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 `2 is convergent ;
q in P ;
V is open ;
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 P ;
x in X ;
Y `2 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 `2 = b ;
assume e in F ;
assume p > 0 ;
assume x in D ;
let i be element ;
assume F is onto ;
assume n <> 0 ;
let x be element ;
set k = z ;
assume o = x ;
assume b < a ;
assume x in A ;
a `2 <= b `2 ;
assume b in X ;
assume k <> 1 ;
f = product l ;
assume H <> F ;
assume x in I ;
assume p is prime ;
assume A in D ;
assume 1 in b ;
y is generated from squares ;
assume m > 0 ;
assume A c= B ;
X is lower ;
assume A <> {} ;
assume X <> {} ;
assume F <> {} ;
assume G is open ;
assume f is dilatation ;
assume y in W ;
y \not <= x ;
A `2 in B `2 ;
assume i = 1 ;
let x be element ;
x `2 = x `2 ;
let X be BCK-algebra ;
assume S is non empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be Vertex of G ;
let G be _Graph , a , b be Vertex of G ;
let a be Int-Location ;
let x be element ;
let x be element ;
let C be FormalContext , f be FinSequence of C ;
let x be element ;
let x be element ;
let x be element ;
n in NAT ;
n in NAT ;
n in NAT ;
thesis ;
let y be Real ;
X c= f . a
let y be element ;
let x be element ;
let i be Nat ;
let x be element ;
n in NAT ;
let a be element ;
m in NAT ;
let u be element ;
i in NAT ;
let g be Function ;
Z c= NAT ;
l <= a ;
let y be element ;
r2 < r ;
let x be element ;
k1 be Integer ;
let X be set ;
let a be element ;
let x be element ;
let x be element ;
let q be element ;
let x be element ;
assume f is being_homeomorphism ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = / 2 ;
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_on Z ;
the function exp is_differentiable_in x ;
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 <= r1 ;
let e be Real , f be FinSequence of REAL ;
not r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is discrete ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > p ;
let y be ExtReal ;
let a be Int-Location , f be FinSequence of REAL ;
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 ;
R is total ;
cluster uparrow x -> closed ;
X <> { x } ;
x in { x } ;
q , b // M ;
A . i c= Y ;
P [ k ] ;
2 |^ x in W ;
X [ 0 ] ;
P [ 0 ] ;
A = A |^ i ;
j >= us ;
G . y <> 0 ;
let X be RealNormSpace , f be PartFunc of X , Y ;
a in A ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in M ;
k < s . a ;
not t in { p } ;
let Y be empty set , X be set ;
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 ;
G c= L ;
assume x in Cl Q ;
m in dom P ;
i <= len Q ;
len F = 3 ;
Free p = {} ;
z in rng p ;
lim b = 0 ;
len W = 3 ;
k in dom p ;
k <= len p ;
i <= len p ;
1 in dom f ;
b `1 = a `1 + 1 ;
x `2 = a * y `2 ;
rng D c= A ;
assume x in K1 ;
1 <= ii ;
1 <= ii ;
px c= PI ;
1 <= ii ;
1 <= ii ;
w in L ;
1 in dom f ;
let seq ;
set C = a * B ;
x in rng f ;
assume f is_differentiable_on X ;
I = dom A ;
u in dom p ;
assume a < x + 1 ;
s-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
F-14 is continuous ;
dom g = X ;
len q = m ;
assume A2 is closed ;
cluster R \ S -> real-valued ;
ex_sup_of D , S ;
x << sup D ;
b1 >= Z ;
assume w = 0. V ;
assume x in A . i ;
g in |. the carrier of X .| ;
y in dom t ;
i in dom g ;
assume P [ k ] ;
op C c= f ;
x4 is increasing ;
let e2 be element ;
- b divides b ;
F c= \mathclose { \tau }
G1 is non-decreasing ;
G1 is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non empty ManySortedSign , f be FinSequence of S ;
assume P [ n ] ;
assume union S is simplex-like & 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 be set , f be FinSequence of X ;
b `2 c= b1 `2 ;
assume not x in Q + Q ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in B-24 ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_in x ;
assume y in rng S ;
let x , y be element ;
i2 < i1 ;
a * h in a * H ;
p , q in Y ;
cluster sqrt I -> ideal ;
q1 in A1 ;
i + 1 <= 2 + 1 ;
A1 c= A2 & A2 c= A1 ;
\hbox { n , m } < n ;
assume A c= dom f ;
Re f is_integrable_on M ;
let k , m ;
a , a \equiv b , b ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_in x0 ;
g is_differentiable_in x0 ;
assume O is transitive & transitive ;
let x , y be element ;
let j2 be Nat ;
[ y , x ] in R ;
let x , y be element ;
assume y in conv A ;
x in Int V ;
let v be Vector of V ;
P3 is_closed_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 ;
let Y be Subset of S ;
let X be non empty TopSpace , f be Function of X , Y ;
[ a , b ] in R ;
x + w < y + w ;
{ a , b } >= c ;
let B be Subset of A , A , B be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of v1 , v2 ;
let b be Element of X , c be Element of X ;
R [ x , y ] ;
x ` ` = x ;
b \ x = 0. X ;
<* d *> in D ^ ;
P [ k + 1 ] ;
m in dom ( n + 1 ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> preobject ;
let R be non empty doubleLoopStr , f be FinSequence of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is lower ;
x in rng go ;
let x be Element of B ;
let t be Element of D ;
assume x in Q .vertices() ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p `2 ;
let M be maid set , f be FinSequence of M , g be FinSequence ;
let N be non empty Subset of M ;
let R be finite non empty transitive antisymmetric RelStr ;
let n , k be Nat ;
let P , Q be reflexive non empty 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 ;
assume I is not destroy a ;
let n , k be Nat ;
let x be Point of T ;
f c= f +* g ;
assume m < v2 ;
x <= c2 . x ;
x in F " { x } ;
cluster S --> T -> N -defined ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be Integer ;
assume F1 <> F2 ;
c in Intersect ( ( union R ) /\ Y ) ;
dom p1 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 ;
set i1 = i + 1 ;
assume a1 = b1 ;
dom g1 = 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 & 1 <= s2 ;
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 ;
order ( i , n ) = i ;
X1 c= dom f ;
h . x in h . a ;
let G be \times j1 in Abelian G , H :] ;
cluster m * n -> square ;
let k9 be Nat ;
i - 1 > m - 1 ;
R is transitive implies R is transitive
set F = <* u , w *> ;
pP c= P ;
I is_closed_on t , Q ;
assume [ S , x ] is : 1 -element ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R = n * r ;
cluster f . x -> complex-valued ;
x in dom ( f1 + f2 ) ;
assume [ X , p ] in C ;
BX c= XX ;
n2 <= ( n2 - 1 ) / ( n2 - 1 ) ;
A /\ ( P ` ) c= A ` ;
cluster x -element for Function ;
let Q be Subset-Family of S , X be non empty Subset of S ;
assume n in dom g2 ;
let a be Element of R ;
t `2 in dom ( e , f ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be SigmaField of X , f be PartFunc of X , Y ;
i . y in rng i ;
REAL c= dom f ;
f . x in rng f ;
\mathbb t <= sqrt ( r ^2 - 2 ) ;
s2 in < r2 & s2 in < r2 ;
let z , z be number ;
n <= [: N . m , N . m :] ;
LIN q , p , s ;
f . x = -' x /\ B ;
set L = [ S \to T ] ;
let x be non negative ExtReal ;
m be Element of M ;
f in union rng F1 ;
let K be add-associative right_zeroed right_complementable associative associative non empty doubleLoopStr , f be Polynomial of K , L ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom x ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster <* X *> -> On ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng ( Sb | [. 0 , 1 .] ) ;
b = sup dom f ;
x in Seg len q ;
reconsider X = D as set ;
[ a , c ] in E ;
assume n in dom h2 ;
w + 1 = a1 ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 ;
let i be Element of NAT ;
Support u = Support p ;
assume X is complete |. m .| ;
assume f = g & p = q ;
n1 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng s2 ;
x0 < x0 + 1 ;
len ( L ^ LF ) = W ;
P c= Seg ( len A ) ;
dom q = Seg n ;
j <= width M ^ B ;
let r3 be real-valued sequence of REAL ;
let k be Element of NAT ;
\int P + d < + \infty ;
let n be Element of NAT ;
assume z in at_set ( 0 , A ) ;
let i be set ;
n - 1 = n - 1 ;
len ( n2 , n ) = n ;
measurable ( Z , c ) c= F
assume x in X or x = X ;
x is midpoint of b , c ;
let A , B be non empty set , f be FinSequence of B ;
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E ^ ;
let B1 be Basis of x , x be Element of X ;
Carrier ( L ) /\ L2 = {} ;
L1 /\ L2 = {} ;
assume \mathopen { x , y } = \mathopen { x , y } ;
assume b , c , b is_collinear ;
LIN q , c , a ;
x in rng ( f | Amax ) ;
set n8 = n + j ;
let D1 be non empty set , D2 be non empty set , D1 , D2 be non empty set ;
let K be add-associative right_zeroed right_complementable associative associative non empty doubleLoopStr , f be Polynomial of K , K ;
assume f `2 = f & h = g ;
R1 - ( R + L ) is total ;
k in NAT & 1 <= k ;
let a be Element of G ;
assume x0 in [. a , b .[ ;
K1 ` is open ;
assume a , b are_maximal in C ;
a , b be Element of S ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f ;
cluster -> <* -> -> -> -> -> -> -> -> -> -> |. s|. sseesseeseseseseeses
not u in { \hbox { \boldmath $ g , h } } ;
the support of f c= B ;
reconsider z = x as Vector of V ;
cluster empty for non empty doubleLoopStr ;
r (#) H is U ;
s . intloc 0 = 1 ;
assume x in C & y in C ;
let U2 be strict non-empty MSAlgebra over S , A be MSAlgebra over S ;
[ x , Bottom T ] is compact ;
i + 1 in dom p ;
F . i is stable Subset of M ;
r-35 in reconsider ry as Element of extended y ( ) ;
let x , y be Element of X ;
A , I be \mathclose { 0. X } ;
[ y , z ] in O ;
Shift ( goto i , 1 ) = goto i ;
rng Sgm A = A ;
q |- p => All ( y , q ) ;
for n holds X [ n ] ;
x in { a } & x in d ;
for n holds P [ n ] ;
set p = [ x , y , z , x , y , z ] ;
LIN o , a , b ;
p . 2 = Z ^ Y ;
( D , 0 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: Al ( ) , Al ( ) :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be ideal of L ;
set g = f1 + f2 , h = f2 + h , i = f1 + h ;
a <= max ( a , b ) ;
i < len G + 1 ;
g . 1 = f . i1 ;
x `2 , y `2 ] in [: A2 , A2 :] ;
( f /* s ) . k < r ;
set v = VAL g ;
i - k + 1 <= S ;
cluster non empty multiplicative for FinSequence ;
x in support ( t ) ;
assume a in [: G , G :] ;
i `2 <= len ( y1 /. i ) `2 ;
assume p divides b1 + b2 ;
M . 0 <= sup ( M1 + M2 ) ;
assume x in ( W-min X ) `1 ;
j in dom ( z | ( Seg n ) ) ;
let x be Element of [: D , D :] ;
IC Comput ( P3 , s3 , l ) = l1 ;
a = {} or a = { x } ;
set uG = Vertices G , CG = G , CG = G ;
seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
hK c= ( h . O ) . ( h . O ) ;
]. a , b .[ c= Z ;
X1 , X2 , X3 , X3 , .|| ;
a in Cl ( union F \ G ) ;
set x1 = [ 0 , 0 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1
k + 1 - 1 = k - 1 ;
cluster binary for Relation ;
ex v st C = v + W ;
let Gbe non empty doubleLoopStr , f be FinSequence of REAL ;
assume V is Abelian add-associative right_zeroed right_complementable associative associative distributive non empty doubleLoopStr ;
[: X , Y :] \/ Y in relational ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
upper_bound B is upper ;
let L be non empty reflexive transitive reflexive transitive antisymmetric reflexive transitive RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E , g |= ( H '&' ( H '&' F ) ) ;
dom G ' /. y = a ;
sqrt ( 1 - 4 * r ) >= - r ;
G . x0 in rng G ;
let x be Element of F , y be Element of F ;
D [ ( P1 , 0 ) --> 1 ] ;
z in dom id ( B ) ;
y in the carrier of N ;
g in the carrier of H & h in the carrier of H ;
rng ( f | [: X , Y :] ) c= [: X , Y :] ;
j `2 + 1 in dom s1 ;
A , B be strict Subgroup of G ;
let C be 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 . 0 ) in REAL ;
ind [. a , b .] = b ;
assume the distance of V is Subspace of v & Q is Subspace of v ;
let a be Element of ^ V ;
let s be Element of P ;
let PL be non empty reflexive transitive transitive RelStr ;
