thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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contradiction ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
thesis ;
contradiction ;
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contradiction ;
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thesis ;
contradiction ;
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thesis ;
contradiction ;
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thesis ;
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contradiction ;
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contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
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thesis ;
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thesis ;
contradiction ;
contradiction ;
thesis ;
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thesis ;
thesis ;
thesis ;
contradiction ;
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thesis ;
contradiction ;
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thesis ;
thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
contradiction ;
thesis ;
thesis ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
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thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
thesis ;
assume not thesis ;
assume not thesis ;
thesis ;
assume not thesis ;
x <> b
D c= S
let Y ;
S ` is convergent
q in A ;
V in W ;
y in N ;
x in T ;
m < n ;
m <= n ;
n > 1 ;
let r ;
t in I ;
n <= 4 ;
M is finite ;
let X ;
Y c= Z ;
A // M ;
let U ;
a in D ;
q in Y ;
let x ;
1 <= l ;
1 <= w ;
let G ;
y in N ;
f = {} ;
let x ;
x in Z ;
let x ;
F is one-to-one ;
e <> b ;
1 <= n ;
f is special ;
S misses C
t <= 1 ;
y divides m ;
P divides M ;
let Z ;
let x ;
y c= x ;
let X ;
let C ;
x _|_ p ;
o is monotone ;
let X ;
A = B ;
1 < i ;
let x ;
let u ;
k <> 0 ;
let p ;
0 < r ;
let n ;
let y ;
f is onto ;
x < 1 ;
G c= F ;
a >= 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 A ;
let x ;
Y ` in Y ;
assume 0 < g ;
not c in Y ;
not v in L ;
2 in z ` ;
assume f = g ;
N c= b ` ;
assume i < k ;
assume u = v ;
I = J ;
B ` = b ` ;
assume e in F ;
assume p > 0 ;
assume x in D ;
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 `1 <= b `1 ;
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 ` in B ` ;
assume i = 1 ;
let x be element ;
x ` = x ` ` ;
let X be BCK-algebra ;
assume S is not empty ;
a in REAL ;
let p be set ;
let A be set ;
let G be _Graph , a , b be element ;
let G be _Graph , a , b be element ;
let a be Element of V ;
let x be element ;
let x be element ;
let C be FormalContext , D be non empty set ;
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 ;
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 <= REAL ;
let y be element ;
r2 in X ;
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 one-to-one ;
let z be element ;
a , b // K ;
let n be Nat ;
let k be Nat ;
B ` c= B ` ;
set s = \mathclose { ^ \smallsmile } ;
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= card G ;
Cl B = B ;
let j be Nat ;
1 <= j + 1 ;
the arccot of arccot is_differentiable_on Z ;
the function exp is differentiable ;
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 <> b ;
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 , x be Real ;
not r in G . l ;
c1 = 0 ;
a + a = a ;
<* 0 *> in e ;
t in { t } ;
assume F is not empty ;
m1 divides m ;
B * A <> {} ;
a + b <> {} ;
p * p > 1 ;
let y be ExtReal ;
let a be Int-Location , f be FinSeq-Location ;
let l be Nat ;
let i be Nat ;
let r ;
1 <= i2 ;
a "\/" c = c ;
let r be Real ;
let i be Nat ;
let m be Nat ;
x = p2 ;
let i be Nat ;
y < r + 1 ;
rng c c= E ;
Cl R is open ;
let i be Nat ;
R2 is connected ;
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 ;
indices >= s ;
G . y <> 0 ;
let X be RealLinearSpace , f be PartFunc of X , Y ;
let a ;
H . 1 = 1 ;
f . y = p ;
let V be RealUnitarySpace , W be Subspace of V ;
assume x in Neighbourhood M ;
k < s . a ;
t in { p } ;
let Y be -valued set , X be Subset of Y ;
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 ` = a * y ;
rng D c= A ;
assume x in K ;
1 <= j0 & j0 <= len G ;
1 <= j0 & j0 <= len G ;
{ p } c= ]. p , q .[ ;
1 <= i & i <= len -15 ;
1 <= i & i <= len -15 ;
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 ;
-7 is bounded ;
assume I c= P1 ;
n in dom I ;
let Q ;
B c= dom f ;
b + p _|_ a ;
x in dom g ;
FF is continuous ;
dom g = X ;
len q = m ;
assume A2 : A is closed ;
cluster R \ S -> real-valued ;
ex_sup_of D , S ;
x \ll 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 ;
{ x } is increasing ;
let e2 be element ;
- b divides b ;
F c= \tau ( F ) ;
G1 is non-decreasing ;
G1 is non-decreasing ;
assume v in H . m ;
assume b in [#] B ;
let S be non void ManySortedSign , p be FinSequence of S ;
assume P [ n ] ;
assume union S is finite & not S is finite ;
V is Subspace of V
assume P [ k ] ;
rng f c= NAT ;
assume X is_>=_than L ;
y in rng f ;
let s , I , J ;
b ` c= b ` ;
assume not x in { Q + + 1 } ;
A /\ B = { a } ;
assume len f > 0 ;
assume x in dom f ;
b , a // o , c ;
B in { BH } ;
cluster product p -> non empty ;
z , x // x , p ;
assume x in rng N ;
cosec is_differentiable_on Z ;
assume y in rng S ;
let x , y ;
i2 < i1 + 1 ;
a * h in a * H ;
p , q ] in Y ;
cluster sqrt I -> non empty ;
q1 in A & q2 in A ;
i + 1 <= 2 + 1 ;
A1 c= A & A2 c= B ;
\hbox { \boldmath $ b $ } < n ;
assume A c= dom f ;
Re ( f ) is_integrable_on M & M is_integrable_on M ;
let k , m ;
a , b // b , c ;
j + 1 < k + 1 ;
m + 1 <= n1 ;
g is_differentiable_on Z ;
g is continuous PartFunc of REAL , REAL ;
assume O is symmetric & O is symmetric ;
let x , y ;
let j1 be Nat ;
[ y , x ] in R ;
let x , y ;
assume y in conv A ;
x in Int V ;
let v be VECTOR of V ;
P3 halts_on s , P3 = P +* I ;
d , c // a , b ;
let t , u ;
let X be set ;
assume k in dom s ;
let r be non negative Real ;
assume x in F | M ;
Y be Subset of S ;
let X be non empty TopSpace , Y be non empty TopSpace ;
