) = { j + len ( q1 ^ q2 ) where j is Nat : j in dom ( q2 ^ q2 ) & len ( q2 ^ q2 ) = len ( q2 ^ q2 ) + j & j in dom ( q2 ^ q2 ) } ;
consider G1 , G2 being Element of V such that G1 <= G1 & G2 <= G2 & G1 <= G2 & G2 <= G2 & G1 is Subset of G2 & G1 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G1 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G1 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G1 is Subset of G2 & G1 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2 is Subset of G2 & G2
redefine func - f -> PartFunc of C , COMPLEX means : Def5 : for c being element st c in dom it holds it . c = - f . c ;
consider \varphi such that \varphi is increasing and for a st \varphi . a = a & for b st b in dom L holds L . b = H ( b ) ;
consider i1 , i2 such that [ i1 , i2 ] in Indices GoB f and [ i1 , i2 ] in Indices GoB f and f /. ( i1 + 1 ) = G * ( i1 , i2 ) ;
consider i , j such that n <> 0 and n = i and j = j + 1 and i <> 0 implies i = j + 1 & j = i + 1 & j = j + 1 & i <> 0 implies i = j + 1 & j = j + 1 & j = k + 1 & i = k + 1 & j = k + 1 & j = k + 1 implies i = j + 1 & j = k + 1 & j = k + 1 implies i = k + 1 & j = k + 1 & j = k + 1 & i = k + 1 & j = k + 1 & j = k + 1 implies i = k + 1 & j = k + 1 & j = k + 1 implies i = k + 1 & j = k + 1
assume that not 0 in Z and for x st x in Z holds ( ( ( ( id Z ) ^ ) (#) ( ( id Z ) ^ ) ) `| Z ) = ( - 1 ) (#) ( ( id Z ) ^ ) ) & for x st x in Z holds ( ( ( id Z ) ^ ) `| Z ) . x > - 1 & ( ( id Z ) ^ ) . x > 0 ;
cell ( G1 , i1 -' 1 , j2 ) \ ( i1 -' 1 ) c= BDD L~ f & ( for i2 st i2 < len f & i2 < len f holds ( ( f | i2 ) | ( i2 -' 1 ) ) \ ( i2 -' 1 ) ) \ ( ( f | i2 ) \ ( i2 -' 1 ) ) \ ( i2 -' 1 ) ) c= ( L~ f ) \ ( i2 -' 1 ) \ ( i2 -' 1 ) ) ;
ex G1 being Subset of X st s = G1 & G1 is open & for G1 being Subset of X st G1 is open & G1 is open holds G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open & G1 is open ;
gcd ( ( the InternalRel of A ) . ( ( the carrier of A ) . ( len ( the InternalRel of A ) , ( the carrier of A ) . ( len ( the InternalRel of A ) , ( the InternalRel of A ) . ( len ( the InternalRel of A ) , ( the carrier of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) ) ) ) , ( the carrier of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) ) ) ) ) = 1 / ( len ( the InternalRel of A ) . ( len ( the InternalRel of A ) )
R1 = ( the \mathopen { - } ( s2 ) ) . ( m + 1 ) .= [ ( the InternalRel of ( s2 ) ) . ( m + 1 ) , ( the InternalRel of s2 ) . ( m + 1 ) ] .= [ 3 , ( the InternalRel of s2 ) . ( m + 1 ) ] ;
CurInstr ( P3 , Comput ( P3 , s3 , 1 + 1 ) ) = halt SCMPDS .= halt SCMPDS .= ( halt SCMPDS ) + IC SCMPDS .= IC SCMPDS .= IC SCMPDS + ( IC SCMPDS ) .= IC SCMPDS + ( IC SCMPDS ) + 1 ) .= IC SCMPDS + ( IC SCMPDS + 1 ) ;
P1 /\ P2 = ( { p1 , p2 } \/ LSeg ( p1 , p2 ) ) \/ ( { p1 , p2 ) /\ LSeg ( p1 , p2 ) ) \/ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) \/ LSeg ( p2 , p1 ) /\ LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) ) = { p1 , p2 } ;
func not bound ( f ) -> Subset of the carrier of Al ( ) means : where p is Element of the carrier of Al ( ) : p in the carrier of it } ;