n be Nat ;
the support of g c= B ;
I = halt SCM R .= ( halt R ) . IC SCM R ;
consider b being element such that b in B ;
set BK = conv K ;
l <= Sup ( F . j ) ;
assume x in \mathopen { [ s , t ] } ;
( x - t ) . t in ]. t - t , t + t .[ ;
x in InsCode ( T ) ;
let h be Morphism of c , a , b be Object of c ;
Y c= [: Y , Y :] ;
A2 \/ LSeg ( LSeg ( LSeg ( p10 , p2 ) , p2 ) c= L1 ;
assume LIN o , a , b ;
b , c // d1 , d2 ;
x1 , x2 , x3 , x4 , x5 , 8 , 6 , 8 , 7 , 8 , 8 , 8 , 6 , 6 , 8 , 8 , 6 , 6 , 8 , 7 , 8
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| ;
[ x , x ] in X [: X , Y :] ;
for n be Nat holds 0 <= x . n
[. a , b .] = [. a , b .] ;
cluster -> [ T ] for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 , q2 is_collinear ;
dom M1 = Seg n & dom M2 = Seg n ;
x = [ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 8 , x5 , 8 , 8 , 8 , 6 , M , M , N , M , N , M ,
R , Q be ManySortedSet of A ;
set d = sqrt ( 1 + n ) ;
rng ( ( g2 - g1 ) /* q ) c= dom W ;
P . ( [#] Sigma \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of M ;
let I be Program of SCM+FSA , a be Int-Location ;
assume x in rng ( R * R ) ;
let b be Element of lattice T ;
dist ( e , z ) > rr ;
u1 + v1 in W2 & v1 in W2 ;
assume the support of L misses rng G ;
let L be lower-bounded antisymmetric transitive antisymmetric non empty reflexive transitive antisymmetric antisymmetric antisymmetric antisymmetric RelStr ;
assume [ x , y ] in [: a , b :] ;
dom ( A * e ) = NAT ;
a , b // G * ( i , 1 ) ;
let x be Element of Bool ( M ) ;
0 <= 2 * PI / 2 ;
o , a9 // o , y ;
{ v } c= the support of l ;
let x be variable of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( product f ) |^ X ;
assume D2 . k in rng D ;
f " . p1 = 0 ;
set x = the Element of X ;
dom Ser G = NAT ;
n be Element of NAT ;
assume LIN c , a , e1 ;
cluster finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . X <= 1 ;
assume x in the support of f ;
conv @ S c= conv @ A ;
reconsider B = b as Element of the carrier of T ;
J , v |= ( P ! l ) ;
cluster J . i -> non empty for TopStruct ;
ex_sup_of Y \/ ( X \/ Y ) , T ;
W1 is_Lin ( W1 ) , R ;
assume x in the carrier of R ;
dom ( nn --> 0 ) = Seg n ;
s4 misses s4 & s4 in { s } ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( f ) ;
assume that that that that that that that that that that that that that I c= J and I c= K and J c= K and I c= K ;
Im ( ( lim seq ) ^\ k ) = 0 ;
( ( ( - 1 ) (#) sin ) `| Z ) . x <> 0 ;
sin is_differentiable_on Z & cos is_differentiable_on Z ;
t2 . n = t2 . n ;
dom ( F | Z ) c= dom F ;
W1 . x = W2 . x ;
y in W .vertices() \/ W .vertices() ;
k9 <= len ( v | ( Seg n + 1 ) ) ;
x * a \equiv y * a . ( m mod a ) ;
proj2 ( S ) c= proj2 .: ( P ) ;
h . p3 = g2 . I . ( I . p3 ) ;
G = ( U /. 1 ) `1 & G * ( 1 , 1 ) `1 = G * ( 1 , 1 ) `1 ;
f . r1 in rng f ;
i + 1 <= len One ;
rng F = rng ( F . i ) ;
mode double Morphism of N , L is Abelian non empty doubleLoopStr ;
[ x , y ] in A [: { a } , { a } :] ;
x1 . o in LSeg ( x2 , o ) ;
the support of \lbrace m , n } c= B ;
not [ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k is convergent & lim ( seq ^\ k ) = x0 ;
len ( F . -12 ) = len I & len ( F . c ) = len I ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) = s ;
k <= k + 1 ;
reconsider c = {} as Element of L ;
let Y be \kern1pt Subset of T ;
cluster -> monotone for Function of L , L ;
f . j1 in K . j1 ;
cluster J => y -> total for Function ;
K c= 2 -tuples_on the carrier of T ;
F . b1 = F . b2 ;
x1 = x or x1 = y ;
attr a <> {} means : Def3 : a = 1 ;
assume cf a c= b & b in a ;
s1 . n in rng ( s1 . n ) ;
{ o , b2 } on C2 & on C2 , C2 ;
LIN o , b , b ;
reconsider m = x as Element of [: V , V :] ;
let f be non trivial FinSequence of D ;
let F2 be non empty topological space ;
assume h is being_homeomorphism & y = h . x ;
[ f . 1 , w ] in [: F , F :] ;
reconsider p2 = x as Subset of m ;
A , B , C is_collinear ;
cluster strict non empty for normal for Element of \rho ;
rng c misses rng ( e ^ c ) ;
z is Element of gr ( { x } ) ;
not b in dom ( a .--> p1 ) ;
assume k >= 2 & P [ k ] ;
Z c= dom ( ( ( cot * cot ) `| Z ) ;
the component of Q c= UBD ( A ) & ( ( A \/ B ) \/ ( A \/ B ) c= UBD ( A ) ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( 1 / 2 ) ;
attr f = u * f means : Def3 : a * f = a * u ;
for n holds P1 [ n ] ;
{ x . O : x in L } <> {} ;
x be Element of V . s ;
a , b be Nat ;
assume S = S2 & p = S2 ;
gcd ( n1 , n2 ) = 1 ;
set o = ( 1 / 2 ) * ( ( 1 / 2 ) * ( 1 / 2 ) ) ;
seq . n < |. r1 .| ;
assume seq is increasing & r < 0 ;
f . y1 <= a ;
ex c be Nat st P [ c ] ;
set g = { n / 1 where n is Nat : n in NAT } ;
k = a or k = b ;
a1 , \hbox { \boldmath $ g $ } , b1 , b2 , b3 is_collinear ;
assume Y = { 1 } & s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W4 .last() = W . 1 ;
cluster -> -> -> trivial for of G , n be Nat ;
reconsider u = u as Element of Bags X ;
A in B ^ implies A , B are_\kern1pt
x in { [ 2 * n + 3 , k ] } ;
1 - sqrt ( ( q `2 / |. q .| - sn ) ^2 ) >= 0 ;
f1 is_subarc from x0 , x0 & f2 is_convergent implies f1 + f2 is convergent
( f . q ) `2 <= ( q `2 ) ^2 ;
h is_the carrier of Cage ( C , n ) ;
( b - b ) / ( p - a ) <= ( p - a ) / ( p - a ) ;
let f , g be transitive Function of X , Y ;
S /. ( k , k ) <> 0. K ;
x in dom ( f - g ) ;
p2 in [: N , I :] . ( p1 , p2 ) ;
len ( H ) < len ( H ) ;
F [ A , F . A ] ;
consider Z such that y in Z and Z in X ;
attr 1 in C means : Def3 : A c= C & A c= C ^ B ;
assume r1 <> 0 or r1 <> 0 ;
rng q1 c= rng ( C1 ^ C2 ) ;
A1 , L , L is_collinear ;
y in rng f & y in { x } ;
f /. ( i + 1 ) in L~ f ;
b in LSeg ( p , S ) ;
then S is negative implies P [ S ] ;
Cl [#] ( T | P ) = [#] ( T | P ) ;
( f | A2 ) | A2 = f2 | A2 ;
0. M in the carrier of W & 0. M in the carrier of V ;
v , v be Element of M ;
reconsider K = union rng K as non empty set ;
X \ V c= Y \ V
let X be Subset of S , T be non empty TopSpace ;
consider H1 such that H = 'not' H1 ;
1_ t c= c1 * t ;
0 * a = 0. R .= a * 0 .= a * 0 ;
A |^ 2 = A |^ ( 2 + 1 ) ;
set v = ( v /. n ) `1 , w = v /. n , y = v /. m ;
r = 0. ( \langle \cal E , \Vert * \Vert *> ) ;
( f . p3 ) `2 >= 0 ;
len W = len ( W | ( m + 1 ) ) ;
f /* ( s * G ) is divergent & f . ( s * G ) < f . ( s * G ) ;
consider l be Nat such that m = F . l ;
t14 . b1 in { b1 , b2 , b1 , b2 , b3 , b1 , b2 , b3 , b3 , 6 , 6 , b1 , b1 , b1 , b2 , b3 } ;
reconsider Y1 = 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 - 1 as non zero Element of NAT ;
c . x >= id ( L . x ) ;
-2 ( T ) \/ omega ( T ) is Basis of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 ,
downarrow a /\ \mathopen { t } is ideal of T ;
let X be card NAT -defined set , N be non empty set , f be non empty FinSequence of NAT ;
rng f = Pro\rm Free ( S , X ) ;
let p be Element of B , x be the sort Element of S ;
max ( N1 , 2 ) >= N1 & max ( N1 , 2 ) >= N1 ;
0. X <= b |^ m * ( m * ( m - 1 ) ) ;
assume i in I & R . i = R . i ;
i = j1 & p1 = q1 & p1 = q2 ;
assume gR in the right of g ;
let A1 , A2 be Point of S ;
x in h " ( P /\ [#] T1 ) /\ [#] T2 ;
1 in Seg 2 & 1 in Seg 3 ;
reconsider X= X as non empty Subset of [: T , T :] ;
x in ( the Arrows of B ) . i ;
cluster ( G . n ) -element for Subset of G ;
n1 <= i2 + ( i2 + 1 ) & n2 <= len ( g2 ) ;
( i + 1 ) + 1 = i + ( 1 + 1 ) ;
assume v in the carrier' of G2 & v in the carrier' of G2 ;
y = Re ( y . i ) + ( Im ( y . i ) * i ) ;
( ( ( - 1 ) * p ) gcd ( - 1 ) ) = 1 ;
x2 is_differentiable_on ]. a , b .[ ;
rng ( M . ( len ( D2 ) ) ) c= rng ( ( M . ( len D2 ) ) ^ ( M . ( len D2 ) ) ;
for p be Real st p in Z holds p >= a
\bf X \bf Y * f = proj1 * f & f is continuous ;
( seq ^\ m ) . k <> 0 ;
s . ( G . ( k + 1 ) ) > x0 ;
( p \! \mathop { p } ) . ( 2 + 1 ) = d ;
A \oplus ( B \oplus C ) = ( A \oplus B ) \ominus C
h \equiv ( gg . ( mod P ) ) . ( h . ( len P ) ) ;
reconsider i1 = i - 1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
for V being VectSp of V holds V is \mathopen { 0. V }
reconsider ii = i - 1 as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the inferior of B ) . n ;
len f2 in Seg len ( f1 ^ f2 ) ;
p1 c= the topology of T & p1 in the topology of T ;
]. r , s .[ c= [. r , s .] ;
let B2 be Basis of T2 , X be non empty Subset of T ;
G * ( B * A ) = id o1 ;
assume p , u , v , u is_collinear & u , v , v is_collinear ;
[ z , z ] in union rng ( F . z ) ;
'not' b . x 'or' b . x = TRUE ;
deffunc F ( set ) = $1 .. S & $1 in S ;
LIN a1 , a2 , a3 & LIN a1 , a2 , a3 ;
f " ( f .: x ) = { x } ;
dom w2 = dom ( w2 . i ) ;
assume that 1 <= i and i <= n and j <= n and i <= n ;
( ( g2 . O ) `2 ) ^2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
Ij * ( i , j ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q1 . x in rng ( q1 ^ q2 ) ;
L-43 misses ( LSeg ( g-43 , i ) \/ LSeg ( g\rbrack , i ) ) ;
consider c being element such that [ a , c ] in G ;
assume N19 = N19 & N19 = N19 ;
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= ( F ^ G ) ^ ( F ^ G ) ;
P . ( B2 \/ D2 ) <= 0 + 0 ;
f . j in [. f . j , f . j .] ;
pred 0 <= x & x <= 1 & x <= 1 ;
p `2 <> 0. TOP-REAL 2 & p `2 <> 0. TOP-REAL 2 ;
cluster ProaaaaaaaaaaaaaaaaaaS ;
x be Element of S , T be Element of T ;
<^ F , a ^> is one-to-one ;
|. i - 1 .| <= - ( - 2 |^ n ) ;
the carrier of I[01] = dom P & P is compact ;
n * ( n + 1 ) > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // a3 , b1 & a3 , a4 // a3 , b2 ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 5 ;
x Joins X , Y , G ;
set v2 = ( v /. ( i + 1 ) ) `1 , v1 = v /. ( i + 1 ) ;
x = r . n .= ( r . n ) . n ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] & dom h = the carrier of I[01] ;
p in Upper_Arc ( P ) /\ Lower_Arc ( P ) ;
dom ( d , A ) = [: A , A :] & dom ( d , A ) = [: A , A :] ;
0 < sqrt ( p `1 - z `2 ) + 1 ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X c= B \ominus X /\ B
- - +infty < \int ( g | B ) + E ;
cluster O \tt F -> \tt for Element of X ;
let U1 , U2 be non-empty MSAlgebra over S , U2 be MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y , z be Point of X , f be PartFunc of X , Y ;
reconsider px = p . x as Subset of V ;
x in the carrier of Lin ( A ) & x in Lin ( A ) ;
let I , J be Program of SCM+FSA , a be Int-Location ;
assume - a is lower & - b is lower ;
Int ( Cl A ) c= Cl ( Int ( Cl 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 ;
Cl Q ` = [#] ( T | P ) ;
set S = the carrier of T ;
set If = Product ( f , n ) , If = f |^ n , If = f |^ ( n + 1 ) , If = f |^ ( n + 1 ) , If = f |^ ( n + 1 ) , If
len p - n = len p - n ;
A is permutation of Funcs ( A , x , y ) ;
reconsider ni = ni - 1 as Element of NAT ;
1 <= j + 1 & j + 1 <= len ( seq . j ) ;
q9 , sthesis Element of M ;
a1 in the carrier of S1 & a2 in the carrier of S2 & a3 in the carrier of S2 ;
c1 /. ( n + 1 ) = c1 . ( n + 1 ) ;
let f be FinSequence of TOP-REAL 2 , p , q be Point of TOP-REAL 2 ;
y = ( ( f * S ) . x ) . x ;
consider x being element such that x in s\mathbb \mathbb linearly A A ;
assume r in ( dist ( o , r ) ) .: P ;
set i2 = ( TOP-REAL n ) .. h , h = ( TOP-REAL n ) .. h , i = ( TOP-REAL n ) .. h , i = ( TOP-REAL n ) .. h , j = ( TOP-REAL n ) .. h , j = ( TOP-REAL
h2 . ( j + 1 ) in rng h2 ;
Line ( M , k ) . i = M . i ;
reconsider m = sqrt ( x ^2 - 2 ) as Element of REAL ;
U1 , U2 , U2 , U2 , U1 , U2 , U2 , U2 be non-empty st U1 , U2 , U2 , U2 , U1 , U2 , U2 , U2 , U2 , U2 , U2 , U2 , U2 , U2 be non-empty MSAlgebra over (
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p1 = len p2 + 1 ;
T1 , T2 be complete topological TOP-REAL of L , f be topological of L ;
then x <= y & Shift ( x , y ) c= MaxADSet ( y ) ;