[ a , b ] in R ;
x + w < y + w ;
not a >= c ;
B be Subset of A , A be Subset of B ;
let S be non empty ManySortedSign ;
let x be variable , f be Function of x , y ;
let b be Element of X , a , b be Element of X ;
R [ x , y ] ;
x ` = x ;
b \ x = 0. X ;
<* d *> in D |^ 1 ;
P [ k + 1 ] ;
m in dom ( n |-> 0 ) ;
h2 . a = y ;
P [ n + 1 ] ;
cluster G * F -> with_-F1 ;
let R be non empty multMagma , a , b be Element of R ;
let G be _Graph ;
let j be Element of I ;
a , p // x , p ;
assume f | X is bounded ;
x in rng ( the_arity_of o ) ;
let x be Element of B ;
let t be Element of D ;
assume x in Q { \rm .vertices ( ) } ;
set q = s ^\ k ;
let t be VECTOR of X ;
let x be Element of A ;
assume y in rng p ;
let M be mamaid id id X , X ;
let N be non empty multMagma ;
let R be relational structure ;
let n , k ;
let P , Q ;
P = Q /\ [#] S ;
F . r in { 0 } ;
let x be Element of X ;
let x be Element of X ;
let u be VECTOR of V ;
reconsider d = x as Int-Location ;
assume I does not destroys a ;
let n , k ;
let x be Point of T ;
f c= f +* g ;
assume m < Partial_Sums ( seq ) . m ;
x <= c2 . x ;
x in F " { x } ;
cluster S --> T -> ManySortedSet ;
assume t1 <= t2 & t2 <= t2 ;
let i , j be odd Nat ;
assume F1 <> F2 & F1 <> F2 ;
c in Intersect ( R ) ;
dom p1 = c ;
a = 0 or a = 1 ;
assume A1 <> A2 & A <> B ;
set i1 = i + 1 ;
assume a1 = b1 & b1 = b2 ;
dom ( g1 * g2 ) = A ;
i < len M + 1 ;
assume not - \infty in rng G ;
N c= dom ( ( f1 + f2 ) `| Z ) ;
x in dom ( sec | X ) ;
assume [ x , y ] in R ;
set d = sqrt ( x , y ) ;
1 <= len g1 & g1 <= len g2 ;
len s2 > 1 ;
z in dom ( ( f1 + f2 ) `| Z ) ;
1 in dom D2 ;
( p `1 ) ^2 = 0 ;
j1 <= width G ;
len cos > 1 + 1 ;
set n1 = n + 1 ;
|. q .| = 1 ;
let s be SortSymbol of S ;
\mathop ( i , n ) . i = i ;
X1 c= dom f ;
h . x in h . a ;
let G be _! non empty SubSpace of G ;
cluster m * n -> invertible ;
let k1 be Nat ;
i -' 1 > m ;
R is transitive ;
set F = <* u , v *> ;
{ p } c= P ;
I is_closed_on t , Q ;
assume [ S , x ] is Galois ;
i <= len ( f2 | i ) ;
p is FinSequence of X ;
1 + 1 in dom g ;
Sum R2 = n * r ;
cluster f . x -> complex-valued ;
x in dom ( ( f1 + f2 ) `| Z ) ;
assume [ X , p ] in C ;
{ B } c= X1 ;
n2 <= n2 + 1 ;
A /\ { { x } c= A `
cluster x -valued -> 0 -element ;
let Q be Subset-Family of S , S be Subset-Family of T ;
assume n in dom ( g2 ) ;
let a be Element of R ;
t `1 in dom ( e ) ;
N . 1 in rng N ;
- z in A \/ B ;
let S be Subset-Family of X , Y ;
i . y in rng i ;
REAL c= dom f ;
f . x in rng f ;
REAL <= sqrt ( r ) ;
s2 in { |[ r , s ]| : r < s & s < b } ;
let z , y be z being element ;
n <= N . ( m + 1 ) ;
LIN q , s , q ;
f . x = \twoheaddownarrow x /\ B ;
set L = [ S \to T ] ;
let x be non negative Real ;
m be Element of M ;
f in union rng ( F ) ;
let K be add-associative right_zeroed right_complementable associative associative non empty doubleLoopStr ;
let i be Element of NAT ;
rng ( F * g ) c= Y
dom f c= dom ( x | X ) ;
n1 < n1 + 1 ;
n1 < n1 + 1 ;
cluster { {} } -> U -total ;
[ y2 , 2 ] = z ;
let m be Element of NAT ;
let S be Subset of R ;
y in rng Sy ;
b = sup rng f ;
x in Seg len q ;
reconsider X = { {} } as set ;
[ a , c ] in E ;
assume n in dom ( h2 ) ;
w + 1 = { a } ;
j + 1 <= j + 1 ;
k2 + 1 <= k1 + 1 ;
i be Element of NAT ;
Support u = Support p ;
assume X is complete and X is b |^ m ;
assume that f = g and p = q ;
n1 <= n1 + 1 ;
let x be Element of REAL ;
assume x in rng s2 ;
x0 < x0 + r ;
len ( L ) = len L ;
P c= Seg ( len A ) ;
dom q = Seg n ;
j <= width M ;
let { r } be real-valued FinSequence of D ;
let k be Element of NAT ;
\int P . M < + \infty ;
let n be Element of NAT ;
assume z in \mathbin { \tt where 0 is Element of NAT : 0 <= 0 & 0 <= n } ;
i be set ;
n -' 1 = n - 1 ;
len ( ( n , m ) |-> ( n , m ) ) = n ;
diff ( Z , c ) c= F ;
assume x in X or x = X ;
x is LIN b , c , d ;
let A , B , C , D , E , F , J , J , F , J , J , F , J , F , J , F , J , J
set d = dim ( p ) ;
let p be FinSequence of L ;
Seg i = dom q ;
let s be Element of E |^ \omega ;
B1 be Subset of T ;
3 /\ L2 = {} ;
L1 /\ L2 = {} ;
assume ]. x , y .[ = ]. x - y .[ ;
assume b , c // b , c ;
LIN q , c , c9 ;
x in rng ( ( ( ( f2 ) | X ) | X ) ) ;
set N8 = n + j ;
let D1 be non empty set , D2 be Subset of X ;
let K be add-associative right_zeroed right_complementable associative associative non empty addLoopStr , F be FinSequence of K ;
assume f ' = f & h = h ;
R1 - R2 is total ;
k in NAT & 1 <= k & k <= n ;
let a be Element of G ;
assume x0 in [. a , b .] ;
K ` is open ;
assume a , b // b , c ;
a , b // a , b ;
reconsider d = x as Vertex of G ;
x in ( s + f ) .: A ;
set a = \int f ;
cluster -> [ -> with_Bs) -> with_T) ;
not u in { \hbox { \boldmath $ g $ } } ;
the carrier of f c= B ;
reconsider z = x as VECTOR of V ;
cluster strict -element for RelStr ;
r (#) H is convergent ;
s . intloc 0 = 1 ;
assume that x in C and y in C ;
let U1 be strict universal algebra , U2 be non-empty MSAlgebra over S ;
[ x , [#] T ] is compact ;
i + 1 + k in dom p ;
F . i is stable Subset of M ;
|[ r , s ]| in cell ( G , y , z ) ;
let x , y be Element of X ;
A , I , J , D , E , F , J , J , M ;
[ y , z ] in [: O , { y } :] ;
LE IncAddr , i , L~ f ;
rng Sgm A = A ;
q |- |[ y , q ]| ;
for n holds X [ n ] ;
x in { a } & x in { a } ;
for n holds P [ n ] ;
set p = [ x , y ] ;
LIN o , a , b ;
p . 2 = Z |^ 2 ;
( ( D ) `2 ) `2 = {} ;
n + 1 + 1 <= len g ;
a in [: A , B :] ;
u in Support ( m *' p ) ;
let x , y be Element of G ;
let I be Subset of L ;
set g = f1 + f2 + g2 ;
a <= max ( a , b ) ;
i-1 < len G + 1 ;
g . 1 = f . ( i1 + 1 ) ;
x ` \ y in A2 ;
( f /* s ) . k < r ;
set v = VAL g ;
i -' k + 1 <= S ;
cluster non empty multiplicative loop structure -> associative for associative non empty multMagma ;
x in support ( EmptyBag n ) ;
assume a in [: { {} } , { {} } :] ;
i `1 <= len ( y `1 ) ;
assume p divides b1 + b2 ;
{ 0 } <= sup ( { 0 } \/ { 1 } ) ;
assume x in \mathop { \rm W \hbox { - } bound ( X ) : not contradiction } ;
j in dom ( z | ( n + 1 ) ) ;