let a , b be Element of COMPLEX ( ) , f , g , h be Function of COMPLEX ( ) , COMPLEX ( ) , h ) , h , i , i , f be Element of COMPLEX ( ) , g , h ) ;
defpred P [ Nat ] means for i st 1 <= i & i <= len g & 1 <= i & i < len g & g /. i = g /. ( i + 1 ) holds i = j & j < len g implies i < len g & j < len g & i < len g & i < len g implies i < len g & j < len g & i < len g implies i < len g & i < len g & i < len g implies i < len g & i < len g & i < len h & i < len h & j < len h & i < len h & j < len h & i < len h implies i < len h & i < len h & j < len h & j < len h & i < len h & j < len h &
assume that C1 , C2 , f , g is_collinear and g , h , h is_collinear and h = h * f & h = g * f & g = h * f & h = h * f & f = g * f , h * g ) ;
( ||. f .|| ) | X = ||. ( f | X ) . x - f /. x .|| .= ||. ( f | X ) . x - f /. x .|| ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) + ( q `2 ) ^2 + ( q `2 ) ^2 ) + ( q `2 ) ^2 + ( q `2 ) ^2 ) ;
for F being Subset-Family of T st F is open & for A being Subset of T st A in F holds A is open & A is open & A is open & A is open holds A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & A is open & Cl ( A c= Cl Cl A c= Cl Cl A is open & A is open & Cl ( A c= Cl Cl A ) & Cl Cl ( A c= Cl A ) & Cl ( A c= Cl Cl A ) implies Cl A is open & Cl ( A c= Cl A ) ) implies Cl A is open & Cl ( A c= Cl A
assume that len F >= 1 and len F = k + 1 and for k st k in dom F holds F . k = G ( k ) ;
i |^ ( Radix ( n ) ) * ( i |^ ( n ) ) = ( i |^ ( n ) ) * ( i |^ ( n ) ) .= i |^ ( n ) * ( i |^ ( n ) ) .= i |^ ( n ) * ( i |^ ( n ) ) .= i |^ ( n ) * ( i |^ ( n ) ) ;
consider q being OChain of G such that r = q and len q = len ( p ^ q ) and len q = len q + 1 and len q = len q + 1 and len q = len q + 1 ;
defpred P [ Element of NAT ] means for g st g <= $1 holds ( ( g , Z ) . $1 ) . $1 = ( ( g , Z ) . $1 ) . $1 + ( g , Z ) . $1 ) . $1 = ( ( g , Z ) . $1 + ( g , Z ) . $1 ) . $1 ;
let A being square Matrix of n , REAL , K , n , k , n , Nat , a , b , c , d be Nat , b , c , d be Nat st len ( A * B ) = n & len ( A * B ) = n & len ( A * B ) = n & len ( A * B ) = n & len ( A * B ) = n ;
consider s being FinSequence of the carrier of R such that Sum s = u * s and for i being Element of NAT st i <= n holds s . i = a * s . i ;
redefine func | ( x , y ) -> Element of REAL equals ( x | ( x , y ) ) | ( x , y ) ;
consider g being FinSequence of ( the carrier of G1 ) | the carrier of G2 such that g = ( the carrier of G1 ) \/ the carrier of G2 and g is continuous and rng g = the carrier of G2 ;
then n1 >= len ( p1 ^ p2 ) & n2 = len ( p1 ^ p2 ) + n2 ;
( - ( q `1 ) * a ) * ( - ( q `1 / q `2 ) ) <= - ( - ( q `1 / q `1 ) ) * ( - ( q `2 / q `1 ) ) & ( - ( q `2 / q `1 ) * ( - ( q `2 / q `1 ) ) * ( - ( q `1 / q `1 ) ) ) * ( - ( q `2 / q `1 ) * ( - ( q `2 / q `1 ) ) ) ) = - ( - ( q `2 / q `1 ) * ( - ( q `2 / q `1 ) ) & ( - ( q `2 / q `1 / q `1 ) ) ^2 ) ;
( F . ( len ( ( p ^ <* ( p ^ q ) ) *> ^ ( F ^ q ) ) ) . ( len ( p ^ q ) + len ( F ^ q ) ) .= ( F ^ q ) . ( len ( p ^ q ) + len q ) .= ( F . ( len p + len q ) + len q ) ;
consider k1 being Nat such that k1 + k = 1 and a = ( <* a *> ^ ( ( a .--> k1 ) | ( k + 1 ) ) ) . ( k + 1 ) ;