set M = n -\hbox { m } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of o ) c= dom H ;
z1 " = z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 " * z1 "
x0 - sqrt ( r ^2 + ( r ^2 ) ) in L /\ dom f ;
then w is \it S & w /\ S <> {} ;
set xZ = ( x ^ Z ) ^ <* Z *> , Z = ( x ^ Z ) ^ <* Z *> ;
len w1 in Seg len w1 & len w2 in Seg len w2 ;
( uncurry f ) . ( x , y ) = g . y ;
let a be Element of PFuncs ( V , { k } ) ;
x . n = sqrt ( |. a . n .| - |. a .| ) ;
( p `1 ) ^2 <= ( G * ( -13 , 1 ) `1 ) ^2 ;
rng ( g | ( L~ g ) ^ h ) c= L~ g /\ L~ h ;
reconsider k = i-1 * ( l + 1 ) as Nat ;
for n be Nat holds F . n is without_- F . n
reconsider x9 = x as Vector of M ;
dom ( f | X ) = X /\ dom f /\ X ;
p , a // p , c & b , a // p , c ;
reconsider x1 = x as Element of REAL m m m -tuples_on REAL ;
assume i in dom ( a * p ^ q ) ;
m . \hbox { \boldmath $ g , p , s } = p . ( \hbox { \boldmath $ g , p , s ) ;
a |^ ( s . m ) - a / ( s . m ) <= 1 ;
S . ( n + k + 1 ) c= S . ( n + k ) ;
assume B1 \/ B2 = B2 \/ ( C2 \/ C2 ) ;
X . i = { x1 , x2 , x3 , x4 , x5 , M } ;
r2 in dom ( h1 + h2 ) /\ dom ( h1 + h2 ) ;
<* 0. R *> = a & bR = b ;
F is closed implies closed & closed & closed & closed & closed closed & closed closed & closed closed & closed closed & closed closed & closed closed ;
set T = <* X , x0 , x1 , x2 , x3 , x4 , x4 , x5 , x5 , x5 , 8 , x5 , 8 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , x5 , M *> ;
Int ( Int ( R ) ) c= Int ( R ) ;
consider y being Element of L such that c . y = x ;
rng ( Fwhere F is Subset-Family of X : F . x = { F . x } ) ;
Ga1 " ( { c } ) c= B \/ S ;
f1 is_differentiable_in X implies f1 is relation & f2 is relation & X c= dom f1 ;
set R = the Point of P , P = the Point of ( TOP-REAL 2 ) | P ;
assume n + 1 >= 1 & n + 1 <= len M ;
let k2 be Element of NAT ;
reconsider pj = u as Element of / ( n + 1 ) ;
g . x in dom f & x in dom g ;
assume that 1 <= n and n + 1 <= len f1 and n + 1 <= len f1 ;
reconsider T = b * N as Element of ( G / N ) / N ;
len ( ( P1 ^ P2 ) ^ ( P1 ^ P2 ) ) <= len ( P1 ^ P2 ) ;
x " in the carrier of A1 & x in the carrier of A2 ;
[ i , j ] in Indices ( A * ( i , j ) ) ;
for m be Nat holds Re ( F . m ) is simple
f . x = a . i .= a1 . i .= a1 . i ;
let f be PartFunc of REAL-NS i , REAL-NS n ;
rng f = the carrier of support A & f . 1 = the carrier of A ;
assume s1 = sqrt ( 2 * ( p `2 - r ^2 ) ) ;
attr a > 1 & b > 0 implies a / b > 1 / a ;
let A , B , C be Subset of [: I , I :] ;
reconsider Y1 = X , Y2 = Y as real set ;
let f be PartFunc of REAL , REAL , x0 be Real ;
r (#) ( v1 , I ) . ( X , I ) . ( X , I ) < r * 1 ;
assume V is Subspace of X & X is Subspace of V ;
let t be binary Function , Q be Subset of t ;
Q [ e \/ { v } , f /. ( e + 1 ) ] ;
g \circlearrowleft ( L~ z ) = z ;
|. [ x , v ] - [ x , v ] .| = vW2 ;
- f . w = - ( L * w ) ;
z - y <= x & y <= z + y implies z - x <= z + x
sqrt ( 7 + ( 1 - e ) ^2 ) > 0 ;
assume X is BCK-algebra of 0 , 0 , 0 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 1 ,
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( tan * tan ) `| Z ) . x in dom ( tan * tan ) ;
i2 = ( f /. ( len f -' 1 ) ) .. ( f /. ( len f -' 1 ) ;
X1 = [: X1 , X2 :] \/ ( X1 \ X2 ) ;
[. a , b .] = 1_ G .= 1_ G ;
let V be non empty VectSpStr over F , W be Subspace of V ;
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] & dom ( f2 | the carrier of I[01] ) = the carrier of I[01] ;
( proj2 | X ) .: X = ( proj2 | X ) .: X ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 - ( a1 . n ) < x0 - r ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A ;
S\vee 19 ^ ( S . g ) = ( S . g ) ^ ( S . g ) ;
reconsider f = v + u as Function of X , Y ;
intloc 0 in dom ( Initialized p ) \/ dom ( Initialized p ) ;
i1 = ( i1 , i2 ) := ( i2 , j2 ) & i2 <> ( i2 , j2 ) := ( i2 , j2 ) ;
sqrt ( r ^2 + sqrt ( 1 + ( 1 + r ^2 ) ^2 ) = sqrt ( 1 + ( 1 + r ^2 ) ^2 ) ;
for x st x in Z holds f2 is_differentiable_in x & f2 . x > 0 ;
reconsider q2 = sqrt ( q ^2 - 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 , ( a * b ) . n ) < r ;
1 in the carrier of [. 0 , 1 .] & [. 0 , 1 .] c= [. 0 , 1 .] ;
( p2 `2 ) ^2 - ( p2 `2 ) ^2 > - ( - ( p2 `2 ) ^2 ) ;
|. r1 .| = |. a1 .| * |. q1 .| ;
reconsider Sj = 8 as Element of Seg 8 ;
( A \/ B ) ^ ( b ^ C ) c= A ^ B ^ C
DDDDDDDW ( ) = DW .last() ( n + 1 ) ;
i1 = [: a , n :] & i2 = [: a , n :] & j2 = [: a , n :] ;
f . a [= f . ( f .: ( O , a ) ) ;
attr f = v & g = u + v & f + g = v + u ;
I . n = \int F . n , M . n , M . n + 1 , M . n + 1 , N . n + 1 ;
( ( \hbox { - 1 , S , T ) ) . s = 1 ;
a = VERUM ( A ) or a = {} ;
reconsider k2 = s . b2 as Element of NAT ;
( Comput ( P , s , 4 ) ) . intpos ( 0 + 4 ) = 0 ;
L~ ( M1 + M2 ) meets L~ ( M2 + M2 ) ;
set h = the continuous Function of X , R ;
set A = { L . ( k + 1 ) : L . ( k + 1 ) < L . ( k + 1 ) } ;
for H st H is negative holds P [ H ] ;
set bi = [: { i } , { i } :] , bi = [: { i } , { i } :] , bi = [: { i } , { i } :] ;
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 + n + 1 ) < sqrt ( 1 + s . n ) ^2 ;
( l , T ) . [ [ l , T ] , T . [ l , T . ( l , T . ( l , T . ( l , T . ( l , T . l ) ) ] ) = [ T . ( l , T . ( l ,
y +* ( i , y /. i ) in dom g ;
let p be Element of [: Al ( ) , Al ( ) :] ;
X /\ X1 c= dom ( f1 - f2 ) /\ ( f1 - f2 ) ;
p2 in rng ( f /^ 1 ) & p1 in rng ( f /^ 1 ) ;
1 <= indx ( D2 , D1 , j1 ) & 1 <= j1 ;
assume x in K1 /\ ( ( TOP-REAL 2 ) | K1 ) ;
- 1 <= ( ( f2 . O ) . O ) `1 & ( ( f2 . O ) . O ) `1 <= - 1 ;
f , g be Function of I[01] , ( TOP-REAL 2 ) | P , b be Real ;
k1 - ( k - 1 ) = ( k - 1 ) - ( k - 1 ) ;
rng ( seq ^\ k ) c= ]. x0 - r , x0 .[ ;
g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p `1 , K ) = - 1_ K ;
consider u being Nat such that b = p |^ y * u ;
ex A being ]. Sum f , g .[ being Element of REAL st a = Sum A ;
Cl ( ( Cl ( H ) \/ { H } ) ) = union ( ( Cl ( H ) \/ { H } ) ;
len t = len ( t1 + t2 ) + len ( t2 + t1 ) ;
then that v = v + w & v in A + ( v + w ) ;
not v <> DataLoc ( t1 . intpos ( 0 + 3 ) , 3 ) ;
g . s = sup ( d " { s } ) ;
( \dot { y , s ) . s = s . ( <* y , s *> . s ) ;
{ s : s < t } in Q & t = {} + s implies t = {} ;
s ` \ s = s ` \ s ` .= s ` \ ( s ` \ s ` ) ;
defpred P [ Nat ] means B + 1 in A ;
( 3 - 4 ) ! = 3 * ( 3 + 4 ) ;
U . ( succ A ) = U ( U , A ) . ( succ A ) ;
reconsider y = y as Element of COMPLEX n -tuples_on ( the carrier of K ) ;
consider i2 being Integer such that y = p * i2 and i2 in dom p ;
reconsider p = Y | Seg k as FinSequence of NAT ;
set f = ( S , U ) \! \mathop ( z , U ) ;
consider Z being set such that lim s in Z and Z in F ;
let f be Function of I[01] , TOP-REAL n , R^1 , a , b be Real ;
[ 'not' ( n + i , i , A ) , 'not' ( n + i , A ) ] <> 1 ;
ex r be Real st x = r & a <= r & r <= b ;
R1 , R2 be Element of REAL n , R be Element of REAL n ;
reconsider l = 0. ( V ) as Linear_Combination of A ;
set r = |. e .| + |. w .| + |. w .| + |. w .| ;
consider y being Element of S such that z <= y and y in X ;
a 'imp' ( b 'imp' c ) = 'not' ( ( a 'or' b ) 'or' ( a 'or' c ) ) ;
||. ( x1 - x2 ) - ( g - x2 ) .|| < r2 ;
b9 , c9 // b9 , c9 & b9 , c9 // b9 , a9 ;
1 <= k2 - ( k + 1 ) & k + 1 <= ( k + 1 ) - ( k + 1 ) ;
sqrt ( ( p `2 / |. p .| - sn ) ^2 ) >= 0 ;
sqrt ( ( q `2 / |. q .| - sn ) ^2 ) < 0 ;
( E-max C ) `1 in LSeg ( ( R /. 1 ) , ( ( R /. 1 ) `1 ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( ( lim F ) | D ) = Re ( lim G ) ;
LIN b , a , c or LIN b , c , a , c ;
p `1 , a // a , b or p `2 = b ;
g . n = a * Sum ( f | n ) .= f . n * f . n ;
consider f being Subset of X such that e = f and f is \rbrace ;
F | ( [: N2 , S :] ) = ( ( F * F ) | [: the carrier of S , S :] ) | [: the carrier of S :] ;
q in LSeg ( q , v ) \/ LSeg ( v , p ) ;
Ball ( m , r ) c= Ball ( m , s ) ;
the carrier of V = { 0. V } ;
rng ( ( ( - 1 ) (#) ( cos - sin ) ) `| Z ) = [. - 1 , 1 .] ;
assume Re ( seq ^\ k ) is summable & Im ( seq ^\ k ) is summable ;
||. ( ||. vseq . n - vseq . m .|| ) . t - ( ||. vseq . n .|| ) . t .|| < e ;
set g = O --> 1 ;
reconsider t2 = t2 as ( 0 + 1 ) -element string of S2 ;
reconsider xn = seq . ( n + 1 ) as sequence of REAL-NS n ;
assume that that Upper_Arc C meets L~ go and ( E-max C ) `1 <= ( E-max C ) `1 ;
- ( ( Cl 1 - 1 ) (#) f ) . x < F . ( n - 1 ) * f . x ;
set d1 = |. ( ( |. x1 - x2 .| ) ^2 - ( |. x1 - x2 .| ) ^2 - |. x1 - x2 .| ) ;
( 2 |^ ( 1 -' 1 ) ) - 1 = ( 2 |^ 1 ) - 1 ;
dom ( v | ( Seg len ( d + 1 ) ) ) = Seg len ( d | ( Seg len ( d + 1 ) ) ) ;
set x1 = - ( k + 1 ) , x2 = - ( k + 1 ) , x3 = ( k + 1 ) - 1 ;
assume for n being Element of NAT holds 0 <= F . n ;
0 <= ( T . i ) . ( i + 1 ) & ( T . i ) . ( i + 1 ) <= 1 ;
for A being Subset of X holds c . c = c . A
the support of ( L1 + L2 ) c= [: the carrier of L1 , the carrier of L2 :] ;
'not' All ( x , p => q ) => All ( x , p => q ) is valid ;
( f | n + 1 ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the carrier of V ;
Z c= dom ( ( ( ( ( arctan * f1 ) + ( arctan * f1 ) ) `| Z ) ;
|. 0. TOP-REAL 2 - ( TOP-REAL 2 ) ^2 - ( TOP-REAL 2 ) ^2 < r ^2 / 2 ;
ConsecutiveSet2 ( A , succ d ) c= ConsecutiveSet2 ( A , succ d ) ;
E = dom ( L . n ) & E is measurable ;
C |^ ( A + B ) = C |^ B * C |^ C ;
the carrier of W2 c= the carrier of V & the carrier of V c= the carrier of V ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
pred x > 0 means : Def3 : sqrt ( 1 - x ^2 ) = x ^2 ;
LSeg ( f ^ g , i ) = LSeg ( f , k ) ;
consider p being Point of T such that C = [. p , q .[ ;
b , c , d is_collinear & a , b , c is_collinear implies a , b , c , d is_collinear
assume f = id [: O , O :] & f is one-to-one ;
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 :] , the carrier of U2 ;
A1 in the topology of ( ( G . k ) . X ) & A2 . k in G . k ;
|. - x .| = - ( - x ) .= - x .= x ;
set S = MajorityIII( x , y , c , d ) ;
sqrt ( n * sqrt ( n ^2 ) ) >= 4 * sqrt ( n ^2 ) ;
v /. ( k + 1 ) = v . ( k + 1 ) ;
0 mod i = - ( i * ( i qua Nat ) ) ;
Indices M1 = [: Seg n , Seg n :] & Indices M2 = [: Seg n , Seg n :] ;
Line ( SIT , j ) . j = SIT . j ;
h . ( x1 , y1 ) = [ y1 , y2 ] ;
|. f .| + ( |. ( |. b .| * ( ( ( ( ( b - b ) * h ) * ( ( b - b ) * h ) ) ) ) .| is nonnegative ;
assume x = ( a1 ^ b1 ) ^ ( b1 ^ b2 ) ;
M is_halting_on IExec ( I , P , s ) , P ;
DataLoc ( t3 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < - x & |. x - y .| = - x + y ;
LIN c , q , b & LIN c , b , c ;
fjoins . ( 1 , t ) = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + ( y1 + z1 ) ;
not cluster f1 . a = ( f1 . a ) . a & v in InputVertices S ;
( p `2 ) ^2 <= ( ( E-max C ) ^2 + ( E-max C ) ^2 ;
set RE = Cage ( C , n ) .. Cage ( C , n ) ;
( p `2 ) ^2 >= ( ( E-max C ) ^2 + ( E-max C ) ^2 ;
consider p such that p = p1 and s1 < 1 and p1 `1 < 0 and p `2 < 0 ;
|. ( f /* s ) . l - ( f /* s ) . l .| < r ;
Segm ( M , p , q ) = Segm ( M , p , q ) ;
len Line ( N , k + 1 ) = width N ;
f1 /* ( f1 /* ( seq ^\ k ) - f2 /* ( seq ^\ k ) is convergent & f2 /* ( seq ^\ k ) is convergent ;
f . x1 = x1 & f . y1 = y1 & f . y2 = y1 ;
len f <= len f + 1 & len f + 1 <> 0 ;
dom ( Proj ( i , n ) * s ) = [: { m } , REAL :] ;
n = k * ( 2 * t ) + ( 2 * t ) ;
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 , Y :] as Subset of [: Y , Y :] ;
1 in the carrier of [. - 1 , 1 .] & [. - 1 , 1 .] c= [. - 1 , 1 .] ;
let L being complete LATTICE ;
[ gi , gj ] in [: I , I :] \ [: I , I :] ;
set S2 = \frac ( x , y , c ) , S2 = \frac ( x , y , c ) ;
assume f1 is_differentiable_in x0 & f2 is_differentiable_in x0 & f1 . x0 < 0 ;
reconsider y = ( a ` ) / ( a ` ) as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 6 , 8 , 8 , 8
( min ( g , h ) . c ) . c <= h . c ;