let x be Element of [: D , D :] ;
IC Comput ( P3 , s3 , 5 ) = l1 ;
a = {} or a = {} ;
set PA = Vertices G , PA = G , PA = G , PA = G , PA = G , PA = G , PA = G , G1 = G , G2 = G , G2
seq " is non-zero ;
for k holds X [ k ] ;
for n holds X [ n ] ;
F . m in { F . m } ;
{ h } c= { h } ;
]. a , b .[ c= Z ;
X1 , X2 , X2 , Y2 is_collinear ;
a in union ( F \ G ) ;
set x1 = [ 0 , 0 ] ;
k + 1 -' 1 = k - 1 ;
cluster binary Relation for Relation of Q ;
ex v st C = v + W ;
let G be non empty multMagma , F be Subset-Family of G ;
assume V is Abelian add-associative right_zeroed right_complementable associative distributive distributive non empty doubleLoopStr ;
{ X } \/ Y in \sigma ( L ) ;
reconsider x = x as Element of S ;
max ( a , b ) = a ;
ex_sup_of B , L ;
let L be non empty RelStr , X be Subset of L ;
R is reflexive & R is transitive implies R is transitive
E |= All ( g , H ) ;
dom G ' = a ;
sqrt ( 1 - 4 ) >= - 4 ;
G . ( p1 , p2 ) in rng G ;
let x be Element of [: F , G :] , F ;
D [ 0 ] ;
z in dom id B ;
y in the carrier of N ;
g in the carrier of H ;
rng ( ( f1 ^ ) ^ ) c= NAT ;
j + 1 in dom ( s1 ^\ k ) ;
A , B are_equipotent ;
let C be non empty Subset of REAL ;
f . z1 in dom h ;
P . k1 in rng P ;
M = { A } +* { A } ;
let p be FinSequence of REAL ;
f . n1 in rng f ;
M . ( F . 0 ) in REAL ;
E-bound [. a , b .] = E-bound A ;
assume the InternalRel of V is total & Q is total & Q is total & Q is total & Q is total & Q is total & Q is total & Q is total & Q is
let a be Element of ^ V ;
let s be Element of P ;
let PI be non empty O ;
let n be Nat ;
the carrier of g c= B ;
I = halt SCM R ;
consider b being element such that b in B ;
set BK = BCS K ;
l <= ( rng F ) . j ;
assume x in ]. s , t .[ ;
( x `1 ) ^2 + ( x `2 ) ^2 in ]. x `1 - r , x + r .[ ;
x in Hom ( T ) ;
let h be Morphism of c , d ;
Y c= { \bf R } ( Y ) ;
A2 \/ { A } c= L1 \/ L2 ;
assume LIN o , a , b ;
b , c // d , e ;
x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 *> in Y ;
dom <* y *> = Seg 1 ;
reconsider i = x as Element of NAT ;
set l = |. ar .| ;
[ x , x ] in X \times X ;
for n be Nat holds 0 <= x . n
[ a , b ] = [. a , b .] ;
cluster -> non empty for Subset of T ;
x = h . ( f . z1 ) ;
q1 , q2 , q1 is_collinear ;
dom ( M1 * M2 ) = Seg n ;
x = [ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 ,
R , Q are_isomorphic ;
set d = sqrt ( 1 + ( n + 1 ) ) ;
rng g2 c= dom ( g | X ) ;
P . ( [#] Omega \ B ) <> 0 ;
a in field R & a = b ;
let M be non empty Subset of V , V be Subset of V ;
I be Program of SCM+FSA ;
assume x in rng ( R * ( id Z ) ) ;
let b be Element of the lattice T ;
dist ( e , z ) < r-r ;
u1 + v1 in W2 + W ;
assume the support of L misses rng G ;
let L be lower-bounded lower-bounded lower-bounded RelStr ;
assume [ x , y ] in [: { a } , { b } :] ;
dom ( A * e ) = NAT ;
a , b // G * ( i , j ) ;
let x be Element of Subset-Family ( M ) ;
0 <= Arg a * PI ;
o , a1 // o , y ;
not v in the carrier of l ;
let x be bound of A ;
assume x in dom ( uncurry f ) ;
rng F c= ( Carrier f ) .: X
assume D2 . k in rng D2 ;
f " . p1 = 0 ;
set x = the Element of X ;
dom Ser G = NAT ;
let n be Element of NAT ;
assume LIN c , a , b ;
cluster finite -> finite for FinSequence of NAT ;
reconsider d = c as Element of L1 ;
( v2 |-- I ) . ( X \ I ) <= 1 ;
assume x in the carrier of ( f | X ) ;
conv ( A @ ) c= conv ( A @ ) ;
reconsider B = b as Element of the lattice of T ;
J |= P ! l ( ) ;
cluster the InternalRel of J . i -> non empty TopSpace ;
ex_sup_of Y , T ;
W1 is_\! \! \smallfrown ( W1 , W2 ) ;
assume x in the carrier of R ;
dom ( ( n + 1 ) |-> ( n + 1 ) ) = Seg ( n + 1 ) ;
seq misses seq " { x0 } ;
assume ( a 'imp' b ) . z = TRUE ;
assume X is open & f = X --> d ;
assume [ a , y ] in Indices ( ( f * g ) * g ) ;
assume that that that that that that that that I c= J and I c= J and J c= K and K c= L and L c= L ;
Im ( seq ) . n = 0 ;
( ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 )
( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( (
t2 . n = t2 . ( n + 1 ) ;
dom ( F `| Z ) c= dom ( F `| Z ) ;
W1 . x = W2 . x ;
y in W { W ( ) } \/ { W ( ) } ;
k1 <= len ( v | ( n + 1 ) ) ;
x * a * y \equiv a * ( ( m mod a ) * y ) ;
proj2 .: S c= proj2 .: ( ( TOP-REAL 2 ) .: P ) ;
h . ( p1 + 4 ) = g2 . ( p1 + 4 ) ;
G1 = G1 /. ( len G1 + 1 ) ;
f . r1 in rng f ;
i + 1 + 1 <= len One ;
rng F = rng ( F | ( n + 1 ) ) ;
mode Ealgebra is associative non empty multMagma ;
[ x , y ] in A \times { a } ;
x1 . o in L2 . o ;
the carrier of TOP-REAL m c= B ;
[ y , x ] in id X ;
1 + p .. f <= i + len f ;
seq ^\ k is convergent & ( seq ^\ k ) . n is convergent implies seq is convergent
len ( F ^ G ) = len F + len G ;
let l be Linear_Combination of B \/ { v } ;
let r1 , r2 be complex number ;
Comput ( P , s , n ) . ( n + 1 ) = s . ( n + 1 ) ;
k <= k + 1 ;
reconsider c = {} as Element of L ;
let Y be with_n) of T ;
cluster strict for Function of L , L ;
f . j1 in K . j1 ;
cluster J => ( I --> J ) -> total ;
K c= 2 |^ ( len K ) ;
F . b1 = F . b1 .= F . b1 ;
x1 = x or y1 = y ;
attr a <> {} means : Def3 : a = {} ;
assume that that a c= b and b c= a and b c= b ;
s1 . n in rng s1 ;
{ o , b } on C ;
LIN o , b , c ;
reconsider m = x as Element of [: V , V :] ;
let f be non constant FinSequence of D ;
let FF be non empty 'or' 'or' |[ a , b ]| ;
assume that h is being_homeomorphism and y = h . ( x , y ) ;
[ f . 1 , f . 2 ] in F ;
reconsider p1 = x as Subset of m ;
A , B , C , D , E , F , G , G , D , F , G , G , F , G , G , F , G , G , F , G , G
cluster non empty strict for sqrt of V ;
rng c misses rng ( e .--> x ) ;
z is Element of gr { x } ;
not b in dom ( a .--> ( p1 . x ) ) ;
assume that k >= 2 and P [ k ] and P [ k + 1 ] ;
Z c= dom ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) `|
the InternalRel of Q c= the InternalRel of A ;
reconsider E = { i } as finite Subset of I ;
g2 in dom ( ( ( ( 1 / 2 ) (#) ( ( ( ( ( ( ( ( ( - 2 ) / 2 ) * ( ( ( 1 + 2 ) / 2 ) ) ) ) ) ) ) ) `| Z
attr f = u * f ;
for n holds P [ n ] ;