consider BB being Subset of ( B1 | ( the carrier of B1 ) ) , the carrier of ( ( B1 | ( the carrier of B1 ) ) | the carrier of B1 ) such that ( B1 | ( the carrier of B1 ) ) = d and ( B1 | ( the carrier of B1 ) ) . ( d , e ) = d ;
v2 . b2 = ( ( the _ of B ) . ( ( the _ of B ) . ( ( the _ of B ) . ( the Arity of B ) . ( the Arity of B ) ) ) . ( ( the Arity of B ) . ( the Arity of B ) . ( the Arity of B ) . ( the Arity of B ) . ( the Arity of B ) ) .= ( ( the _ of B ) . ( the Arity of B ) ) . ( ( the Arity of B ) . ( ( the Arity of B ) . ( ( the Arity of B ) . ( the Arity of B ) ) . ( ( the Arity of B ) . ( the Arity of B ) . ( the Arity of B ) ) .= ( ( the Arity of B ) . ( the Arity of B ) ) . ( the Arity of B ) ) . ( ( the Arity of B )
dom IExec ( I , P , s ) = dom Start-At ( ( ( card I + 2 ) + 2 , SCMPDS ) .= dom ( card I + 2 ) \/ dom Start-At ( card I + 2 , SCMPDS ) .= dom ( card I + 2 ) \/ dom Start-At ( card I + 2 , SCMPDS ) .= dom ( card I + 2 ) \/ dom Start-At ( card I + 2 , SCMPDS ) .= ( card I + 2 , SCMPDS ) \/ dom SCMPDS + card I ;
ex dh be Real st h > 0 & |. h .| < d & |. h .| < d & |. h .| . h - R /. h - R /. h .| < r ;
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) \/ LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int cell ( G * ( len G , 1 ) , G * ( len G , 1 ) ) \/ LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) ;
LSeg ( mid ( h , i1 , i2 -' 1 ) , i2 ) = LSeg ( h , i2 ) .= LSeg ( h , i2 ) ;
A = { q where q is Point of TOP-REAL 2 : LE q `1 , P & q `1 <= 0 } ;
( - x ) | ( ( - x ) | ( - x ) ) = ( - x ) | ( - x ) .= ( - x ) | ( - x ) .= ( - x ) | ( - x ) ) .= ( - x ) | ( - x ) .= ( - x ) | ( - x ) ;
0 * sqrt ( 1 + ( p `1 / p `2 ) * sqrt ( 1 + ( p `2 / sqrt ( 1 + p `2 ) * sqrt ( 1 + ( p `1 / sqrt ( 1 + p `2 / sqrt ( 1 + p `2 / sqrt ( 1 + p `2 ) ) ) ) ) = sqrt ( ( ( p `1 / sqrt ( 1 + p `2 / sqrt ( 1 + p `2 / sqrt ( 1 + p `2 ) ) ) ) ^2 ) ;
sqrt ( ( ( W- ( x + y ) * ( - x ) ) ) * ( ( - ( x + y ) * ( - y ) ) ) ) = ( ( - ( x + y ) * ( - x ) ) * ( - x ) ) * ( - x ) .= ( - ( x + y ) * ( - x + y ) ) * ( - x + y ) .= ( - x + ( - x ) * ( - y ) ) * ( - x ) * ( - x ) * ( - y ) .= ( - x ) * ( - x ) * ( - y ) * ( - y + y ) * ( - y + y ) * ( - y ) * ( - y + y ) * ( - x + y ) * ( - y + y ) * ( - y ) * ( - y + y ) .= ( - ( y + y ) * ( - x ) * ( -
redefine func Shift ( f , h ) -> PartFunc of REAL , REAL means : Def5 : for x be Element of REAL holds it . x = ( - h ) . x ;
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f and [ i , j ] in Indices GoB f and f /. k = f /. k ;
assume not y in Free ( H ) implies x in Free ( H ) \/ { x } & not x in Free ( H ) \/ { x } & x in Free ( H ) \/ { x } ;
defpred P11 ( Element of NAT ) = ( p |^ ( $1 ) ) * ( p |^ ( $1 ) ) & ( p |^ ( $1 + 1 ) ) * ( p |^ ( $1 + 1 ) ) = p |^ ( $1 + 1 ) * ( p |^ ( $1 + 1 ) ) ;
func \sigma ( C ) -> non empty Subset of X means : Def5 : for x being Element of X holds it . x = { x } ;
[#] ( ( dist ( ( dist ( ( ( dist ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( dist ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } or rng ( F | ( S , 2 ) ) = { 1 } ;