set G2 = the empty of G , G2 = the empty Subset of G , e = the Element of G , f = the Subset of G , g = the Subset of G , e = the Subset of G ;
reconsider g = f as PartFunc of REAL , REAL-NS n ;
|. s1 . m - ( s . m ) .| < d / ( p . m - s . m ) ;
for x being element st x in u ( ) holds x in u ( )
P = the carrier of ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( ( TOP-REAL n ) | ( TOP-REAL n ) ) ) ;
assume p1 in LSeg ( p1 , p2 ) /\ LSeg ( p2 , p1 ) ;
( 0. X \ x ) \ ( m + 1 ) = 0. X ;
let g be Element of Hom ( cod f , cod g ) ;
2 * a * b + ( 2 * c ) * d <= 2 * ( 1 + ( 2 * c ) * d ) ;
f , g be PartFunc of the carrier of X , Y , h be PartFunc of X , Y ;
set h = hom ( a , g ) , f = hom ( a , f ) ;
then idseq ( n ) | Seg m = idseq ( m ) ;
H * ( g " * a ) in the carrier of H * ( g * a ) ;
x in dom ( ( ( - 1 / 2 ) * ( ( - 1 ) / 2 ) ) * ( ( - 1 / 2 ) * ( ( - 1 ) / 2 ) ) ;
cell ( G , i1 , j1 -' 1 ) misses C ;
LE q2 , q1 , P & LE q2 , q2 , P & LE q2 , q2 , P ;
attr B is Subset of A means : Def4 : B c= BDD A & B c= BDD A ;
deffunc D ( set , set , set , set ) = union rng ( $2 , $2 ) ;
n + - n < len ( p ^ ( n + 1 ) ) - n ;
attr a <> 0. K means : Def3 : rk ( M ) = rk ( a ) ;
consider j such that j in dom bm and I = len Fm + j ;
consider x1 such that z in x1 and x1 in y1 and y1 in y1 and x1 in y1 ;
for n ex r being Element of REAL st X [ n , r , r ]
set C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2 , i + 1 ) , C1 = Comput ( P2 , s2
set \cal v = 3 -tuples_on { a , b , c , d } , N = { a , b , c } , M = { a , b , c } , N = { b , d } , S = { b , c } , S = { b , d } , S = { b ,
conv ( @ F ) c= union ( ( F .: ( E .: W ) ) ) ;
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) (
s3 <= ( ( s2 - s2 ) * ( 1 - s1 ) * ( 1 - s1 ) ) * ( 1 - s1 ) ;
dom ( f (#) ( f1 + f2 ) ) = dom f /\ dom ( f1 + f2 ) /\ dom ( f2 + f3 ) ;
dom ( f * G ) = dom ( l (#) F ) /\ Seg k .= Seg k /\ Seg k ;
rng ( s ^\ k ) c= dom ( f1 /* s ) \ { x0 } ;
reconsider g1 = gp as Point of TOP-REAL n , g1 = g1 as Point of TOP-REAL n ;
( T * h . s ) . x = T . ( h . ( s . x ) ) ;
I . ( J . ( x , y ) ) = ( I * L ) . ( x , y ) ;
y in dom ( ( the Gij \not ) * ( ( ( Frege A ) . o ) . o ) ;
for I being non degenerated doubleLoopStr , f being Polynomial of I , L holds f is commutative
set s2 = s +* ( intloc 0 , 1 ) , P1 = ( intloc 0 ) .--> 1 ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P1 , s1 , k ) .= P1 . IC Comput ( P1 , s1 , k ) ;
lim ( S1 , a ) in the carrier of [: a , b :] & lim ( S1 , a ) in the carrier of [: a , b :] ;
v . ( l . i ) = ( v *' ( l . i ) . i ;
consider n be element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x and x <> 0 ;
Choose ( X , 0 , 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 , 1 , 0 , 1 , 0 , 1 ,
j + ( 2 * ( k + 1 ) ) + ( 2 * ( k + 1 ) ) > j + ( 2 * ( k + 1 ) ) ;
{ s , t } on on on Q & { s , t } on Q ;
n1 > len ( p2 , n1 , n2 , n2 , n3 , n2 , n3 , n2 , n3 , n4 , n4 , n4 , n4 ) ;
g1 . ( HT ( ( ( ( being Element of L ) , T ) , T ) ) = 0. L ;
then H , H1 , H2 are_fiberwise_equipotent implies card H , H \kern1pt ;
( ( ( N-min L~ f ) .. f ) .. f + ( ( W-min L~ f ) .. f ) .. f > 1 ;
]. s , 1 .[ = ]. s , 1 .[ /\ ]. s , 1 .[ ;
x1 in [#] ( ( TOP-REAL 2 ) | ( L~ g ) | ( L~ g ) ) ;
let f1 , f2 be PartFunc of REAL , REAL , f be PartFunc of REAL , REAL , g be PartFunc of REAL , REAL ;
DigA ( tg1 , z1 ) is Element of k -tuples_on ( the carrier of L ) ;
I . ( d , k1 ) = d & I . ( d , k1 ) = d . ( k1 , k2 ) ;
u `2 = { [ a , u ] , v `2 = [ a , v ] , v = [ a , v ] , w = [ b , w ] ;
( w | p ) | ( p | ( p | p ) ) = p ;
consider v2 such that v2 in W2 and x = v + v2 and v2 in W2 and v1 in W2 + W3 ;
for y st y in rng F ex n st y = a |^ n
dom ( ( g * ( id V ) qua Function ) = K ;
ex x being element st x in ( ( 0. ( U1 \/ U2 ) ) \/ A ) . s ;
ex x being element st x in ( sO \/ A ) . s & x in ( ( O \/ A ) . s ;
f . x in the carrier of [. - r , r .] & f . x in [. - r , s .] ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 } /\ LSeg ( p2 , p1 ) ;
sqrt ( b + ( b-1 ) ^2 ) in { r : a < r & r < b } ;
ex_sup_of { x , y } , L & x "\/" y = sup { x , y } ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of [: G , G :] such that z = y and P [ z , f . z ] ;
( the addF of ( |. vseq .| ) ) . $1 <= e ;
len ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ ( w ^ v ) ) ) ) ) ) = len w + 1 ;
assume q in the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 & q = ( ( TOP-REAL 2 ) | K1 ) | K1 ;
f | [: E , F :] = g | [: E , F :] & f | [: E , F :] = g | [: E , F :] ;
reconsider i1 = x1 , i2 = x2 , j2 = x3 as Element of NAT ;
( a * A ) ^2 = ( a * ( A * B ) ) ^2 ;
assume ex n2 be Element of NAT st f |^ n2 is takmnumber ;
Seg len ( ( Sum ( f2 ) ) | ( len ( ( f1 ^ f2 ) ) | ( Seg len f1 ) ) = dom ( ( f1 ^ f2 ) | ( Seg len f1 ) ) ;
( Complement A1 ) . m c= ( Complement A1 ) . m ;
f1 . p = p1 & g1 . p = p2 & g2 . p = q2 ;
FinS ( F , Y ) = FinS ( F , Y ) ^ ( F , Y ) ;
( x | y ) | z = z | ( y | x ) ;
sqrt ( |. x .| ^2 + n ) <= sqrt ( ( r ^2 + n ^2 ) ^2 ) ;
Sum ( F ) = Sum f & dom ( F . n ) = dom g & dom ( F . n ) = dom f ;
assume for x , y , z being set st x in Y holds x /\ y in Y ;
assume W1 is Subspace of W2 & W2 is Subspace of W3 & W1 is Subspace of W3 implies W1 + W2 is Subspace of W3
||. ( t . x - t . x ) .|| = lim ||. ( x - t . x - t . x ) ;
assume that i in dom D and f | A is lower and g | A is lower ;
sqrt ( ( p `2 ) ^2 + d ^2 ) <= sqrt ( 1 ^2 + d ^2 ) ;
g | Ball ( p , r ) = id ( Ball ( p , r ) ) ;
set NN = ( ( Cage ( C , n ) ) .. Cage ( C , n ) ) .. ( Cage ( C , n ) ) ;
let T be non empty TopSpace ;
width B |-> 0. K = width ( B @ ) .= width ( B @ ) .= width ( B @ ) ;
attr a <> 0 implies ( A +^ B ) +^ a = ( A -- B ) +^ ( A -- C )
then f is_differentiable_in z0 z0 & f is_differentiable_in z0 & f is_differentiable_in z0 , 1 implies pdiff1 ( f , 1 ) is_differentiable_in z0
assume that a > 0 and a <> 1 and b <> 0 and c <> 1 and a <> 0 ;
w1 , w2 , w2 , w1 , w2 is_collinear & w2 , w1 , w2 is_collinear implies w2 , w1 , w2 , w2 is_collinear
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) .= Exec ( I , Comput ( p2 , s2 , k ) ) ;
ind ( T | b ) = ind ( T | b ) .= ind ( T | b ) .= ind ( T | b ) ;
[ a , A ] in Indices Line ( K , 1 ) & [ a , A ] in Indices Line ( K , 1 ) ;
m in ( the Arrows of C ) . ( o1 , o2 ) ;
( ( 'not' a , CompF ( PA , G ) ) . z = TRUE ;
reconsider \varphi = \varphi /. 11 , \varphi = phi /. 2 as Element of z1 ;
len s1 - ( len s2 - 1 ) * ( len s2 - 1 ) > 0 + 1 ;
\delta ( D , f ) . ( sup A ) < r ;
[ f , f1 ] in the carrier of A & [ f , f1 ] in the carrier of B ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = K1 & ( ( TOP-REAL 2 ) | K1 ) /\ K1 <> {} ;
consider z being element such that z in dom g2 and p = g2 . z ;
[#] V1 = { 0. V } .= the carrier of V .= { 0. V } ;
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 x2 - x1 < s and ||. x1 - x0 .|| < s ;
h1 = f ^ ( <* p3 *> ^ <* p3 *> ) .= h ^ ( <* p3 *> ^ <* p3 *> ^ <* p3 *> ) ;
c / [ b , c ] = c / ( [ a , c ] , [ b , c ] ) .= c / ( [ a , c ] , [ b , c ] , [ b , c ] ) ;
reconsider t1 = p1 , t2 = p2 , t2 = p1 as Morphism of C , V ;
sqrt ( 1 - ( 1 - 2 ) * ( 1 - 2 ) ) in the carrier of ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( ( TOP-REAL 2 ) | ( TOP-REAL 2 ) ) ) ) ;
ex W being Subset of X st p in W & W is open & h .: W c= V ;
( h . p1 ) `2 = C * ( ( p1 `2 ) `2 + D ) `2 + D * ( p1 `2 ) `2 ;
R . b = 2 * 2 .= 2 * b .= b * a ;
consider r1 such that B = 1- r1 * ( 1 - r1 ) + r1 * ( 1 - r1 ) and 0 <= r1 ;
dom g = dom ( ( the Sorts of A ) * ( ( the Sorts of A ) * ( the Sorts of A ) ) . o ) ;
[ P . ( l ) , P . ( l + 1 ) ] in => ( T . ( l + 1 ) ) ;
set s2 = Initialize s , P2 = P +* I , P3 = P +* I , s4 = P +* I , s4 = P3 ;
reconsider M = mid ( z , i2 , i1 ) as Matrix of REAL , REAL ;
y in product ( ( Carrier J ) +* ( { 1 , 1 } ) ) ;
1 / ( [ 0 , 1 ] ) = 1 & 0 / ( [ 0 , 1 ] ) = 1 / ( [ 0 , 1 ] ) ;
assume x in the left of g or x in the right of g & x in the LSeg ( g , i ) ;
consider M being strict Subgroup of A such that a = M and T is SubSpace of M and M is SubSpace of M ;
for x st x in Z holds ( ( ( ( ( for x st x in Z holds f + f ) `| Z ) . x ) / ( 1 + x ) ^2 ) <> 0
len ( W1 + ( len W2 ) + m ) = 1 + ( len W1 + m ) ;
reconsider h1 = ( v . n ) - t . n as Lipschitzian from X , Y ;
( i mod len ( p + q ) ) + 1 in dom ( p + q ) ;
assume that s2 is negative and F in the |= of s1 and F in the |= of s2 and F in the |= of s2 ;
( ( ( ( the addF of A ) . ( x , y , z ) ) , 3 ) / ( ( ( ( the addF of A ) . ( x , y , z ) ) , 3 ) ) / ( ( ( ( the addF of A ) . ( x , y , z ) ) / ( 2 * ( x , y , z ) )
for u being element st u in Bags n holds ( p *' m + m ) . u = p . u
for B being Subset of E st B in E holds A = B or A misses B or A misses B
ex a being Point of X st a in A & A /\ Cl { y } = { a } ;
set W2 = [: { p } , { p } :] , C1 = [: p , q :] ;
x in { X where X is Subset of L : X in F } ;
the carrier of W1 /\ W2 c= the carrier of W1 & the carrier of W1 c= the carrier of W2 & the carrier of W1 c= the carrier of W2 ;
( ( 1 + a ) * ( b + a ) ) * ( b + a ) = ( 1 + a ) * ( b + a ) ;
( dom ( X --> f ) ) . x = ( X --> f ) . x ;
set x = the Element of LSeg ( g , n ) /\ LSeg ( g , m ) ;
p => ( q => r ) => ( p => ( p => r ) ) in TAUT ( A ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) , G * ( i2 , k ) ) ;
set PI = LSeg ( G * ( i1 , j ) , G * ( i1 , k ) , G * ( i2 , k ) ) ;
- 1 + 1 <= sqrt ( ( i / 2 |^ n ) - m ) + 1 ;
( reproj ( 1 , z ) ) . x in dom ( f1 (#) ( f1 (#) ( f2 - f1 ) ) ) ;
assume b1 . r = { c1 , c2 } & b2 . r = { c1 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c2 , c1 , c2 , c2 , c2 , c1 , c2 , c2 , c1 , c2 , c2 , c2 , c2 ,
ex P st P on P & P on P & P on P & P on Q ;
reconsider gf = g * f as strict Subgroup of X * f , h = h * f as strict Element of X ;
consider v1 being Element of T such that Q = ( \mathopen { \uparrow ( v1 ) ) ` and v1 in F ;
n in { i where i is Nat : i < n + 1 } ;
( F /. ( i , j ) ) `2 >= ( F /. m ) `2 ;
assume K1 = { p : - 1 <= p & p `2 <= 1 & p `2 <= 1 } ;
ConsecutiveSet2 ( A , succ ( O , succ ( O , succ ( O , O ) ) ) ) = ( ( ConsecutiveSet2 ( A , succ ( O , O ) ) ) ^ ( ( A , O ) ) ^ ( A , O ) ) ;
set I1 = Macro ( a , intloc 0 , intloc 0 ) , i2 = Macro ( a , intloc 0 , intloc 0 ) , i1 = ( a , intloc 0 ) := ( a , intloc 0 ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. 1
X c= ( the carrier of L1 ) \times ( the carrier of L2 ) ;
consider x9 being Element of GF ( p ) such that x9 |^ 2 = a and x9 |^ 2 = a ;
reconsider e1 = e1 , e2 = e1 as Element of D ( ) ;
ex O being set st O in S & C c= O & M c= O & M c= O & M c= O ;
consider n be Nat such that for m be Nat st n <= m holds S . m in U1 ;
f * g is_differentiable_in ( reproj ( i , x ) ) . i ;
defpred P [ Nat ] means A + ( $1 + 1 ) = succ ( A + 1 ) ;
the left of ( - g ) . ( - g ) = the left where g is Element of ( - g ) . ( - g ) ;
reconsider pf1 = x , pf1 = y as Point of TOP-REAL 2 ;
consider g2 such that g2 = y and x <= g2 and g2 in LSeg ( g2 , 0 ) and g2 . x = g2 . ( 0 + 1 ) ;
for n being Element of NAT ex r being Element of REAL st X [ n , r , s ]
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len ( y2 ^ y1 ) .= len ( x2 ^ y2 ) + len ( y2 ^ y1 ) ;
for x being element st x in X holds x in the set of ( the set of K ) . ( n + 1 )
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = {} ;
func RealNormSet ( X ) -> set equals { ( id X ) . ( id X ) ;
len ( ( that that that that that that that that that that that that that ( that that that ( that that ( that ( that ( that ( C /. 1 ) /. 1 ) ) < ( ( C /. 1 ) ) .. ( C /. 1 ) ) and ( C /. 1 ) .. ( C /. 1 ) ) .. ( C /. 1 ) <= ( C /. len ( C /. 1 ) ..