{ x ( O ) : x in L & L ( O ) <> {} } <> {} } ;
let x be Element of V . s ;
a , b are_relative_prime ;
assume S = S2 & p = S2 & p = S2 ;
gcd ( n1 , n1 ) = 1 ;
set o9 = ( 2 * PI ) * ( ( 2 * PI ) * ( 2 * PI ) ) ;
seq . n < |. r1 .| ;
assume that seq is increasing and r < 0 and r < 1 ;
f . y1 <= a ;
ex c being Nat st P [ c ] ;
set g = { n } \ { n } ;
k = a or k = b ;
{ a , b } is open & { b } is open & { a , b } is open } is open ;
assume Y = { 1 } & s = <* 1 *> ;
I1 . x = f . x .= 0 ;
W1 . ( len W ) = W . ( len W + 1 ) ;
cluster -> connected for _Graph of G ;
reconsider u = u as Element of Bags ( X ) ;
A in B ^ implies A is |^ ( n + 1 )
x in { [ 2 * n + 3 , 3 ] } ;
1 >= sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ;
f1 is sredefine by f2 & f2 is not true & f1 is not true & f2 is not true & f2 is not true & f2 is not true & f2 is not true & f2 is not true & f2 is not true &
( f . q ) `2 <= ( q `2 ) `2 ;
h is_Sh_of Cage ( C , n ) ;
( b ) `1 <= ( ( p `1 ) ^2 ) ^2 ;
let f , g be RMembership_Func of X , Y ;
S /. ( k , m ) <> 0. K ;
x in dom ( max ( f , g ) ) ;
p2 in N . ( p1 , p2 ) ;
len ( ( H ) < len ( H ) ;
F [ A , F ( ) ] ;
consider Z such that y in Z and Z in X ;
hence 1 in C & A c= C ;
assume r1 <> 0 or r2 <> 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 crossover ( p , S ) ;
then S is universal & S is universal & S is universal ;
Cl ( ( Cl ( T ) ) ` ) = [#] T ;
( ( f2 | A ) | A ) . ( len f2 + 1 ) = f2 . ( len f2 + 1 ) ;
0. M in the carrier of W ;
v , w |= H ;
reconsider K = union K as non empty set ;
X \ V c= Y \ V
let X be Subset of S ;
consider H1 such that H = 'not' H1 ;
{ {} } c= d1 * ;
0 * a = 0. R .= a ;
A |^ 2 = A |^ ( 2 + 1 ) ;
set vIf = ( ( card I + 1 ) + 1 ) * ( card I + 1 ) ;
r = 0. \langle <* 0. ( ( X , X , 1 ) *> , \Vert \rangle ;
( f . p3 ) `2 >= 0 ;
len W = len ( W ) .= len ( W ) ;
f /* ( s * k ) is divergent_to+infty ;
consider l being Nat such that m = F . l ;
t8 <> b1 & b1 <> b2 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <> b3 & b1 <>
reconsider X1 = X1 as SubSpace of X ;
consider w such that w in F and not w in w ;
let a , b , c , d be Real ;
reconsider i = i as non zero Element of NAT ;
c . x >= id L . x ;
\sigma ( T ) \/ \omega is Subset-Family of T ;
for x being element st x in X holds x in Y
cluster [ x1 , x2 , x3 , x4 , x5 , cin , dp , cin , cin , dp , cin , dp , cin , dp , cin *> -> pair ;
downarrow a /\ { t } is Subset of T ;
let X be non empty set , N be non empty set ;
rng f = union { X , Y } ;
let p be Element of B , r be SortSymbol of S ;
max ( N , 2 ) >= N ;
0. X <= b |^ ( m + 1 ) ;
assume that i in I and R . i = R . i ;
i = j1 & ( i = j1 implies i = j1 ) & ( i = j2 ) & i = j1 or i = j2 ) & j = j2 ) & j = j2 or i = j2 or i = j1 or i = j2 or
assume \mathfrak R in the carrier of R ;
let A1 , A2 , C be Subset of S ;
x in h " ( P /\ Q ) ;
1 in Seg 2 & 2 in Seg 3 & 1 in Seg 3 ;
reconsider X1 = X as non empty Subset of [: T , T :] ;
x in ( the Arrows of B ) . i ;
cluster E . n -> ( the Target of G ) -element ;
n1 <= i2 + len g2 + 1 ;
( i + 1 ) + 1 = i + 1 ;
assume v in the carrier of G2 ;
y = Re ( y . i ) + Im ( y . i ) ;
( ( ( ( ( - 1 ) * ( ( 1 - p ) * ( ( p ) ) ) ) ) ) ) ) mod p = 1 ;
x2 is_differentiable_on ]. a , b .[ ;
rng ( ( D2 | ( len D2 ) ) | ( len D2 ) ) c= rng ( D2 | ( len D2 ) ) ;
for p be Real st p in Z holds p >= a
|[ a , b ]| `2 = |[ a , b ]| `2 ;
( seq ^\ k ) . ( n + k ) <> 0 ;
s . ( k + 1 ) > x0 ;
( p | ( 2 \ { p } ) . 2 = d ;
A \oplus ( B \ C ) = ( A \ B ) \ C
h \equiv gg . ( h . ( len g ) ) ;
reconsider i1 = i-1 as Element of NAT ;
let v1 , v2 be VECTOR of V ;
for V being Subspace of V holds W is Subspace of V
reconsider xx = i as Element of NAT ;
dom f c= [: C , D :] ;
x in ( the distance of B ) . ( n + 1 ) ;
len <* f2 *> in Seg ( len f2 ) ;
{ p } c= the topology of T ;
]. r , s .[ c= [. r , s .] ;
B2 be Subset-Family of T2 ;
G * ( B * A ) = ( the InternalRel of U1 ) * ( B * A ) ;
assume p , u , v is_collinear & q , w is_collinear & q <> w & p , y is_collinear & q <> w ;
[ z , z ] in union rng ( F | ( n + 1 ) ) ;
'not' b . x 'or' b . x = TRUE ;
deffunc F ( set ) = { $1 : $1 in S & $1 in S & not contradiction } ;
LIN a1 , b1 , c1 ;
f " { x } = { x } ;
dom ( ( ( r (#) ( ( ( ( ( ( G ) | X ) | X ) | X ) ) | X ) ) ) = dom ( ( r (#) ( ( ( G | X ) | X ) | X ) ) ;
assume that 1 <= i and i <= n and i <= n ;
( g2 . O ) `2 <= 1 ;
p in LSeg ( E . i , F . i ) ;
I /. ( i + 1 ) = 0. K ;
|. f . ( s . m ) - g .| < g1 ;
q1 . x in rng ( q | ( i + 1 ) ) ;
L1 misses L2 & L1 /\ L2 = ( L1 \/ L2 ) \/ ( L1 /\ L2 ) /\ L2
consider c being element such that [ a , c ] in G ;
assume NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN
q . ( j + 1 ) = q /. ( j + 1 ) ;
rng F c= { F . ( n + 1 ) } ;
P . ( B \/ C ) <= 0 + ( B + C ) ;
f . j in { f . j } ;
pred 0 <= x & x <= y ;
p `1 <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & p <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q <> 0. TOP-REAL 2 & q
cluster { topology ( S , T ) } -> non empty ;
let x be Element of S \times T ;
<^ a , b ^> is one-to-one ;
|. i .| <= - ( i + 1 ) ;
the carrier of I[01] = dom P ;
n * ( n + 1 ) > 0 * n ;
S c= ( A1 /\ A2 ) /\ ( A1 /\ A2 ) ;
a3 , a4 // b , c ;
then dom A <> {} & dom A <> {} ;
1 + ( 2 * k + 4 ) = 2 * k + 4 ;
x joins X & y in Y & x in X & y in Y ;
set v2 = v /. ( i + 1 ) ;
x = r . ( n + 1 ) .= r . ( n + 1 ) ;
f . s in the carrier of S2 & f . s in the carrier of S2 ;
dom g = the carrier of I[01] ;
p in LSeg ( P , p1 ) /\ LSeg ( P , p2 ) ;
dom ( ( A /\ B ) /\ ( A /\ B ) ) = A /\ B ;
0 < sqrt ( ( |. z .| - |. z .| ) / ( 2 |^ ( n + 1 ) ) ) ;
e . ( m + 1 ) <= e . ( m + 1 ) ;
B \ominus X c= B
- - ( g | B ) < M ;
cluster O \tt : where F is \tt Relation of X , Y is \mathbin { \mid \mid of X ;
let U1 , U2 be non-empty MSAlgebra over S , U2 be non-empty MSAlgebra over S ;
Proj ( i , n ) * g is_differentiable_on X ;
x , y // x , y & x , y // y , z ;
reconsider p1 = p . x as Element of V ;
x in the carrier of ( TOP-REAL n ) | K1 ;