( f " ) . ( ( f " ) . i ) = f . ( i ) .= f . i .= f . i .= f . i .= f . i .= f . i .= f . i .= f . i .= f . i .= f . i ;
consider P1 , P2 being Subset of TOP-REAL 2 such that P1 = { p1 , p2 } and P1 = { p1 , p2 } and P2 = { p1 , p2 } and P = { p1 , p2 } and P = { p1 , p2 } and P = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p2 } and Q = { p1 , p1 , p2 , p1 , p2 , p1 , p2 , p2 , p3 , p1 , p1 , p2 , p3 , p1 , p2 , p3 , p1 , p3 , p1 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p3 = p1 , p2 , p3 , p3 , p3 , p3 , p3 , p3 , p3 , p4 , p4 , p4 , p4 , p4 , p4 , p4 , p4 and P
f . p2 = |[ ( ( p2 `1 ) / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 ) , ( p2 `2 / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 + ( p2 `2 / |. p2 .| ) ^2 ) ]| ;
( proj ( a , X ) " ) . x = ( proj ( a , X ) ) . x .= ( proj ( a , X ) ) . x .= ( proj ( a , X ) ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X ) . x .= ( u , X
let T being non empty TopSpace , A , B , C being Subset of T , A being Subset of T , B being Subset of T st A <> B & C c= A & B c= C holds A is open
for i , [#] G st i + 1 in dom F & for G being Subset of G st G in F holds G is Subgroup of G & G is Subgroup of G holds G is Subgroup of G
for x st x in Z holds ( ( ( ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( ( ( ( ( ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan * ( arctan ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) `| Z ) ) = ( ( ( ( ( ( ( ( ( ( ( ( arctan * ( arctan *
synonym f /* ( x0 + r ) -> right & for f . ( x0 + r ) = 0 & for x0 st x0 in dom f holds f . ( x0 + r ) = 0 & f . ( x0 + r ) = 0 & f . ( x0 + r ) = 0 ;
then X1 misses ( X1 union X2 ) & X1 misses ( X1 union X2 ) & X2 is SubSpace of X1 union X2 & X1 is SubSpace of X1 union X2 implies X1 union X2 is SubSpace of X1 union X2 & X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 & X1 is SubSpace of X1 union X2 implies X1 union X2 is SubSpace of X1 union X2 & X1 is SubSpace of X1 union X2 & X2 is SubSpace of X1 union X2 ;
ex N be Neighbourhood of x0 st N c= dom ( f , x0 ) & for r st r in dom ( f , x0 ) holds ( ( f , x0 ) --> r ) . r = ( f , x0 ) --> r ) . r ;
sqrt ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) + ( sqrt ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
( sqrt ( 1 - ( ( ( ( 1 / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
assume for x , h st x = ( ( for x st x in Z holds h . x = ( \HM { the } \HM { function } ) * ( \HM { the } \HM { function } \HM { sin } ) ) . x ) & ( \HM { the } \HM { function } ) . x = ( ( \HM { the } \HM { function } ) * ( \HM { the } \HM { function } \HM { sin } ) . x ) ;
consider X1 being Subset of Y , Y1 being Subset of Y such that ( X1 = the carrier of Y ) & ( Y = the carrier of X ) & ( Y is Subset of X ) and ( Y is Subset of X ) ;
card ( S . n ) = card ( { d , n + 1 } ) + ( card ( { d , n } ) ) .= card ( { d , n } ) + 1 ;
sqrt ( ( ( the carrier of ( TOP-REAL 2 ) ) * ( i + 1 ) ) + ( ( the carrier of ( TOP-REAL 2 ) ) * ( i + 1 ) ) ) = sqrt ( ( the carrier of TOP-REAL 2 ) * ( i + 1 ) ) .= sqrt ( ( the carrier of TOP-REAL 2 ) * ( i + 1 ) ) ^2 + ( ( the carrier of TOP-REAL 2 ) * ( i + 1 ) ) ^2 ) ;