attr K is a \notin K & a <> 0. K implies v . ( a , i ) = i * v . ( a , i ) ;
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 y in X & y in Y & y in Y
IC Comput ( P1 , s1 , k ) in dom ( Comput ( P1 , s1 , k ) ) ;
attr q < s & r < s & s < t implies ]. p , q .[ c= ]. p , s .[ ;
consider c being Element of Class ( f , c ) such that Y = { ( F . c ) . c , c } ;
the ResultSort of S2 = id the carrier' of S2 & the ResultSort of S2 = id the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] , y2 = [ <* z , x *> , f3 ] , z2 = [ <* z , x *> , f3 ] ;
assume x in dom ( ( ( ( ( ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( ln * f ) ) ) `| Z ) ) ) ;
r-7 in Int cell ( GoB f , i , j ) \ cell ( GoB f , i , j ) \ L~ f & ( GoB f ) ` < ( L~ f ) ` ;
( q `2 ) ^2 >= ( ( Cage ( C , n ) ) /. ( i + 1 ) ) ^2 ;
set Y = { a "/\" a : a in X } ;
i - len f <= len f + ( len f - len f ) - len f - len f ;
for n ex x st x in N & x in N1 & h . n = - ( n + 1 )
set s0 = ( ( \mathop { \rm intloc 0 , I , p ) +* Start-At ( 0 , SCM+FSA ) ) . i ;
p . k = 1 or p . ( k + 1 ) = 1 or p . ( k + 1 ) = 1 ;
u + Sum ( L \ { u } ) in ( U \ { u + Sum ( L ) ) \/ { u + Sum ( L ) } ;
consider x9 being set such that x in x9 and x9 in V and x9 in V and x9 in V and x9 in V and x9 in V ;
( p ^ q ) . m = ( q | ( Seg ( len p + 1 ) ) . m ;
g + h = gg + h & h + g = g + h + h ;
L1 is distributive & L1 is distributive implies L1 "\/" L2 is distributive & L1 "\/" L2 is distributive
attr x in rng f & y in rng ( f | X ) implies f | X = f | X ;
assume that 1 < p and p < 1 and p < 1 and 0 <= 1 and p <= 1 and 0 <= 1 and p <= 1 ;
F\circ ( f , t ) = rpoly ( 1 , t ) *' t + ( 1 , t ) *' t ;
let X be set , A be Subset of X , B be Subset of X ;
( ( ( N-min X ) /. 1 ) `1 <= ( ( E-max X ) `1 ) `1 ;
let c being Element of the bound A , a , b be Element of the bound A ;
s1 . intpos ( 0 + 1 ) = ( Exec ( i2 , s1 ) . intpos ( 0 + 1 ) ) . intpos ( 0 + 1 ) .= 0 ;
let a , b be Real , y be Real , x be Element of REAL , y be Real ;
for x , y being Element of X holds x \ y = ( x \ y ) `
mode BCK-algebra of i , j , m , n , m be Nat , i , j be Nat , j be Nat , n be Nat , m be Element of NAT , i , j be Nat ;
set x2 = ( Re ( y . x ) ) ^2 , y2 = ( Im ( y . x ) ^2 ;
[ y , x ] in dom ( u . y ) & u . ( y , x ) = g . y ;
]. lower_bound divset ( D , k ) , upper_bound divset ( D , k ) .[ c= A ;
0 <= \delta ( S . n ) & |. Sum ( S . n ) .| < sqrt ( e / 2 ) ;
( - ( - ( q `2 / |. q .| - sn ) ) ^2 <= ( - ( q `2 / |. q .| - sn ) ) ^2 ;
set A = sqrt 2 ;
for x , y being set st x in field R & y in R holds x , y are_are separated
deffunc F ( Nat ) = b . ( ( M . $1 ) * G . ( M . $1 ) * ( M . $1 ) * ( M . $1 ) ;
for s being element holds s in PreNorms ( f 'or' g ) iff s in |= ( f 'or' g ) \/ ( |= ( f 'or' g ) )
let S being non empty non void non empty holds S is connected iff S is connected
max ( ( ( ( ( z `1 ) ) ^2 + ( z `2 ) ^2 ) , ( z `2 ) ^2 ) >= 0 ;
consider n1 be Nat such that for k holds seq . k < r + s . k ;
Lin ( A /\ B ) is Subspace of Lin ( B ) & Lin ( A ) = Lin ( B ) ;
set n-15 = nnn , M = ( M . ( x qua Element of BOOLEAN ) , x = ( M . ( x qua Element of BOOLEAN ) ) . ( x qua Element of BOOLEAN ) ;
f " ( V ) in [#] ( ( TOP-REAL 2 ) | X ) & f " ( V ) in D ( V ) ;
rng ( ( a , c ) --> ( 1 , b ) ) c= { a , b , c } ;
consider y being subgraph of G1 , e being Vertex of G such that y `1 = y and dom y `2 = WG1 and dom y `2 = WG1 `1 and y `2 = WG1 `1 ;
dom ( ( 1 / f ) (#) ( f | ]. 0 , PI / 2 .[ ) ) c= ]. 0 , PI / 2 .[ ;
an Abelian ( i , j , r , s , n ) is Element of proj ( i , n , r ) ;
v ^ ( ( n |-> 0 ) --> 1 ) in Lin ( ( ( ( B | ( n + 1 ) ) --> 1 ) ;
ex a , k1 , k2 being Nat st i = a := k1 & k2 = b := k2 & k2 = b := k2 ;
t . ( NAT + 1 ) = ( ( NAT --> ( i1 + 1 ) ) . ( NAT + 1 ) ) . ( ( the carrier of S ) \ \lbrace i1 + 1 , i2 ) .= ( ( the carrier of S ) \ \lbrace i1 , i2 , j2 , j2 ) ;
assume F is bbbbbe Subset-Family of X & dom p = Seg ( n + 1 ) & dom p = Seg ( n + 1 ) ;
not LIN b , b , c & not LIN a , b , c ;
( L1 \/ L2 ) over O c= ( L1 \/ L2 ) over O & ( L1 \/ L2 ) . O = ( L1 L1 ) . O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = G ( d ) ;
consider a , b such that a * ( u + w ) = b * ( w + w ) and 0 < a and 0 < b ;
defpred P [ FinSequence of D ] means |. $1 - 1 .| <= Sum ( ( $1 - 1 ) * ( $1 - 1 ) ) ;
u = ( 1 / 2 ) * v + ( 1 / 2 ) * v .= v * u + ( 1 / 2 ) * v .= v * u + ( 1 / 2 ) * v ;
dist ( ( seq . n ) + x , x ) <= dist ( ( seq . n ) . x , x ) + dist ( ( seq . n ) . x , x ) ;
P [ p , |. p .| ] , id ( the carrier of A ) = [ p , id ( the carrier of A ) , id ( the carrier of A ) ] ;
consider X be Subset of [: A ( ) , A ( ) :] such that X c= Y ( ) and X is finite and X is finite and X is finite ;
|. b .| * |. eval ( f , z ) .| >= |. b .| * |. eval ( f , z ) .| ;
1 < ( ( ( Cage ( C , n ) ) .. Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : g <= h & h <= g } ;
( Partial_Sums ( G . n ) . n ) . ( ( Partial_Sums ( G . n ) . m ) . m <= ( Partial_Sums ( G . n ) . m ) . m ;
f . y = x * 1_ L .= x * 1_ L .= ( power L ) . ( ( L ) . y ) * 1_ L .= ( power L ) . ( ( L ) . y ) * ( power L ) . y ;
NIC ( goto i1 , ( goto i2 ) \ { i1 , i2 } ) = { i1 , i2 , i2 , j2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p1 , p2 ) = { p1 } ;
product ( ( Carrier ( I , i ) +* ( i , { 1 } ) ) +* ( i , { 1 } ) ) in [: Z , Z , Z , i , j ) ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s , n ) +* ( the carrier of S2 ) ;
( W-min ( Q ) ) `1 <= ( ( ( q1 ) `1 ) ^2 + ( ( q1 `2 ) ^2 ) ;
f /. ( i2 + 1 ) <> f /. ( i1 + ( i2 + 1 ) ) ;
M , ( ( f . ( x , a ) ) . ( ( ( ( ( ( ( ( x , m ) . m ) . m ) . m ) . m ) . ( ( ( ( ( x , m ) . m ) . m ) . m ) . ( ( ( ( x , m ) . m ) . m ) . m ) ) ;
len ( ( P ^ ( Q ^ ( Q ^ ( P ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^ ( Q ^
A |^ ( m , n ) c= A |^ ( m , n ) & A |^ ( k , n ) c= A |^ ( k , n ) ;
( R |^ n ) \ { q : |. q .| < a } c= a
consider n1 be element such that n1 in dom p1 and y1 = 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 and X c= Z ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA & CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v be VECTOR of l1 holds ||. v .|| = upper_bound rng ( |. v .| ) & ||. v .|| = upper_bound rng ( |. v .| )
for \varphi , \varphi st \varphi in X & phi in X holds not phi in X
rng ( ( Sgm dom ( f | ( dom f ) ) ^ ( ( Sgm ( dom f ) ^ ( ( f | ( dom f ) ) ) ) ) c= dom ( f | ( dom f ) ) ;
ex c being FinSequence of D st len c = k & P [ c ] & a = c ^ ( a ^ c ) ;
the_result_sort_of ( a , b , c ) = <* <* b , c *> , <* b , c *> , <* c , d *> *> , <* b , c *> *> ;
consider f1 be Function of the carrier of X , REAL such that f1 = |. f .| and f1 is continuous and f2 is continuous ;
a1 = b1 & a2 = b2 & a3 = b2 & b1 = b2 & b2 = b1 & b1 = b2 & b2 = b2 & b2 = b1 & b2 = b2 & b1 = b2 & b2 = b1 & b2 = b2 & b1 = b2 & b2 = b2 implies b1 = b2 & b2 = b1 & b2 = b2 & b1 = b2 & b2 = b1 & b2 = b1 & b2 = b2 & b2 = b2 & b1 = b2 & b2 = b2
D2 . indx ( D2 , D1 , n1 ) = D1 . ( n1 + 1 ) ;
f . <* r , s *> = <* r , s *> /. 1 .= <* r , s *> . 1 .= <* r , s *> . 1 .= <* r , s *> . 1 .= <* r , s *> . 1 .= <* r , s *> . 1 .= <* r , s *> . 1 ;
consider n be Nat such that for m be Nat st n <= m holds C . m = C ( m ) ;
consider d be Real such that for a , b be Real st a in X & b in Y holds a <= b ;
||. L /. h - ( L /. h ) .|| <= ||. ( L /. h - L /. h .|| + ||. h .|| ;
attr F is commutative means : Def3 : for b being Element of X holds F . b = f . b ;
p = 1 * ( p1 + 0 ) + 0 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) .= 1 * ( p1 + 0 ) ;
consider z1 such that b , z1 , z2 , z1 is_collinear and o , z1 , z2 is_collinear and o , z1 , z1 is_collinear and o , z1 , z2 is_collinear ;
consider i such that Arg ( ( Rotate ( s , q ) ) . i ) = s + ( 2 * PI * i ) ;
consider g such that g is one-to-one and dom g = card ( f . x ) and rng g = f . x and rng g c= f . x ;
assume A = P2 \/ Q & Q <> {} & P /\ Q <> {} & P /\ Q <> {} implies P /\ Q = {} & P /\ Q = {} & P /\ Q = {} & P /\ Q = {} & P /\ Q = {} & P /\ Q = {} & P /\ Q = {} & P /\ Q = {} ;
attr F is associative means : Def3 : F is associative & F .: ( F .: ( f , g ) ) = F .: ( f .: ( f , g ) ) ;
ex x being Element of NAT st m = x `1 & x in z `1 or m in { i } ;
consider k2 be Nat such that k2 in dom ( P . ( k2 + 1 ) ) and l in dom ( P . ( k2 + 1 ) ) ;
seq = r (#) seq implies for n holds seq . n = r * seq . n
F1 . [ [ a , a ] , [ a , a ] , [ a , a ] ] = [ f * [ a , a ] , [ a , a ] , [ b , a ] , [ b , a ] , [ b , a ] ] , F = [ f * [ a , a ] , F * [ b , a ] , F = [ b , a ] , F = [ b , a ] , G ]
{ p } "\/" D2 = { p "\/" q where q is Element of L : q in D } ;
consider z being element such that z in dom ( ( dom ( F . i ) ) | ( dom F . i ) and ( dom ( F . i ) | ( dom F ) ) . z = y ;
for x , y being element st x in dom f & f . x = f . y holds x = y
cell ( G , i , j ) = { |[ r , s ]| : r <= G * ( 0 , 1 ) `1 & s <= G * ( 1 , 1 ) `2 } ;
consider e being element such that e in dom ( T | ( E . e ) ) and ( T | ( E . e ) ) . e = v ;
( F ' * b1 ) . x = ( Mx2Tran ( J , b1 , b2 ) ) . ( ( ( Mx2Tran ( J , b2 ) ) . b1 ) ;
- ( 1 _ { \mathbb R } ) = ( ( 1 _ { n } ) (#) D ) . n .= ( ( ( 1 , n ) (#) D ) . n ) * ( ( ( 1 , n ) (#) D ) . n ) .= ( ( ( ( 1 , n ) (#) D ) * D ) . n ;
attr x in dom f /\ dom g & g in dom f /\ dom g ;
len ( f1 . j ) = len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) .= len ( f1 . j ) ;
All ( 'not' All ( 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not' 'not'
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl ( ( L~ Cage ( C , n ) ) . ( k + 1 ) ) ;
x \ ( a |^ m ) = x \ ( a |^ k ) .= ( x \ ( a |^ k ) \ a |^ k ) \ a .= x \ a ;
k \hbox { \it th } = ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k .= ( commute I ) . k ;
for s being State of A , n being Nat holds Following ( s , n + 1 ) . ( n + 1 ) is stable ;
for x st x in Z holds f1 . x = a ^2 & ( f1 - f2 ) . x > 0 implies ( f1 - f2 ) . x <> 0
support ( support ( L ) \/ support ( ( support ( L ) ) \/ support ( m ) ) c= support ( ( support ( L ) ) \/ support ( m ) ) \/ support ( m ) ) ;
reconsider t = u as Function of ( the carrier of A ) , the carrier of B ( ) * , the carrier of C ( ) ;
- ( a * sqrt ( 1 + ( a * b ^2 + b ^2 ) ) ) <= - ( a * sqrt ( 1 + ( b ^2 + b ^2 ) ) ;
\varphi . ( succ b1 ) = g . a & \varphi . ( a , a ) = f . ( g . a , g . a ) ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) and i = len ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 7 } } = { x1 , x2 , x3 , x4 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
the Sorts of U1 /\ ( U1 "\/" U2 ) c= the Sorts of U1 & the Sorts of U1 /\ ( U2 "\/" U2 ) c= the Sorts of U2 ;
( - ( 2 * a * b + b * c ) ) / ( 2 * a * b + b * c ) > 0 ;
consider W such that for z being element holds z in [: N , N :] iff z in [: N , I :] & P [ z , W , I , J , K , f , f , g , h , h , h , h , h , h , i , f , g , h , h , h , i , i , f , g , h , h , i , i , f , g , h , h , i , h ,
assume ( the Arity of S ) . o = <* a , b *> & ( the ResultSort of S ) . o = r ;
Z = dom ( ( ( arctan + arccot ) (#) ( ( arctan + arccot ) (#) ( arctan + arccot ) ) ) /\ dom ( ( arctan + arccot ) (#) ( arctan + arccot ) ) ;
integral ( f , S ) is convergent & lim ( f , S ) = integral ( f , S ) ;
X . ( a => f ) => ( ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) => ( 'not' f ) in TAUT ( X ) ;
len ( ( - M2 ) * ( M @ ) ) = n & width ( - M ) = n & width ( - M ) = n ;
attr X1 union X2 is open means : Def3 : X1 is SubSpace of X1 & X1 is SubSpace of X2 & X1 is SubSpace of X1 & X2 is SubSpace of X1 implies X1 union X2 is SubSpace of X2 & X1 union X2 is SubSpace of X1 & X1 union X2 is SubSpace of X2 ;