let I , J be Program of SCM+FSA ;
assume - a is lower & b is lower implies - a is lower
Int ( A /\ ( A /\ B ) ) c= Cl ( A /\ B ) ;
assume for A being Subset of X holds A is open implies A is open ;
assume q in Ball ( x , r ) ;
( p2 `2 ) ^2 <= ( p2 `2 ) ^2 + ( p2 `2 ) ^2 ;
Cl Q ` = [#] T ;
set S = the carrier of T ;
set I1 = [ I , J ] , J = [ I , J ] , J = [ I , J ] , I = [ I , J ] , J = [ I , J ] , J = [ I , J ] ;
len p -' n = len wn - n ;
A is Permutation of A , B ;
reconsider nI = nI as Element of NAT ;
1 <= j + 1 & j <= len s ;
q , m , k is_collinear ;
a1 in the carrier of S1 & a2 in the carrier of S1 & a2 in the carrier of S2 ;
c1 /. ( n + 1 ) = c1 . ( n + 1 ) ;
let f be FinSequence of TOP-REAL 2 ;
y = ( ( ( ( ( ( ( ( ( ( ( ( ( ( N ) * ( ( N ) ) * ( ( N ) ) * ( ( N ) ) * ( ( N ) * ( ( N * ( N ) ) * ( (
consider x being element such that x in \mathop { \rm support \hbox { - } A } ;
assume r in ( dist o ) .: P ;
set i2 = [ n , h ] , j2 = [ n , h ] ;
h2 . ( j + 1 ) in rng h2 ;
Line ( M , k ) = M * ( i , j ) ;
reconsider m = sqrt ( x ^2 + 2 * PI ) as Element of REAL n ;
U1 , U2 , U2 , U2 , U1 , U2 , U2 , U2 , U2 , U2 , U2 , U2 , U2 ;
set P = Line ( a , d ) ;
len p1 < len p2 + 1 & len p2 = len p2 ;
T1 , T2 be T T non empty TopSpace ;
then x <= y & ( x <= y ) implies x is Element of L
set M = n -\hbox { - } ;
reconsider i = x1 , j = x2 as Nat ;
rng ( the_arity_of o ) c= dom ( the_arity_of o ) ;
z1 " * z " = z1 " * z " * z " ;
x0 - r in L /\ dom f ;
then w is which is which -string of S ;
set x9 = { x where x is Element of X : <* x *> in Z } ;
len ( ( - w ) + ( len w ) ) in Seg len w ;
( uncurry f ) . ( x , y ) = g . ( y , z ) ;
a be Element of PFuncs ( V , { V } ) ;
x . n = sqrt ( a . n ) .= |. a . n .| ;
( p `1 ) ^2 <= ( G * ( 1 , j ) `1 ) ^2 ;
rng ( g | ( len g ) ) c= L~ g ;
reconsider k = i-1 * ( l + 1 ) as Nat ;
for n be Nat holds F . n is Seg ( n + 1 )
reconsider x9 = x as VECTOR of V ;
dom ( f | X ) = X /\ dom ( f | X ) ;
p , a // p , c & b , c // p , b ;
reconsider x1 = x as Element of REAL m ;
assume i in dom ( a * p ) ;
m . ( \hbox { \boldmath $ g $ } ) = p . ( len g + 1 ) ;
a |^ ( m + 1 ) - ( n + 1 ) <= 1 - ( n + 1 ) ;
S . ( n + k ) c= S . ( n + k ) ;
assume B \/ { A } = { B } \/ { A } ;
X . i = { x1 , x2 } . i ;
r2 in dom ( ( ( ( h + c ) (#) ( ( h + c ) (#) ( ( h + c ) - ( h + c ) - ( h + c ) ) ) ) `| Z ) ;
||. a .|| = a & b-Point ( R ) = b ;
{ F } is closed & { t } is closed & t in Q & t in Q & t in Q & t in Q & t in Q & t <> Q & t <> Q & t <> Q & t <> Q & Q is closed & Q
set T = \langle X , { x0 } , { x0 } , { x0 } , { x0 } , { x0 } , { x0 } , { x0 } } ;
Int ( R /\ ( R /\ ( R /\ S ) ) c= R /\ ( R /\ S )
consider y being Element of L such that c . y = x ;
rng ( F | ( X \/ { x } ) ) = { F . ( X \/ { x } ) } ;
G1 " { c } c= B \/ S " { c } ;
{ f where f is PartFunc of X , Y : for n be Nat holds f . n = f . n } ;
set R1 = the Point of TOP-REAL 2 , R2 = the Point of TOP-REAL 2 ;
assume n + 1 >= 1 & n + 1 <= len M ;
k2 be Element of NAT ;
reconsider pI = u as Element of TOP-REAL n ;
g . x in dom f & g . x in dom f ;
assume that 1 <= n and n + 1 <= len ( f1 ^ <* x *> ) and n <= len f1 + 1 ;
reconsider T = b * N as Element of G ;
len ( ( ( ( ( ( ( /. ( len G ) ) ) ) /. len G ) ) `1 <= ( ( ( GoB f ) /. 1 ) `1 ;
x " in the carrier of ( ( A ) | ( the carrier of ( ) | ( the carrier of ) ) ) ;
[ i , j ] in Indices A & [ i , j ] in Indices A ;
for m be Nat holds Re ( F . m ) is simple function
f . x = a . i .= a1 . i ;
let f be PartFunc of REAL , REAL , i be Element of NAT , r be Real st r = f . ( i + 1 ) holds r = r * ( i + 1 )
rng f = the carrier of ( TOP-REAL n ) | K1 ;
assume s1 = sqrt ( 2 * ( ( 1 - r ) / 2 ) ) ;
attr a > 1 & a > 0 & b > 1 implies a * b + b * c + d * c + d * a + d * c + d * a * c + d * a * c + d * b + d * c +
let A , B , C , D , E , F , J , J , M , N , N , F , J , M , N , F , J , M , N , F , J , J , M , N , F , J ,
reconsider X1 = X , Y1 = Y as Subset of X ;
f be PartFunc of REAL , REAL , REAL ;
r * ( v1 |-- I ) . ( n + 1 ) < r * ( v1 |-- I ) . ( n + 1 ) ;
assume V is Subspace of X & X is Subspace of X ;
t-3 , -3 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t1 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t2 , t1 , t2 , t2 , t2 ,
Q [ e ] \/ { [ e , f ] } ] ;
g \circlearrowleft ( L~ z ) = z /. ( 1 + 1 ) ;
|. [ x , y ] - [ x , y ] .| = v-f ;
- f . w = - ( L . w ) ;
z -' y <= x iff z <= x & z <= y
sqrt ( 7 * ( 1 + 7 ) ) > 0 ;
assume X is BCK-algebra of 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 1 , 0 , 0 ,
F . 1 = v1 & F . 2 = v2 ;
( f | X ) . x2 = f . x2 ;
( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( - 1 ) (#) ( ( ( ( 1 / 2 ) ) (#) ( ( ( ( 1 / 2 ) ) (#) ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) )
i2 = ( GoB f ) * ( i1 , j1 ) `1 .= ( GoB f ) * ( i1 , j1 ) `1 ;
X1 = X2 \/ X1 \/ X2 .= X1 \/ X2 ;
[. a , b .] = { 1_ G } ;
let V , W be non empty addLoopStr ;
dom g2 = the carrier of I[01] ;
dom f2 = the carrier of I[01] ;
( proj2 | X ) .: X = ( proj2 | X ) .: X ;
f . ( x , y ) = h1 . ( x , y ) ;
x0 < a1 . n & a1 . n < a1 . n ;
|. ( f /* s ) . k - ( f /* s ) . k .| < r ;
len Line ( A , i ) = width A ;
SIf ^ .: ( S . ( g . x ) ) = { S ( ) } ^ ( g . x ) } ;
reconsider f = v + u as Function of X , Y ;
intloc 0 in dom ( Initialized p ) ;
i1 <> i2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2 & i2 <> j2
#Z r + r * PI = cos . ( r + PI ) ;
for x st x in Z holds ( ( f2 * f1 ) `| Z ) . x = 1 / ( ( exp_R . x ) ^2 )
reconsider q2 = sqrt ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) as Element of REAL n ;
( 0 qua Nat ) + 1 <= i + 1 ;
assume f in the carrier of X & g in the carrier of Y ;
F . a = H . ( ( { x } \leftarrow { y } ) ) ;
true ( T ) = TRUE ;
dist ( a * seq . n ) < r ;
1 in the carrier of \lbrack 0 , 1 .] ;
( p2 `1 ) ^2 - ( p2 `1 ) ^2 > - ( p2 `1 ) ^2 - ( p2 `2 ) ^2 ;
|. r1 .| = |. a .| * |. r1 .| ;
reconsider Sseq = 8 as Element of Seg len seq ;
( A \/ B ) .: ( A \/ B ) c= A ^ B