let L being lower-bounded antisymmetric antisymmetric non empty reflexive transitive antisymmetric antisymmetric antisymmetric transitive antisymmetric RelStr , X be Subset of L ;
reconsider f29 = F . ( F . ( b , c ) ) as Function of [: M , M :] , M . ( b , c ) , M . ( b , c ) as Function of M , M . ( b , c ) , M . ( b , c ) ;
consider w being FinSequence of I such that the initial { s } ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ q ^ w ^ q ^ w ^ w ^ q ^ w ^ w ^ w ^ q ^ w ^ w ^ w ^ q ^ w ^ w ^ w ^ q ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ q ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w ^ w
g . ( a |^ 0 ) = g . ( 1_ G ) .= g . ( 1_ G ) |^ ( a |^ 0 ) .= g . ( 1_ G ) |^ ( a |^ 0 ) .= g . ( 1_ 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 K being Subset of X st K in L holds K /\ L <> {} ;
( the carrier of C1 ) /\ ( the carrier of C2 ) c= the carrier of C1 /\ C2 ;
reconsider o9 = o `1 , p = p `2 as Element of TS ( ( the Sorts of A ) . v , r = r as Element of ( the Sorts of A ) . v ;
1 * ( ( 0 * ( x + h ) + 0 * ( x + h ) ) + 0 ) = x1 + 0 * 0 .= x1 + 0 * 0 .= 0 ;
E " ( 1 - 1 ) = ( E qua Function ) . 1 .= ( E qua Function ) . 1 .= E . 1 ;
reconsider u1 = the carrier of U1 /\ ( U1 "\/" U2 ) as non empty Subset of ( U1 /\ U2 ) ;
( x "/\" z ) "\/" ( x "/\" z ) <= ( x "/\" ( x "/\" z ) "\/" ( x "/\" z ) ;
|. f . ( ( l + 1 ) - 1 ) .| < sqrt ( 1 - sqrt ( 1 - ( M . l ) ) ^2 ) ;
LSeg ( ( Cage ( C , n ) ) * ( i , j ) , ( Gauge ( C , n ) ) * ( i + 1 ) ) is vertical ;
( f | Z ) /. x = L /. ( - ( f /. x ) / ( f /. x ) ^2 + R /. ( - x ) ^2 .= L /. ( - x ) / ( f /. x ) ^2 + R /. ( - x ) ^2 ;
g . c * ( f . c ) * f . c <= h . c * ( f . c ) * f . c ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) + g | divset ( D , i ) ;
assume that ColVec2Mx ( f ) in the carrier of A and ColVec2Mx ( f ) = space ( A ) and width ( f ) = width A and width ( f ) = width A and width ( f ) = width A ;
len ( - ( - ( - ( - ( - ( - ( - ( - ( - ( - - - ( - ( - - ( - - - ( - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - - ( - ( - - ( - ( - - - ( - - ( - - ( - - - ( - - ( - - - ( - - ( - - - - ( - - - ( - - ( - -
let n , i be Nat , n be Nat , i be Nat , n be Element of NAT , x be Element of ( the carrier of G ) . i ;
pdiff1 ( f1 , 2 ) is_partial_differentiable_in z , 1 & pdiff1 ( f2 , 2 ) . 1 = { x0 } & pdiff1 ( f2 , 2 ) . 1 = { x0 } ;
attr a <> 0 & b <> 0 & Arg ( a - b ) = Arg ( a - b ) implies Arg ( a - b ) = Arg ( a - b ) & Arg ( a - b ) = Arg ( a - b ) & Arg ( a - b ) = Arg ( a - b ) ;
for c being set st c in [. a , b .[ holds not c in Intersection ( ( the carrier of a , b ) , ( the carrier of b , c ) )
assume V1 is linearly closed & V1 in { v + u where v , u is Element of V : v in V1 & u in V1 & v in V1 & u in V1 & v in V1 + V2 & u in V1 + V2 & v in V1 + V2 ;
z * ( x1 + 1 ) * ( y1 + 1 ) * ( x1 + 1 ) in M & z * ( y1 + 1 ) * ( y1 + 1 ) * ( y1 + 1 ) * ( y1 + 1 ) * ( y1 + 1 ) in N ;
rng ( ( ( P qua Function ) * ( S qua Function ) ) = Seg ( card ( ( P qua Function ) * ( S qua Function ) ) ) .= ( ( P * ( S qua Function ) ) * ( S * ( S qua Function ) ) ) .= ( ( P * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( S * ( T *
consider s2 be Real_Sequence such that s2 is convergent and b = lim s2 and for n holds s2 . n <= lim s2 ;
h2 " . n = ( h2 " ) . n & 0 < ( ( h " ) . n ) " & 0 < ( h " ) . n ;
( Partial_Sums ( |. ( r .| ) | ( X ) ) . m = |. ( ( r .| ) | ( X ) ) . m - ( |. r .| ) . m .| .= |. ( r .| ) . m - ( r .| ) . m - ( r .| ) . m .= 0 ;
( Comput ( P1 , s1 , 1 ) ) . b = 0 .= ( Comput ( P1 , s1 , 1 ) ) . b .= Comput ( P1 , s1 , 1 ) . b .= Comput ( P1 , s1 , 1 ) . b ;
- v = ( - 1 ) * v & - v * w = - 1 * v + 1 * v & - 1 * v = 1 * v + 1 * v & 1 - 1 * v = 1 * v + 1 * v ;
sup ( ( k .: D ) .: ( k , D ) ) = sup ( ( k .: D ) .: ( k , D ) ) .= sup ( k .: ( k .: D ) ) .= sup ( k .: ( k .: D ) ) ;
A |^ ( k , l ) |^ ( n + 1 ) = ( A |^ ( k + 1 ) ) |^ ( k + 1 ) .= ( A |^ ( k + 1 ) ) |^ ( k + 1 ) ;
let R being add-associative right_zeroed right_complementable associative associative non empty doubleLoopStr , I , J be Subset of R ;
( f . p ) `1 = sqrt ( ( p `1 ) ^2 + ( p `2 ) ^2 ) .= sqrt ( ( p `2 ) ^2 + ( p `2 ) ^2 ) ;
let a , b be non zero Nat , n be Nat , a be Element of NAT , b be Element of NAT ;
consider A5 being set such that r is countable and r is Element of [: Al ( ) , D ( ) :] and ( ex i being Nat st i in Seg n & A ( ) ) = { i } ;
for X being non empty addLoopStr , M being Subset of X , x being Point of X st x in M holds x + M in M + M
{ [ 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 , \Vert , x5 , x5 , \Vert , x5 , \Vert , x5 , \Vert , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
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 * ( f . O
( Gauge ( C , n ) * ( i , j ) ) /. k in L~ Cage ( C , n ) /\ L~ Cage ( C , n ) ;
cluster m gcd n -> prime for Nat ;
( f * F ) . x1 = f . x1 & ( f * F ) . x2 = f . x2 ;
let L be lattice , a , b , c be Element of L , a , b be Element of L holds a \ b <= b implies a <= b
consider b being element such that b in dom ( H / ( { x } ( { y } ) ) ) and z = H / ( { x } ) ;
assume that x in dom ( F * g ) and y in dom ( F * g ) and z in dom ( F * g ) and ( F * g ) . x = ( F * g ) . y ;
assume ex e being element st e Joins W . 1 , G & e in G . ( e + 1 ) & e in G . ( e + 1 ) ;
( ( ( ( f (#) h ) (#) ( h + c ) ) (#) ( f + c ) ) . n = ( ( ( f (#) h ) (#) ( h + c ) ) (#) ( h + c ) ) . n ;
j + 1 = j + ( len ( h1 ^ h2 ) - 1 ) .= i + ( len ( h1 ^ h2 ) - 1 .= i + ( len h1 + 1 ) - 1 .= i + ( len h1 + 1 ) - 1 ;
( S *' S ) . f = S *' ( S *' ( f , T ) ) .= S *' ( f , T ) .= S *' ( f , T ) ;
consider H such that H is one-to-one and rng H = Carrier ( L ) and Sum ( L ) = Sum ( L ) and Sum ( L ) = Sum ( L ) and Sum ( L ) = Sum ( L ) ;
attr R is maximal_arc means : Def3 : for p , q st p in R & q in R & p in R holds p , q // q , r ;
dom ( ( X --> f ) +* ( X --> f ) ) = meet ( ( X --> f ) +* ( X --> f ) ) .= meet ( ( X --> f ) +* ( X --> f ) ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom ( X --> f ) .= dom f /\ dom f ;
upper_bound ( ( proj2 .: ( Upper_Arc C /\ Upper_Arc C ) /\ Vertical_Line ( w ) ) <= upper_bound ( ( proj2 .: ( w /\ w ) /\ Vertical_Line ( w ) ) ;
for r be Real st 0 < r ex n be Nat st for m be Nat st n <= m holds |. S . m - 0 .| < r
i * ( f - h ) = i * ( f - h ) .= i * ( f - h ) .= i * ( f - h ) .= i * ( f - h ) .= i * ( f - h ) ;
consider f being Function of X , Y such that dom f = 2 -tuples_on X and for Y being set st Y in 2 -tuples_on X holds f . Y = F ( Y ) ;
consider g1 , g2 being element such that g1 in [#] ( Y | C ) and g2 in ( X /\ Y ) and g = [ g1 , g2 ] ;
func d \! \mathop { n } -> Nat means : Def3 : for n being Nat holds it . n = n & it . n = n + 1 & it . n = 1 ;
fmax . [ 0 , t ] = f . [ 0 , t ] .= ( f . [ 0 , t ] ) . [ 0 , t ] .= ( f . [ 0 , t ] ) . [ 0 , t ] .= ( f . 0 , t ) . [ 1 , t ] ;
t = h . D or t = h . E or t = h . E ;
consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( seq . n ) - ( seq . n ) ) < 1 / ( 1 / ( n + 1 ) ) ;
sqrt ( ( q `2 / |. q .| - sn ) ^2 ) <= sqrt ( ( q `2 / |. q .| - sn ) ^2 ) ^2 ;
h . ( i + 1 ) = h . ( i + 1 ) .= h . ( i + 1 ) ;
consider o being Element of S , x1 being Element of { [ o , x2 ] } such that a = [ o , x1 , x2 ] and not contradiction and not contradiction ;
let L be RelStr , a , b be Element of L , x be Element of L , y be Element of L ;
||. h1 . n - h1 . n .|| = ||. h1 . n - h1 . n .|| .= ||. h1 . n - h1 . n .|| .= ||. h1 . n - h1 . n .|| .= ||. h1 . n - h1 . n .|| .= ||. h1 . n - h1 . n .|| ;
( ( - ( ( f + g ) (#) f ) `| Z ) . x = f . x - ( - ( f + g ) (#) f ) . x .= ( - ( f + g ) (#) f ) . x - ( - ( f + g ) (#) f ) . x .= - ( f + g ) . x ;
attr r = F ^ ( p , q ) means : Def3 : len r = len ( p ^ q ) & len r = len ( p ^ q ) ;
sqrt ( r ^2 + ( r ^2 + ( r ^2 + ( r ^2 ) ) ^2 ) <= sqrt ( r ^2 + ( r ^2 ) ^2 ) ;
let i being Nat , M be Matrix of n , K , K be Matrix of n , K ;
then a <> 0. R & a * ( a * v ) = 1 * v & a * v = 1 * v ;
p . ( j -' 1 ) * ( q . ( j + 1 ) ) = Sum ( p . ( j + 1 ) ) * ( q . ( j + 1 ) ) ;
deffunc F ( Nat ) = L . 1 + ( R /* ( h ^\ n ) ) . $1 * ( R /* ( h ^\ n ) ) . $1 ;
assume the carrier of H = f .: ( the carrier of H1 ) & the carrier of H = f .: ( the carrier of H2 ) & the carrier of H = f .: ( the carrier of H2 ) ;
Args ( o , Free ( X , X ) ) = ( ( the Sorts of Free ( S , X ) ) * ( the Arity of S ) ;
H1 = n + 1 .= n + ( 2 to_power ( n + 1 ) ) .= n + ( 2 to_power ( n + 1 ) ) .= n + 1 .= n + 1 ;
( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . O ) ) ) ) ) ) ) ) ) . ( ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O . ( O .
F1 .: ( dom ( F1 /\ F2 ) ) = ( F1 .: ( { f . n } ) .= { f . ( n + 1 ) } .= { f . ( n + 1 ) } ;
attr b <> 0 & d <> 0 & b <> 0 & d <> 0 implies sqrt ( b / a ) = sqrt ( b / sqrt ( b / sqrt ( b / sqrt ( b / sqrt ( b ^2 ) ) ^2 ) ) ;
dom ( f +* g +* h ) = dom ( f +* g +* h ) /\ D .= dom ( f +* g +* h ) /\ D .= D /\ D .= D /\ D ;
for i being set st i in dom g ex a being Element of L st a = g * v & a in B * v
g `2 * P `2 = g `2 * ( g `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) * ( g `2 * P `2 ) .= g `2 * ( g `2 * P `2 ) ;
consider i , s1 such that f . i = s1 & not ( not thesis & s1 in LSeg ( s1 , i ) & not s1 in LSeg ( s1 , i ) & not s1 in LSeg ( s1 , i ) ) ;
h | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ .= g | ]. a , b .[ .= g | ]. a , b .[ ;
[ s1 , t1 ] , [ s2 , t2 ] , [ s2 , t2 ] , [ s2 , t2 ] , [ s2 , t2 ] , [ s2 , t2 ] , [ s2 , t2 ] ] , [ s2 , t2 ] , [ s2 , t2 ] ] , [ s2 , t2 ] , [ s2 , t2 ] ] , [ s2 , t2 ] ] ] , [ s2 , t2 ] ] , [ s2 , t2 ] ] } , [ s2 , t2 ] ] is connected ;
then H is negative & H is negative & H is negative & H is negative implies H is negative & H is negative & H is negative & H is negative & H is negative & H is negative ;
attr f1 is total means : Def3 : f1 is total & f1 is total & f2 is total & f1 is total & f1 is total implies f1 + f2 is total & f1 + f2 is total & f1 + f2 is total & f1 + f2 is total ;
z1 in [: [: { z1 , z2 } , { z2 } :] or z1 in [: { z1 , z2 } , { z2 } :] & z2 in [: { z1 , z2 } , { z2 } :] & z1 in [: { z1 , z2 } & z2 in [: { z1 , z2 } , { z2 } :] ;
p = 1 * p .= a " * a * p .= a " * p " * q .= a " * ( b * q ) .= a " * ( b * p ) ;
for rseq be Real_Sequence , K be Nat st for n be Nat holds K . n <= K . n holds upper_bound rng K <= K . n
<* E-max ( C ) , E-max ( C ) *> meets L~ Cage ( C , n ) or LSeg ( \mathfrak o , W-min ( C ) ) /\ L~ Cage ( C , n ) c= { W-min ( C , n ) } ;
||. f . ( g . ( k + 1 ) ) - f . ( g . ( k + 1 ) .|| <= ||. g . ( k + 1 ) .|| * ||. ( g . ( k + 1 ) - g . ( k + 1 ) ) .|| ;
assume h = ( ( B .--> C ) +* ( D .--> E ) +* ( E .--> F ) +* ( F .--> F ) +* ( F .--> J ) +* ( F .--> F ) +* ( F .--> J ) +* ( F .--> F ) +* ( F .--> E ) +* ( F .--> J ) +* ( F .--> F ) +* ( F .--> J ) ;
|. ( ( ( Carrier ( H . n ) ) . k ) - ( ( ( H . n ) . k ) - ( ( H . n ) . k ) .| <= e * ( e * ( H . n ) ) - ( e * ( H . n ) ) ;
( ( the Sorts of Free ( S , X ) ) . v ) . e = [ [ ( the Sorts of Free ( S , X ) ) . v , ( the Sorts of Free ( S , X ) ) . e , ( the Sorts of Free ( S , X ) ) . e ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , 8 , x5 , 8 , 8 , 8 , x5 , 8 , x5 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 } } = { x1 , x2 , x3 , x4 , x5 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 ,
assume that A = [. 0 , PI / 2 .[ and f is_integrable_on A and bounded and f is_integrable_on A and f | A is bounded ;
p `2 is Permutation of dom ( f1 /. i ) & p `2 = ( ( Sgm Y ) . i ) `2 ;