D00W . ( len DW ) = DW . ( len W ) + 1 ;
i1 = [: NAT + n , Seg n :] & j1 = [: { n } , { n } :] ;
f . a [= f . ( O O O ) "\/" f . ( O O ) ;
attr f = v & g = u + v ;
I . n = \int ( F . n ) | E ;
\raise .4ex : are _of ( T , S ) . s = 1 ;
a = VERUM ( A ) or a = VERUM ( A ) ;
reconsider k2 = s . ( k2 + 1 ) as Element of NAT ;
( Comput ( P , s , 4 ) ) . intpos ( 0 + 4 ) = 0 ;
L~ ( M1 , j ) meets L~ M2 ;
set h = the continuous Function of X , Y , ( ) | X ;
set A = { L . ( n + 1 ) where n is Nat : n <= len L } ;
for H st H is negative holds P [ H ]
set bOne = { S where S is Subset of T : S is open & V c= S } ;
Hom ( a , b ) c= Hom ( a , b ) ;
sqrt ( 1 + ( n + 1 ) ^2 ) < sqrt ( 1 + ( n + 1 ) ^2 ) ;
( l ) `1 = [ [ l , l ] `1 , l ] `1 ;
y +* ( i , y ) . i in dom g ;
let p be Element of [: A , B :] ;
X /\ X1 c= dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1
p2 in rng ( f ^ ) ;
1 <= indx ( D2 , D1 , j1 ) + 1 ;
assume x in K /\ ( K /\ L ) ;
- 1 <= ( ( f2 . O ) `1 ) ^2 + ( f2 . O ) ^2 ;
f , g be Function of I[01] , ( TOP-REAL 2 ) | K1 , ( ( TOP-REAL 2 ) | K1 ) | K1 ;
k1 -' k2 = k1 - k2 + 1 .= k1 - k2 + 1 ;
rng seq c= ]. x0 - r , x0 + r .[ ;
g2 in ]. x0 - r , x0 + r .[ ;
sgn ( p , K ) = - ( - 1 ) .= - 1 ;
consider u being Nat such that b = p |^ u and u in { y } ;
attr a is ]. a , b .[ means : Def3 : a < b ;
Cl ( H ) = union ( { H } ) ;
len t = len ( ( len t ) + 1 ) + len <* t *> .= len t + 1 ;
Sseq = v + w .= ( v + w ) * ( ( - w ) * ( ( - w ) * ( ( - w ) * ( ( - w ) * ( ( - w ) * ( ( - w ) * ( ( - w ) * ( (
v <> DataLoc ( t . a , 3 ) ;
g . s = sup ( { d } ) ;
( <* y *> . s ) . ( s . y ) = s . ( s . y ) ;
{ s : s < t } = { { s }
s ` \ s = s ` .= s ` \ s ` .= s ` ;
defpred P [ Nat ] means B + 1 in A ;
( 329 + 1 ) ! = 329 * ( 329 + 1 ) ;
U = { [ A , B ] } ;
reconsider y1 = y as Element of REAL n ;
consider i2 being Integer such that y1 = p * ( i2 + 1 ) ;
reconsider p = Y | ( Seg k ) as FinSequence of NAT ;
set f = ( S , U ) -TruthEval , g = ( S , U ) -TruthEval , z = ( S , U ) -TruthEval , m = ( S , U ) -TruthEval , m = S , z = m , m = m , m = m , z = m ,
consider Z being set such that lim s = Z and Z in F ;
let f be Function of I[01] , ( TOP-REAL 2 ) | K1 , ( ( TOP-REAL 2 ) | K1 ) | K1 ;
M . ( n + i ) <> 1 ;
ex r be Real st x = r & a <= b & r <= b & b <= r & r <= b & r <= b & r <= b & r <= b & r <= b & r <= b & r <= b & r <= b & b <= b & r <=
R1 , R2 be Element of REAL n , R2 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 ` = 'not' ( a 'or' b ) ;
||. x1 - y1 .|| < r / 2 ;
b , c // b , c ;
1 <= k2 & k2 <= k & k + 1 <= len ( k2 , k ) + 1 & k + 1 <= len k2 + 1 ;
sqrt ( ( ( ( p `1 / |. p .| - cn ) / ( 1 + cn ) ) ^2 ) - ( ( p .| ) ^2 ) ) >= 0 ;
sqrt ( ( ( q `1 / |. q .| - cn ) / ( 1 + cn ) ) ^2 ) < 0 ;
Cage ( C , n ) /. ( i + 1 ) in LSeg ( Gauge ( C , n ) * ( i , j ) , G * ( i , j ) ) ;
consider e being Element of NAT such that a = 2 * e + 1 ;
Re ( F | D ) = Re ( F | D ) ;
LIN b , a , c or LIN b , c , c ;
p `1 <= a `1 or p `1 <= b ;
g . n = a * Sum ( f ) .= a * Sum ( f ) ;
consider f being Subset of X such that e = f and f is open ;
F | ( N2 \times N2 ) = F * ( N1 , N2 ) ;
q in LSeg ( q , v ) \/ LSeg ( q , v ) ;
Ball ( m , r ) c= Ball ( m , r ) ;
the carrier of V = { 0. V } ;
rng ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#)
assume that Re ( seq ) is summable and Im ( seq ) is summable and Im ( seq ) is summable ;
||. ( ||. seq .|| . n ) - ( ( seq . n ) - ( seq . n ) ) .|| < e / 2 ;
set g = O --> 1 ;
reconsider t2 = t2 as 0 -element string of S2 , t = 0 as 0 -started string of S ;
reconsider x9 = seq . ( n + 1 ) as Element of REAL n ;
assume that Index ( E-max C , E-max C , E-max C ) meets L~ Cage ( C , n ) and not Cage ( C , n ) meets L~ Cage ( C , n ) ;
- ( card { 1 } - ( n + 1 ) ) < F . ( n + 1 ) - F . ( n + 1 ) ;
set d = dist ( x1 , x2 ) , e = dist ( x2 , x3 ) , f = dist ( x2 , x3 ) , f = dist ( x2 , x3 ) , g = dist ( x2 , x3 ) , g = dist ( x2 , x3 ) , h = dist ( x2 , x3 )
2 |^ ( 2 -' 1 ) -' 1 = 2 |^ ( 2 -' 1 ) - 1 ;
dom ( ( ( len d ) |-> ( len d ) ) ) = Seg len d ;
set x1 = ( - ( k + 1 ) ) * ( ( k + 1 ) - ( k + 1 ) ) + ( k + 1 ) * ( ( k + 1 ) ) ;
assume for n being Element of NAT holds 0 <= F ( n ) ;
assume 0 <= ( ( ( ( G . i ) `1 ) ^2 & ( G * ( i , j ) `1 ) ^2 <= 1 & ( G * ( i , j ) `1 ) ^2 ) ^2 <= 1 ;
for A being Subset of X holds c . ( A /\ B ) = c . ( A /\ B )
the carrier of ( ( TOP-REAL 2 ) | K1 ) \/ K1 c= K1 ;
'not' p => ( 'not' p ) is valid
( f | n ) /. ( k + 1 ) = f /. ( k + 1 ) ;
reconsider Z = { [ {} , {} ] } as Element of the carrier of K ;
Z c= dom ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * (
|. 0. TOP-REAL 2 - ( q `2 ) .| < r - ( q `2 ) ;
zeroed \ B c= zeroed ( A , B ) ;
E = dom ( ( ( ( G . n ) | E ) | E ) ) & ( ( G . n ) | E ) | E ) | E ) | E is E ;
C |^ ( A + B ) = C |^ ( A + B ) ;
the carrier of W2 c= the carrier of V & the carrier of V c= the carrier of W ;
I . IC Comput ( P , s , 2 ) = P . IC Comput ( P , s , 2 ) ;
attr x > 0 implies ( x / ( x + 1 ) ) ^2 = x / ( x + 1 ) ;
LSeg ( f , i ) = LSeg ( f , i ) ;
consider p being Point of T such that C = { [. p , q .] : p <= q & q <= p } ;
b , c // b , c & b , c // b , c implies b , c // b , c
assume f = id the carrier of G & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & f is one-to-one & g
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 [: U , U :] , { U } ;
A1 in the carrier of ( ( G . k ) | X ) ;
|. x - y .| = |. x - y .| .= |. x - y .| .= |. x - y .| ;
set S = 1GateCircStr ( <* x , y *> , '&' ) ;
Fib ( n ) * ( Fib ( n ) ) >= 4 * sqrt 5 ;
v1 /. ( k + 1 ) = v . ( k + 1 ) ;
0 mod i = Re ( i * ( 0 qua Nat ) ) ;
the carrier of ( ( TOP-REAL n ) | K1 ) = [: the carrier of ( TOP-REAL n ) | K1 , the carrier of ( TOP-REAL n ) | K1 :] ;
Line ( S1 , j ) = Line ( S1 , j ) ;
h . x1 = [ y1 , y2 ] ;
|. f .| ^ ( |. ( f . ( card b ) .| * ( ( card b ) * ( ( b . d ) ) ) ) .| ) .| is non-negative ;
assume x = ( a1 ^ <* b1 *> ) . ( len <* b1 *> ) ;