for x , y , z being Real st x in A & y in A holds |. ( 1 - f . x ) - f . y .| <= 1 * |. f . x - f . y .|
( ( p2 `2 ) ^2 - ( - ( 1 - sn ) ) ^2 = |. ( ( p2 `2 / |. p2 .| - sn ) ) ^2 - ( ( - sn ) ^2 ) * ( 1 - sn ) ) ^2 - sn ) ;
let f be PartFunc of the carrier of C , REAL , g be PartFunc of C , REAL ;
assume for x being Element of Y st x in EqClass ( z , CompF ( B , G ) ) holds ( 'not' Ex ( u , G ) ) . x = TRUE ;
consider F3 such that dom F = n1 and for k being Nat st k in dom F holds P [ k , F . k , F . ( k + 1 ) ] ;
ex u , u1 st u <> u1 & u1 , v1 , u1 , v1 , v2 is_collinear & u , u1 , v1 is_collinear & u1 , v1 , v2 is_collinear & u1 , v1 , v2 is_collinear & u1 , v1 , v2 is_collinear & u1 , u1 , v1 is_collinear & u1 , v1 , v2 is_collinear & u1 , v1 , v2 is_collinear & u1 , v1 , v2 is_collinear ;
let G be Group , A , B be Subset of G , N be normal Subgroup of G , A be Subset of G holds ( N ` ) ` = N ` * ( N ` )
for s be Real st s in dom F holds F . s = \int ( R ^ ( f + g ) ) . s
width ( ( ( f1 + f2 ) (#) ( f1 + f2 ) ) = len ( ( f1 + f2 ) (#) ( f2 + g2 ) ) .= len ( ( f1 + f2 ) (#) ( f1 + f2 ) ) .= len ( ( f1 + f2 ) (#) ( f2 + g2 ) ) .= len ( ( f1 + f2 ) (#) ( f1 + f2 ) ) .= len ( ( f1 + f2 ) (#) ( f1 + f2 ) ) ;
f | ]. - PI / 2 , 0 .[ = f & f | ]. - PI / 2 , 0 .[ = f | ]. - PI / 2 , 0 .[ ;
assume that X is closed and a in X and a in X and a in X and a in X and b in X and a in X and x in X and y in X and x in X and y in X and x in X and y in X and x in X and y in X and x in X and y in X and x in X and y in X and x in X ;
Z = dom ( ( ( ( ( ( ( ( - arctan ) * ( arctan + arccot ) ) + ( ( - arctan ) * ( arctan + arccot ) ) + ( ( - arctan ) * ( arctan + arccot ) ) ) `| Z ) ) ;
func [: V , l :] -> Subset of V equals { l . k where l is Nat : 1 <= l & l <= len l & l . l = l . ( k + 1 ) } ;
let L be non empty reflexive transitive antisymmetric reflexive transitive antisymmetric reflexive RelStr , M be Subset of L , N be net of L , M be Subset of L , f being Function of N , M holds f . ( f . ( f . ( g . g ) ) = f . ( g . ( f . g ) )
for s being Element of NAT holds ( ( ( ( id Creal ( V ) ) + ( ( Creal ( V ) ) + ( Creal ( V ) ) ) . s ) . s = ( ( ( ( ||. ( V .|| ) + ( ||. V .|| ) + ( ||. V .|| ) ) . s ) . s ) . ( ( ||. V .|| ) . s ) . ( ( ||. V .|| + ( ||. V .|| ) . s )
then z /. 1 = ( W-min L~ z ) .. z & ( W-min L~ z ) .. z < ( W-min L~ z ) .. z ;
len ( p ^ <* 0 qua Real *> ) = len p + len <* 0 qua Real *> .= len p + 1 .= len p + 1 ;
assume that Z c= dom ( ( - 1 / ( ( ln * f ) ) * f ) and for x st x in Z holds f . x = 1 / ( x + a ) ^2 ) and f . x > 0 ;
let R being add-associative right_zeroed right_complementable associative associative associative non empty doubleLoopStr , I , J be Subset of R , I be Subset of R ;
consider f being Function of [: B1 , B2 :] , B2 such that for x being Element of [: B1 , B2 :] holds f . x = F ( x ) ;
dom ( ( x2 + y2 ) ^ ( y1 + y2 ) ) = Seg len ( x ^ y1 ) .= Seg len ( x ^ y1 ) .= Seg len ( x ^ y1 ) .= Seg len ( x ^ y1 ) ;
for S being transitive functor of C , B being D , C being D of E holds ( id C ) . ( id C ) = id ( ( the carrier of C ) . ( the carrier of C ) )
ex a st a = a2 & a in [: f , g :] /\ [: f , g :] & { f . a , g . a , h . b } = { f . a , g . b } ;
a in Free ( H , ( ( ( ( ( ( ( H , ( ( ( ( ( ( ( ( H , H ) / ( ( H , H ) . ( ( ( H , H ) . m ) ) . m ) ) . m ) ) ) ) ) ) ;
let C1 , C2 be C2 , f being stable Function of C1 , C2 , g be stable Function of C1 , C2 holds f = g iff f = g
( W-min L~ go \/ L~ pion1 ) /\ ( L~ pion1 \/ L~ pion1 ) = ( L~ go \/ L~ pion1 ) /\ ( L~ pion1 \/ L~ pion1 ) ;
consider u , x0 , y0 , z0 , z0 being Real such that u = <* x0 , y0 , z0 , z0 , z0 *> and f . 3 = SVF1 ( 3 , f , z0 ) . 1 and u <> 0 ;
then ( t . {} ) `1 in Vars & ( t . {} ) `2 in Vars implies ex x being Element of C st x in Vars & t . {} = [ x , s ] & t . {} = [ x , s ] ;
( 'not' p '&' p ) . v = ( 'not' p ) . v '&' ( 'not' p ) . v .= ( 'not' p ) . v ;
assume for x , y being Element of S st x <= y & y in T . x holds a >= f . y ;
func Class ( R , a ) -> Subset-Family of R means : Def3 : for A being Subset of R holds it = Class ( R , a ) ;
defpred P [ Nat ] means ( ( the Partial_Sums of G ) . $1 c= G . ( the carrier of G ) & ( the Partial_Sums of G ) . $1 c= G . ( the carrier of G ) ;
assume that dim ( U1 ) = 0 and dim ( U2 ) = 0 implies dim ( U1 ) = 0 & dim ( U2 ) = 1 & dim ( U2 ) = 0 & dim ( U2 ) = 1 ;
not ( ex m st m in dom ( m . t ) & ( m in dom ( m . t ) implies m = ( m . t ) . t ) . {} ) ;
d = ( ( x ^ y ) ^ ( y ^ d ) ) . ( ( y ^ d ) . ( y ^ d ) ) .= f . ( ( y ^ d ) ^ d ) . ( y ^ d ) .= f . ( ( y ^ d ) ^ d ) . ( y ^ d ) .= f . ( ( y ^ d ) ^ d ) . ( y ^ d ) ;
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 ) |^ ( len x ) = x + ( x + ( x + ( len x ) ) .= x + ( x + ( x + ( x + ( len x ) ) ) .= x + ( x + ( x + ( len x ) ) ) .= x + ( x + ( x + ( x + ( len x ) ) ) .= x + ( x + ( x + ( len x ) ) ) ;
k9 - ( ( f /. ( k + 1 ) ) + ( f /. ( k + 1 ) ) in dom ( f /. ( k + 1 ) ) ;
assume P1 is_an_arc_of p1 , p2 & P2 is_an_arc_of p1 , p2 & P1 is_an_arc_of p2 , p1 & P c= LSeg ( p1 , p2 ) & P /\ Q = { p1 } & P /\ Q = { p1 } & P /\ Q = { p1 } & P /\ Q = { p1 } & P /\ Q = { p1 } ;
reconsider a1 = a , b1 = b , c1 = c , c2 = b , c2 = c , c2 = d , c2 = d , c2 = b , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = c , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , c2 = d , 6 = d d d , 6 = d d d = d d , c1 = c , c2 = d d d d = d d , c2 = d , c2 = d d d = c , c2 = d d , c2 = d d d d = d d = d d , c2 = d d , c1 = d d , c2 = d d , c1 = d d
reconsider sf = G1 . ( t , f . t ) as Morphism of ( G1 * F . t ) , ( G1 * F ) . t as Morphism of ( G1 * F ) . t , ( G1 * F ) . t ;
LSeg ( f , i + 1 ) = LSeg ( f /. ( i + 1 ) , f /. ( i + 1 ) ) ;
\int P . m , ( P . m ) -( P . m ) . n , ( P . m ) . n + ( P . m ) . n , ( P . m ) . n + ( P . m ) . n + ( P . m ) . n + ( P . m ) . n <= ( P . m ) . m + ( P . m ) . n ;
assume that dom f1 = dom f2 and for x being element st x in dom f1 holds f1 . x = f2 . x and f2 . x = F ( x , y ) ;
consider v such that v = y and dist ( u , v ) < min ( ( ( G * ( i , 1 ) + 1 ) ) ;
let G be Group , H be Subgroup of G , a , b be Element of G , i be Element of NAT holds a |^ i = b |^ ( i + 1 ) * a & b |^ ( i + 1 ) = b |^ ( i + 1 ) * a |^ ( i + 1 )
consider B being Function of Seg ( S + L ) , the carrier of V such that for x being element st x in Seg ( S + L ) holds P [ x , B . x , L . x ] ;
reconsider K1 = { p1 where p1 is Point of TOP-REAL 2 : p1 `1 <= sn & p1 `2 <= 0 } as Subset of ( TOP-REAL 2 ) | P ;
sqrt ( ( ( ( TOP-REAL 2 ) | ( L~ Cage ( C , m ) ) ) ^2 - ( ( ( TOP-REAL 2 ) | ( L~ Cage ( C , m ) ) ) ^2 - ( ( ( TOP-REAL 2 ) | ( L~ Cage ( C , m ) ) ) ^2 ) ) ^2 ) <= sqrt ( ( ( ( TOP-REAL 2 ) `2 ) ^2 - ( ( TOP-REAL 2 ) ^2 ) ^2 ) - ( ( TOP-REAL 2 ) ^2 ) ) ^2 - ( ( TOP-REAL 2 ) ^2 ) ;
for x be Element of X , n be Nat st x in E holds |. ( Re F . n ) . x .| <= P . x & |. ( Im F . n ) . x .| <= P . x
len ( F ^ <* 2 *> ^ q ) = len ( ( F ^ <* 2 *> ^ q ) + len ( F ^ <* 2 *> ) .= len ( ( F ^ <* 2 *> ) + q ) + 1 .= len ( F ^ <* 2 *> ) + 1 .= len ( F ^ <* 2 *> ) + 1 ;
v / ( ( x , m ) / ( x , m ) ) / ( ( x , m ) / ( x , m ) ) / ( x , m ) / ( x , m ) = 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 , ( ( { x } \leftarrow ( { x } ) ) . m ) |= r ;
func w1 \ ( w , w2 ) -> Element of ( Union G ) . ( ( the carrier of G ) \ { w } ) \/ ( { w } ) \/ ( { w } ) ;
s2 . b2 = ( Exec ( n2 , s1 ) ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 .= Exec ( n2 , s2 ) . b2 ;
for n being Nat , k be Nat holds 0 <= ( Partial_Sums ( |. seq .| ) . n ) . k
set F = S \! \mathop { {} } ;
( Partial_Sums ( seq ) . ( n + 1 ) ) . ( n + 1 ) >= ( Partial_Sums ( seq ) . n + 1 ) . ( n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x = L . ( - R . x ) + R . ( - R . x ) ;
the closed subset of Closed-Interval-TSpace ( a , b , c , d ) = ( the carrier of Closed-Interval-TSpace ( a , b , c , d ) ) ` .= ( the carrier of Closed-Interval-TSpace ( a , b , c ) ) ` ;
a * b ^2 + ( a * c ) ^2 + ( a * b ) ^2 >= a * b * c + ( a * c ) ^2 ;
v / ( x1 , m ) / ( x2 , m ) = v / ( x2 , m ) / ( x1 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m ) / ( x2 , m , m ) / ( x2 , m )
Ex ( Q ^ <* x *> , M ^ <* x *> ) = ( Ex ( Q , M , 1 , 0 , 1 , 0 ) +* ( ( Q ^ <* x *> , M ^ <* x *> , M ^ <* x *> ) ) +* ( ( Q ^ <* x *> , M ^ <* x *> ) ) +* ( ( Q ^ <* x *> , M ^ <* x *> ) ) .= ( ( ( Q ^ <* x *> ^ <* x *> ) --> ( ( ( ( <* x *> ) --> ( ( ( <* x *> ) --> ( ( <* x *> ) --> ( ( ( Q ^ <* x *> ) --> ( ( <* x *> ) --> TRUE ) ) +* ( ( <* x *>
Partial_Sums ( F . n ) = r |^ ( n + 1 ) .= ( r |^ ( n + 1 ) ) * ( ( r |^ ( n + 1 ) ) .= ( r |^ ( n + 1 ) ) * ( ( r |^ ( n + 1 ) ) .= ( r |^ ( n + 1 ) ) * ( r |^ ( n + 1 ) ) .= ( r |^ ( n + 1 ) ) * ( r |^ ( n + 1 ) ) ;
( ( GoB f ) * ( len GoB f , 1 ) + ( GoB f ) * ( len GoB f , 1 ) ) `1 = ( ( GoB f ) * ( 1 , 1 ) ) `1 + ( ( GoB f ) * ( 1 , 1 ) ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums s ) . $1 = ( ( a * ( $1 + 1 ) + 1 ) * ( ( a * ( $1 + 1 ) ) + 1 ) * ( ( a * ( $1 + 1 ) ) + 1 ) * ( a * ( $1 + 1 ) ) ;
the_result_sort_of ( g . I ) = ( the Arity of S ) . I .= ( [ g , the carrier' of S ] ) . I .= [ g , the carrier' of S ] ;
( X \times Y ) \/ ( X \/ Y ) c= X \/ Y & card ( X \/ Y ) = card ( X \/ Y ) ;
for a , b being Element of S , s being Element of F st s = n & a = F . n holds b = F ( n + 1 ) \ G ( n + 1 )
E , f |= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( of of of , m ) ) ) ) | ( ( ( m , m ) ) | ( ( ( m , m ) ) | ( ( ( m , m ) ) | ( ( ( m , m ) ) | ( ( ( m , m ) ) | ( ( ( m , m ) ) | ( ( m , m ) ) ) | ( ( ( m , m ) ) | ( ( ( ( m , m ) ) | ( ( ( ( m , m ) ) | ( ( m , m ) ) | ( ( m , m ) ) | ( ( m , m ) ) |
ex R being 1-sorted st R = ( p | ( n + 1 ) ) & ( ( p | ( n + 1 ) ) . i = ( the carrier of R ) . i ) . i ;
[. a , b + sqrt ( 1 + k + 1 ) , b + sqrt ( 1 + k ) .] is Element of the qua set , a , b , c be Element of the set , f , g be Function of the carrier , REAL ;
Comput ( P , s , 2 + 1 ) = Exec ( P . ( 2 + 1 ) , Comput ( P , s , 2 ) ) .= Exec ( goto ( 2 + 1 ) , Comput ( P , s , 2 ) ) .= Exec ( goto 2 , Comput ( P , s , 2 ) ) ;
( ( h1 . k ) . k ) . k = ( power ( L ) . k ) . k .= ( ( power ( L ) ) . k ) . k .= ( ( power ( L ) ) . k ) . k .= ( ( ( power ( L ) ) . k ) . k .= ( ( power ( L ) ) . k ) . k ;
sqrt ( ( f /. c ) * ( g /. c ) ^2 ) = f /. c * ( g /. c ) ^2 .= ( f (#) g ) /. c * ( g /. c ) ^2 .= ( f (#) g ) /. c * ( g /. c ) .= ( f (#) g ) /. c * ( f (#) g ) /. c ;
len ( ( ( Gauge ( C , n ) ) * ( len Gauge ( C , n ) , 1 ) ) - ( ( ( Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) ) + ( ( Gauge ( C , n ) * ( len Gauge ( C , n ) , 1 ) ) - ( ( Gauge ( C , n ) ) * ( len Gauge ( C , n ) , 1 ) ) ) = len ( ( ( Gauge ( C , n ) ) - ( ( Gauge ( C , n ) ) - ( ( Gauge ( C , n ) ) - ( ( Gauge ( C , n ) ) - ( ( C , n ) ) - ( ( Gauge ( C , n ) ) - ( ( Gauge ( C , n ) ) - ( ( C
dom ( r (#) f ) = 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 f /\ X /\ X /\ X .= dom f /\ X /\ dom f /\ X .= dom f /\ dom f /\ dom f /\ dom f /\ X /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ X /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f /\ dom f
defpred P [ Nat ] means for n being Nat holds 2 * n + 1 - $1 * n + 1 * n + 1 * n + 1 * n + 1 * n * n + 1 * n * n + 1 * n * n + 1 * n + 1 * n * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n + 1 * n * n * n * n * n + 1 * n * n + 1 * n + 1 * n + 1 * n * n * n * n * n + 1 * n * n + 1 * n * n * n * n + 1 * n + 1 * n +
consider f being Function of [: { n + 1 } , { k + 1 } :] , k + 1 } such that f = f and f is increasing and f is increasing and f is increasing & f is increasing & f is increasing ;
consider c being Function of S , BOOLEAN such that c = IExec ( A , B , C ) +* ( B , C ) and E = IExec ( B , C , D ) +* ( B , D ) and E = IExec ( C , D , E ) +* ( C , D ) ;
consider y being Element of Y such that a = "\/" ( { F . x , y } , L ) and y in { F . x , F . y } and P [ y , L . y , L . y , L . y ] ;
assume that A c= Z and f = ( ( \HM { the } \HM { function } ) * ( ( \HM { the } \HM { function } ) + ( \HM { the } \HM { function } ) ) ^2 ) and f is continuous ;
( f /. i ) `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 q1 + 1 ) = { j + len q1 where j is Nat : j in dom q1 + len q2 } ;
consider G1 , G2 being Element of V such that G1 <= G1 and G2 <= G1 and G1 <= G2 and G2 <= G2 and G1 is there exists a strict Subspace of V st G1 is Subspace of G2 & G2 is there exists a st G1 being Subset of V st G1 is open & g in G1 & f . G1 = G & g . ( G1 \/ G2 ) ;
func - f -> PartFunc of C , V means : Def1 : dom it = dom f & 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 a in a & a in b holds not ( ex L st L in L & L in a & L in a & L in b ) implies L . a in { H . a } ;
consider i1 , i2 such that [ i1 , j1 ] in Indices ( GoB f ) and f /. ( i1 + 1 ) = ( ( GoB f ) * ( i1 , i2 ) ) * ( i1 , i2 ) and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 + 1 ) ;
consider i , n such that n <> 0 and n <= i and n <= n and n <= i and i <= n and n <= n and i <= n and n <= n and i <= n and n <= n and n <= i and i <= n and n <= n and i <= n implies i <= n + 1 ;
assume that not 0 in Z and Z c= dom ( ( ( 1 / 2 ) (#) ( ( 1 + 2 ) * ( f1 + f2 ) ) ^2 ) and for x st x in Z holds ( ( 1 / 2 ) (#) ( f1 + f2 ) ) ^2 < 1 / 2 * ( ( 1 + 2 ) * ( f1 + f2 ) ^2 ) ;
cell ( G1 , i1 -' 1 , j1 -' 1 ) \ ( ( L~ f ) \ ( L~ f ) ` ) c= BDD ( L~ f ) \ ( ( L~ f ) ` ) ;
ex Q being open Subset of X st s = Q & ex F being Subset-Family of X st F in Q & F is open & F is open & F is open & union F c= union ( F ) & union ( F ) is finite & union ( F ) c= union ( F ) ;
gcd ( A , ( r / A ) * ( 1 / 2 ) , ( r / 2 ) * ( 1 / 2 ) ) = 1 / 2 * ( ( r / 2 ) * ( 1 / 2 ) ) ;
( ( the LSeg ( ss , 2 ) ) . ( m + 1 ) = ( ( the LSeg ( ss , 2 ) ) . ( m + 1 ) ) . ( m + 1 ) .= [ ( the LSeg ( ss , 2 ) ) . ( m + 1 ) , ( the Carrier s ) . ( m + 1 ) ] .= [ 3 , 1 ] ;
CurInstr ( P3 , Comput ( P3 , s3 , m + 1 ) ) = CurInstr ( P3 , Comput ( P3 , s3 , m + 1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m + 1 ) ) .= CurInstr ( P3 , Comput ( P3 , s3 , m ) ) .= CurInstr ( P3 , Comput ( P3 , Comput ( P3 , s3 , m + 1 ) ) ;
P1 /\ P2 = ( { p1 , p2 , p3 , p4 } ) /\ LSeg ( p1 , p2 , p3 ) \/ LSeg ( p2 , p3 , p4 ) /\ LSeg ( p1 , p3 , p4 ) ;
func not f in the carrier of A ( ) means : Def3 : for i being Nat holds it . i = f . i & ex p st p in dom p & not i in dom p & not i in dom p & not i in dom p & p . i in the carrier of A ( ) ;
let a , b be Element of COMPLEX , f be Polynomial of COMPLEX , a , b be Integer ;
defpred P [ Nat ] means ( for i being Nat st 1 <= i & i <= len g holds G * ( i , j ) `1 = g /. ( j + 1 ) `1 & ( G * ( i + 1 , j + 1 ) `1 ) `1 = g /. ( j + 1 ) `1 ;
assume that C1 , C2 , f is_collinear and g , h , h being stable and f = s * f and h = s * f and g = s * f and h = s * f ;
( ||. f .|| ) . c = ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .= ||. f .|| . c .|| .= ||. f .|| . c ;
|. q .| ^2 = ( ( q `1 ) ^2 + ( ( q `2 ) ^2 ) ^2 & 0 <= ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 & 0 <= ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ;
for F being Subset-Family of T st F is open & not {} in F & for i being Nat st i in dom F holds not i in F . i & not i in F . i holds not i in F . ( i + 1 )
assume that len F >= 1 and len F = k + 1 and for k being Nat st k in dom F holds F . k = g . k and for k being Nat st k in dom F holds F . k = g . k ;
i |^ ( n + 1 ) - i |^ ( n + 1 ) = i |^ ( s |^ ( n + 1 ) - i ) .= i |^ ( n + 1 ) - i .= i |^ ( n + 1 ) - i .= i |^ ( n + 1 ) - i .= i |^ ( n + 1 ) - i .= i |^ ( n + 1 ) ;
consider q being oriented Subset of G such that r = q and q <> {} and q in rng ( p ^ q ) and not q in rng ( p ^ q ) and not q in rng ( p ^ q ) and not q in rng ( p ^ q ) and not q in rng ( p ^ q ) ;
defpred P [ Element of NAT ] means for g being Element of ( len ( f , Z ) --> Z ) . $1 holds ( ( ( f , Z ) +* ( $1 , Z ) ) . $1 = ( ( ( f , Z ) +* ( $1 , Z ) ) . $1 ) . $1 ;
let A be Matrix of n , K , B , C be Matrix of n , K , K , f be Matrix of n , K ;
consider s being FinSequence of the carrier of R such that Sum s = u and for i being Element of NAT st i in dom s holds s . i = a * s . i and ex a being Element of R st a in I & s . i = b * s . i ;
func | ( x , y ) -> Element of REAL equals Sum ( x , y ) + ( y | ( x , y ) ) * ( x , y ) ;
consider g1 be FinSequence of F such that g1 is continuous and rng g1 = A and rng g1 c= A and g1 . 0 = A and g1 . 1 = A and g1 . ( len g1 ) = B and g1 is continuous and g1 is continuous and g1 is continuous and for i st i in dom g1 & i < len g1 holds g1 . i = A . i ;
then n1 >= len p1 & n2 >= len p2 & n2 >= len p1 & n2 >= len p1 & n1 >= len p2 & n2 >= len p1 & n1 >= len p2 implies n1 + n2 = n1 + n2 + n2
( q `2 ) * a <= ( q `2 ) * a & ( - q `2 ) * a <= ( - q `2 ) * a & ( - q `2 ) * a <= - ( - q `2 ) * a & ( - q `2 ) * a <= - ( - q `2 ) * a ;
( F . ( len p + 1 ) ) . ( len p + 1 ) = F . ( len p + 1 ) .= F . ( len p + 1 ) .= F . ( len p + 1 ) ;
consider k1 being Nat such that k1 + 1 = k and k = 1 and k <= ( ( a := intloc 0 ) .--> 1 ) and k = ( a := intloc 0 ) --> 1 ;
consider BB being Subset of [: { 1 } , { 1 } :] , A being Subset of [: { 1 } , { 1 } :] , B being Subset of [: { 1 , 1 } , { 1 , 1 } :] such that D = { 1 , 2 } and B is finite and D is finite and 1 = card { 1 , 2 } ;
v2 . b2 = ( curry F ) . b2 .= ( ( curry F ) . b2 ) . b2 .= ( ( ( curry F ) * b2 ) . b2 ) . b2 .= ( ( ( curry F ) * b2 ) . b2 ) . b2 .= ( ( ( curry F ) * b2 ) . b2 ) . b2 .= ( ( ( curry F ) . b2 ) . b2 .= ( ( ( ( curry F ) * b2 ) . b2 ) . b2 ;
dom IExec ( I , P , s ) = ( the carrier of SCMPDS ) \/ ( ( I + card I ) \ Start-At ( card I + card J + 3 ) ) .= dom Start-At ( card I + 3 , SCMPDS ) .= card I + 3 ;
ex d1 be Real st h . ( h . ( h . t ) ) < 0 & |. h . t - h . t .| < e ;
LSeg ( G * ( len G , 1 ) , G * ( len G , 1 ) ) c= Int cell ( G , len G , 1 ) \/ { G * ( len G , 1 ) } ;
LSeg ( mid ( h , i1 , i2 ) , i ) = LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) .= LSeg ( h /. ( i + 1 ) , h /. ( i + 1 ) ) ;
A = { q where q is Point of TOP-REAL 2 : LE q , q , P & LE q , q , P & LE q , q , P & LE q , q , P } ;
( - x ) | ( - x ) = ( - 1 ) * ( ( - x ) | ( - x ) .= ( - 1 ) * ( - x ) .= ( - 1 ) * ( - x ) .= ( - 1 ) * ( - x ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) .= ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 - 1 ) * ( - 1 ) * ( - 1 - 1 ) * ( - 1 ) * ( - 1 ) * ( - 1 * ( - 1 ) * ( - 1 * ( - 1 ) * ( - 1 - 1 * ( - 1
0 * sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + 1 ) ^2 ) ) ^2 ) ) ^2 ) ) ^2 ) ^2 ) = sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2 / sqrt ( 1 + ( p `2
sqrt ( ( - ( - ( - ( - ( - ( - ( p `2 / |. p .| - sn ) ) / ( 1 + sn ) ) ) ^2 ) = ( ( - ( - ( p `2 / |. p .| - sn ) / ( 1 + sn ) ) ^2 ) ) * sqrt ( 1 + sn ) ^2 ) .= ( - ( ( - TOP-REAL 2 ) * ( 1 + sn ) ) ^2 ) .= sqrt ( 1 + sn ) ^2 ) ;
func Shift ( f , h + h ) -> PartFunc of REAL , REAL means : Def3 : for x being Element of REAL , y being Element of REAL st x in dom it holds it . x = - ( f . y ) * ( f . x ) & it . y = - ( f . y ) * ( f . y ) * ( f . y ) & it . y = - ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) * ( f . y ) & for x be Element of REAL st x - h . x ) = f . x + f . x ) & for x
assume that 1 <= k and k + 1 <= len f and [ i , j ] in Indices G and [ i , j ] in Indices G and [ i , j ] in Indices G and f /. k = G * ( i + 1 , j + 1 ) and f /. k = G * ( i + 1 , j + 1 ) and f /. k = G * ( i + 1 , j + 1 ) ;
assume that not y in Free H and not x in Free ( H ) and not ( x in Free H ) & ( x in Free H implies ( x in Free H ) \/ { y } ) ;
defpred P [ Element of NAT ] means $1 < p implies ex n being Nat st n <= $1 & n <= $1 & n <= len p & p . $1 = F ( n ) ;
func Gauge ( C ) -> non empty Subset of X means : Def3 : for A being Subset of X holds it . A c= it iff for C being Subset of X st C in it holds it . C c= A \ B & for C being Subset of X st C in it holds C c= C \ B & C is open & C is bounded ;
[#] ( ( dist ( ( dist ( P , Q ) ) | Q ) = ( dist ( ( ( ( dist ( P , Q ) ) ) | Q ) ) .: Q & lower_bound ( ( dist ( Q , Q ) ) | Q ) = lower_bound ( ( dist ( Q , Q ) ) .: Q ) ;
rng ( F | ( S , T ) ) = { 1 , 2 , 3 , 4 , 5 , 6 , 6 , 7 , 8 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 8 , 6 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 6 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , 8
( f " * ( f . i ) ) . i = f . i .= f . i .= f . i .= f . i .= f . i ;
consider P1 , P2 being non empty Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 , p2 , p3 } and P1 /\ P2 = { p1 , p2 , p3 } and P /\ Q = { p1 , p2 , p3 } and P /\ Q = { p1 , p2 } and P /\ Q = { p1 , p2 , p3 } ;
f . p2 = |[ ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 ]| ;
( proj ( a , X ) ) . x = ( proj ( a , X ) ) . x .= ( proj ( a , X ) ) . x .= ( proj ( a , X ) ) . x .= ( proj ( a , X ) ) . x .= 0 .= 0 ;
let T be non empty TopSpace , G be non empty Subset of T , p be Point of T , r be Real , p be Point of T , q be Point of T , r be Real ;
for i being strict Subgroup of G , G being strict Subgroup of G st i + 1 in dom F & G is strict holds F is strict Subgroup of G & G is strict Subgroup of G & F is Subgroup of G & G is strict implies F is Subgroup of G
for x st x in Z holds ( ( ( ( arctan + arccot ) `| Z ) . x ) / ( ( arctan + arccot ) . x ) ^2 ) = ( ( arctan + arccot ) . x ) ^2 * ( ( arctan + arccot ) . x ) ^2
func f /* a -> right of S means : Def4 : for x being Element of S holds it . x = lim ( f /* a ) & for n being Nat st n in dom f holds it . n = - f . x ;
then X1 , Y1 , Y2 , Y1 , Y2 , Y1 , Y2 being non empty SubSpace of X , Y1 , Y2 be non empty SubSpace of X ;
ex N be Neighbourhood of ( f . x0 ) . ( x - x0 ) st N c= dom ( SVF1 ( 1 , f , u ) ) & ex L st for x st x in N holds L . x - f . x = L . ( x - x0 ) ;
sqrt ( ( ( 1 - ( ( ( ( TOP-REAL 2 ) ) * ( 1 + sn ) ) ^2 ) ) ^2 + ( 1 - sn ) ^2 ) >= sqrt ( 1 + ( ( ( TOP-REAL 2 ) * ( 1 + sn ) ) ^2 ) ^2 + ( 1 + sn ) ^2 ) ^2 ;
( ( 1 / ( ( ( ( ( ( ( ( f1 - g1 ) * ( #Z n ) * ( #Z n ) ) * ( #Z n ) ) ) ^ ( ( 1 / ( ( ( ( f1 + g1 ) * ( #Z n ) ) * ( #Z n ) ) ) ) ) ) ^ ( ( ( 1 / ( ( ( f1 + g1 ) * ( #Z n ) ^ ( #Z n ) ) ) ) ) ) = ( ( ( 1 / ( ( ( ( ( ( ( ( ( ( ( ( f1 + g1 ) ) ) ) ^ ( ( ( ( ( f1 + g1 ) ) ^ ( ( ( n + g1 ) ) ) ) ^ ) ) ) ^ ) ) ^ ) ) & ( ( ( ( ( n + g1 ) ) ) ^ ( ( ( n + g1 ) ) ) ^ ) ) ) . ( ( ( ( ( n + g1 ) ) ) . ( ( n + g1 ) ) ) . ( ( ( n + g1 ) ) ) ) ) . ( ( n + g1 ) ) ) ) . ( n + g1 ) ) ) . ( n + g1 ) ) .
assume that for x holds f . x = ( ( - 1 / 2 ) * ( sin . x ) ^2 - sin . x & ( - 1 / 2 ) * ( sin . x ) ^2 < ( - 1 / 2 ) * ( sin . x ) ^2 & ( - 1 / 2 ) * ( sin . x ) ^2 < ( - 1 / 2 ) * ( sin . x ) ^2 ;
consider [: [: X , Y :] , [: Y , Z :] such that t = [: [: X , Y :] and [: X , Z :] is open and [: X , Y :] is open and [: Y , Z :] is open and [: X , Z :] is open and [: Y , Z :] is open ;
card ( S . n + 3 ) = card { { { [ d , n ] , b ] } .= card { [ d , n ] , b , c ] } .= card { [ d , n ] , d ] } ;
sqrt ( ( ( ( TOP-REAL n ) * ( i - 1 ) - ( i - 1 ) ) - ( i - 1 ) * ( i - 1 ) ) ^2 ) = sqrt ( ( ( ( ( TOP-REAL n ) - ( i - 1 ) ) - ( i - 1 ) ) * ( i - 1 ) ) ^2 ) .= sqrt ( ( ( ( TOP-REAL n ) - 1 ) - 1 ) ^2 ) .= sqrt ( ( ( TOP-REAL n ) - 1 ) ^2 ) ;