M is_closed_on IExec ( I , P , s ) , P , Initialize s , Initialize s ) ;
DataLoc ( t2 . a , 4 ) = intpos ( 0 + 4 ) ;
x + y < x + y & |. x - y .| = - x + y ;
LIN c , q , c9 & q , c9 // q , c9 ;
{ f . 1 } = f . ( 0 , t ) .= a ;
x + ( y + z ) = x1 + y1 + y2 .= x1 + y1 + y2 ;
f1 . a = f1 . a & f2 . a = f2 . a & f2 . a = f2 . a ;
( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ^2 <= ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) ^2 ;
set R1 = Cage ( C , n ) /. ( i + 1 ) ;
( p `1 ) ^2 >= ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) ^2 ;
consider p such that p = { p1 : p1 < p & p < 1 } and p in { p1 } ;
|. ( f /* ( s ) ) . l - ( f /* ( s ^\ k ) ) .| < r ;
Segm ( M , p ) = Segm ( M , p ) ;
len Line ( N , k + 1 ) = width N ;
f1 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /* seq is convergent & f2 /*
f . x1 = x1 & f . x2 = y1 & f . x2 = y2 ;
len f <= len f + 1 & len f + 1 = len f + 1 ;
dom ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) * ( ( Proj ( i , n ) *
n = k * ( 2 * ( n + 1 ) ) + ( 2 * ( n + 1 ) ) ;
dom B = 2 \ { {} } ;
consider r such that r _|_ a and r _|_ b ;
reconsider B1 = the carrier of Y , B2 = the carrier of X , Y = the carrier of Y , Z = the carrier of X , Z = the carrier of Y , Y = the carrier of Z ;
1 in the carrier of [. - 1 , 1 .] ;
let L being complete LATTICE , L be lower-bounded LATTICE ;
[ q1 , q2 ] in [: I , I :] \ { I } ;
set S2 = 1GateCircStr ( x , y , c ) ;
assume f1 is_differentiable_on Z & f2 is_differentiable_on Z & f2 is_differentiable_on Z & f2 is_differentiable_on Z ;
reconsider y = ( a ` ` ) ` as Element of L ;
dom s = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 7 , 8 , 7 , 8 , 7 , 8 , 8 , 7 } & s . 1 = <* 5 , 6 , 6 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8 , 7 , 8
( min ( g , c ) ) . ( c + 1 ) <= h . ( c + 1 ) ;
set G2 = the InternalRel of G , G1 = the InternalRel of G , G2 = the InternalRel of G , G2 = the InternalRel of G , G2 = the InternalRel of G , G1 = the InternalRel of G , G2 = the InternalRel of G , G2 = the InternalRel of G , G1 = the InternalRel of G , G2
reconsider g = f as PartFunc of REAL , REAL n ;
|. s1 . m - p .| / ( n + 1 ) .| < d / ( n + 1 ) ;
for x being element st x in u . ( u + v ) holds x in v . ( u + x )
P = the carrier of ( TOP-REAL n ) | K1 .= K1 ;
assume p1 in LSeg ( p01 , p2 ) /\ LSeg ( p01 , p2 ) ;
( 0. X \ ( k + 1 ) ) |^ ( m + 1 ) = 0. X ;
let g be Element of hom ( cod f , g ) ;
2 * a * ( 2 * b + c ) <= 2 * ( ( 2 * a ) * b ) ;
f , g be PartFunc of X , Y , h be PartFunc of X , Y ;
set h = hom ( a , f ) ;
then idseq ( n ) | Seg ( m + 1 ) = idseq ( m + 1 ) ;
H * ( g " * a ) in the carrier of H ;
x in dom ( ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 )
cell ( G , i1 , j1 -' 1 , j1 ) misses C ;
LE q2 , p2 , P , p1 , p2 & LE q2 , p2 , P , p2 , p1 , p2 & LE q2 , p2 , P , p1 , p2 , p2 & q2 , p2 , p1 , p2 , p2 & P is closed ;
attr B is Subset of A means : Def3 : B c= A & B is open & A is open & B is open ;
deffunc D ( set , set ) = union rng ( $1 ^ <* f . ( $1 + 1 ) *> ) ;
n + 1 - ( n + 1 ) < len ( p1 + p2 ) - ( n + 1 ) ;
attr a <> 0. K means : only : card ( a * M ) = card ( a * M ) ;
consider j such that j in dom Fhom and I = len b1 + j and I = len b1 + j ;
consider x1 such that z in x1 and x1 in { x1 , x2 } and x1 in { x2 } ;
for n , r being Element of REAL n st X [ n , r ] holds X [ n , r ]
set Cs2 = Comput ( P2 , s2 , i ) , P2 = Comput ( P2 , s2 , i ) , P2 = Comput ( P2 , s2 , i ) , s2 = P2 ;
set v = 3 / ( a * b ) , w = 3 / ( a * b ) , w = - ( b * c ) ;
conv ( F .: ( { E } ) c= ( F .: ( { E } ) ) .: ( { E } )
1 in [. - 1 , 1 .] /\ dom ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) (
s3 <= s1 + 0 + 1 - ( n + 1 ) ;
dom ( f * ( ( f * ( ( g ) `| Z ) ) `| Z ) = dom f /\ dom ( g `| Z ) ;
dom ( f * F ) = dom ( l * F ) /\ Seg ( k + 1 ) ;
rng ( s ^\ k ) c= dom ( f1 /* s ) \ { x0 } ;
reconsider g1 = g1 as Point of TOP-REAL n ;
( T * h ) . ( s . ( n + k ) ) = T . ( s . ( n + k ) ) ;
I . ( J . ( x , y ) ) = ( I * J ) . ( x , y ) ;
y in dom <* <* 19 *> *> ^ <* o *> ;
for I being non degenerated commutative Ring , J being non empty doubleLoopStr , I being Program of L holds I is commutative associative
set s2 = s +* ( ( intloc 0 ) .--> 1 ) , P2 = P +* I +* I , s2 = P +* I +* I , P2 = P +* I +* I +* I ;
P1 /. IC Comput ( P1 , s1 , k ) = P1 . IC Comput ( P2 , s2 , k ) ;
lim S1 in the carrier of ( ]. a , b .[ | [. a , b .[ ) ;
v . ( i + 1 ) = ( v *' ) . ( i + 1 ) ;
consider n being element such that n in NAT and x = seq . n ;
consider x being Element of c such that F1 . x <> F2 . x ;
\langle X , 0 , 0 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0
j + ( 2 * ( ( 2 * ( k + 1 ) ) + 1 ) / ( 2 * ( 2 * ( 2 * ( 2 * ( k + 1 ) ) ) ) ) ) > j + 2 ;
{ s , t } on { s , t } & { t , s } on Q & { t , s } on Q } on Q ;
n1 > len ( p2 , n1 ) + len ( p2 , n1 ) ;
( g1 . ( HT ( g2 , T ) ) . ( HT ( g1 , T ) ) = 0. L ;
then H , H1 , H2 , H , H , H , N , H , N , H , H , N , H , H , N , H , H , H , N ;
( ( E-max L~ Cage ( C , n ) ) .. f > 1 ;
]. s , 1 .[ = ]. s , 1 .[ /\ ]. s , 1 .[ ;
x1 in [#] ( TOP-REAL 2 ) | K1 ;
let f1 , f2 be PartFunc of REAL , REAL , f2 be PartFunc of REAL , REAL , i be Element of NAT st f1 = f2 & f2 is PartFunc of REAL , REAL , j be Nat holds f1 /. ( i + 1 ) = f2 /. ( i + 1 )
DigA ( tmax , tmax , tmax ) is Element of k -tuples_on BOOLEAN ;
I . ( 122222222222222222222222222222) = I . ( k + 1 ) & I . ( k + 1 ) = I . ( k + 1 ) ;
{ u } \/ { v } = { [ a , b ] } ;
( w | ( p | ( p | ( p | ( p | ( p | ( p | ( p | ( p | ( p | p ) ) ) ) ) ) ) ) ) ) | ( ( ( | ( p | ( p | ( p | ( p | ( p | ( p | (
consider v2 such that v2 in W2 and x = v + v2 and x = u + v2 and y in W2 and x in W2 and y in W2 and x in W2 ;
for y st y in rng F ex n st y = a |^ n & F . n = a |^ n
dom ( ( g * J ) | ( K /\ L ) ) = K ;
ex x being element st x in ( Sub \/ { 0 } ) . s & A = ( U1 \/ { 0 } ) . s ;
ex x being element st x in ( ( w \/ { A } ) \/ { A } ) . x ;
f . x in the carrier of ( TOP-REAL n ) | K1 ;
( the carrier of X1 union X2 ) /\ ( the carrier of X1 union X2 ) <> {} ;
L1 /\ LSeg ( p1 , p2 ) c= { p1 } ;
sqrt ( b + b-2 ) in { r : a < b & b < b } ;
ex_sup_of { x , y } , L ;
for x being element st x in X ex u being element st P [ x , u ]
consider z being Point of G such that z = y and for n being Nat holds P [ n , z ] ;
( the carrier of ( \langle L , L *> ) . xx <= e ;
len ( w ^ <* w *> ) + 1 = len w + 1 ;
assume q in the carrier of ( TOP-REAL 2 ) | K1 ;
f | ( E \ { E } ) = g | ( E \ { E } ) ;
reconsider i1 = x1 , i2 = x2 as Element of NAT ;
( a * A ) ` = ( a * A ) ` .= ( a * A ) ` ;
assume pred f |^ ( n + 1 ) is not zero means : Def3 : for n be Element of NAT holds f |^ n is not zero ;
Seg len ( ( f2 ^ ) ) = dom ( ( f2 ^ ) ^ <* d *> ) ;
( ( Complement A ) . ( m + 1 ) ) . ( n + 1 ) c= ( ( Complement A ) . ( n + 1 ) ) . ( n + 1 ) ;
f1 . p = p1 & f2 . p = p2 & f2 . p = p2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p = q2 & f2 . p
FinS ( F , Y ) = FinS ( F , Y ) ;
( x | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | | ( y | | ( y | | ( y | | ( y | | ( | ( | ( | ( y | | ( | | ( ) ) ) ) ) ) ) ) ) ) ) ) ) = z ;
sqrt ( x ^2 + ( r ^2 ) / ( 2 * ( 2 * ( 2 * ( 2 * ( ( 2 * ( ( 2 * ( 1 + r ) ) ) ) ) ) ) ) <= sqrt ( r * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
Sum ( f ) = Sum ( f ) & dom ( g | ( dom f ) ) = dom ( g | ( dom f ) ) ;
assume for x , y being set st x in Y & y in X holds x /\ y in Y ;
assume W1 is Subspace of W2 & W2 is Subspace of V & W is Subspace of V ;
||. t . x - ( ||. x .|| ) . n .|| = ||. ( ( x - t ) . n ) - ( ( x - t ) . n ) .|| ;
assume that i in dom D and f | A is bounded and g | A is bounded ;
sqrt ( ( ( ( p `1 ) ^2 ) ^2 ) <= ( ^2 + ( |. p .| ) ^2 ) ^2 ) ;
g | Sphere ( p , r ) = id ( the carrier of S ) ;
set N8 = ( E-max L~ Cage ( C , n ) ) `1 , ( E-max L~ Cage ( C , n ) ) `1 , ( E-max L~ Cage ( C , n ) ) `2 ;
let T being non empty TopStruct , A be Subset of T ;
width B |-> ( 0. K ) = len B .= len A ;
attr a <> 0 implies ( A \ B ) \ ( A \ B ) = ( A \ B ) \ ( A \ B )
then f is_differentiable_on ]. - 1 , 1 .[ ;
assume that a > 0 and a <> 1 and b <> 0 and a <> 0 & b <> 0 & a <> 0 & b <> 0 & b <> 0 & a <> 0 & b <> 0 & b <> 0 & a <> 0 & b <> 0 & b <> 0 & b <> 0 & a <> 0 & b <> 0 & b
w , y // w , y ;
p2 /. IC Comput ( p2 , s2 , k ) = p2 . IC Comput ( p2 , s2 , k ) ;
ind ( ( b1 | b ) | b ) = ind ( b1 | b ) .= ind ( b1 | b ) ;
[ a , A ] in Indices M & A c= B implies A is being_line
m in ( the Arrows of C ) . ( ( the Morphism of C ) . ( m , n ) , ( the Morphism of C ) . ( m , n ) ) ;
( 'not' a , PA ) . z = TRUE ;
reconsider \varphi = \varphi , \varphi = \varphi , \varphi = \varphi , \varphi = l , \varphi = l , \varphi = l , \varphi = ( l , u ) , \varphi = u , u = u , u = l , u1 = u , u2 = u , FF = u , FF = u , FF = u , FF = u , FF = u , FF =
len s1 * ( len s2 ) - 1 > 0 ;
x0 * ( ( f . ( ( ( ( ( ( ( ( ( ( ( ( ( ( - / 2 ) * ( ( f / 2 ) * ( ( f / 2 ) * ( ( ( ( f / 2 ) * ( ( ( ( f / 2 ) * ( ( ( ( ( f ) ) * ( ( f / 2 ) * ( ( ( f / 2 )
[ f1 , f2 ] in [: the carrier of A , { the carrier of A } :] ;
the carrier of ( TOP-REAL 2 ) | K1 = K1 & ( TOP-REAL 2 ) | K1 = K1 or ( TOP-REAL 2 ) | K1 = K1 & ( TOP-REAL 2 ) | K1 = K1 or ( TOP-REAL 2 ) | K1 = K1 or ( TOP-REAL 2 ) | K1 = K1 & ( ( TOP-REAL 2 ) | K1 ) | K1 = K1 ) & ( TOP-REAL 2 ) | K1 = K1 )
consider z being element such that z in dom g2 and p = g2 . z ;
[#] V = { 0. V } .= { 0. V } ;
consider P2 being FinSequence such that rng P2 = M and for n being Nat st n in dom P2 holds P2 . n = F ( n ) ;
assume that x1 in dom ( f | X ) and f | X is bounded ;
h1 = f ^ <* ( <* p *> ^ <* q *> ) /. ( len f + 1 ) *> .= f /. ( len f + 1 ) ;
c /. [ b , c ] = c /. [ b , c ] .= c /. [ d , c ] ;
reconsider t1 = p1 , t2 = p2 , t2 = p3 as Element of the carrier of C ;
sqrt ( 1 - ( 2 * ( 1 + ( 2 * ( 1 + ( 2 * ( 1 + ( 2 * ( 1 + 1 ) ) ) ) ) ) ) in the carrier of ( TOP-REAL n ) | K1 ) ;
ex W being Subset of X st p in W & h .: W c= V
( h . p1 ) `2 = C * ( ( 1 - l ) `2 ) `2 + D * ( 1 - l ) `2 ) `2 .= C * ( 1 - l ) `2 + D * ( 1 - l ) `2 ;
R . b = 2 * PI .= 2 * PI ;
consider \lambda such that B = 1- C + ( r1 + 1 ) and r1 <= r1 + 1 ;
dom g = dom ( ( the Sorts of A ) * ( the ResultSort of S ) ) ;
[ P . ( l + 1 ) , P . ( l + 1 ) ] in { [ l , l ] } ;
set s2 = Initialize s1 , P2 = P +* I , s2 = P +* I , P2 = P +* I , P2 = P +* I , s2 = P +* I , P2 = P +* I , s2 = P +* I , P2 = P +* I , s2 = P +* I , P2 = P +* I , s2 = P +* I , P2 = P +* I ,
reconsider M = mid ( z , i2 , j1 ) as Matrix of len z , len z ;
y in product ( ( Carrier ( J ) ) ) ;
1 / ( 0 , 1 ) = 1 & ( 1 - ( 0 , 1 ) / ( 0 , 1 ) ) / ( 1 - ( 1 - ( 1 - ( 1 - 1 ) ) / ( 1 - ( 1 - ( 1 - 1 ) ) ) / ( 1 - ( 1 - 1 ) ) = 1 ;
assume x in the carrier of g or x in the carrier of g ;
consider M being strict Subspace of A such that a = M and M is Subspace of A ;
for x st x in Z holds ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 ) /
len W1 + len W2 + len W1 = 1 + len W2 + len W2 ;
reconsider h1 = { v where v is VECTOR of X : v in W & w in W & v in W } as Lipschitzian of X ;
( i1 mod ( len p ) + 1 ) + 1 in dom ( p mod ( len p ) ) ;
assume that s2 is convergent and F is convergent and F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & F is convergent & ( ( for n st n >= k holds F . n <= ( ( for n holds F . n
( ( ( mod ( x , y ) ) , ( ( mod ( x , y ) ) , ( ( mod ( x , y ) , ( ( y , z ) , ( x , z ) ) ) ) ) mod ( ( x , y ) mod ( ( x , y ) mod ( ( y , z ) mod ( ( y , z ) mod
for u being element st u in Bags n holds ( p *' u ) . ( u + u ) = p . u
for B being Subset of { u } st B in E holds A /\ B = B or A /\ B = B
ex a being Point of X st a in A & A /\ B = { a } ;
set W2 = [: p , q :] \/ { p } ;
x in { X where X is Subset of L : X is directed & X is directed } ;
the carrier of W1 /\ ( the carrier of W2 ) c= the carrier of W1 /\ ( the carrier of W2 ) ;
[ a + b , b ] in [: { a , b } , { b } :] ;
( dom ( X --> ( f . x ) ) . n = ( X --> ( f . x ) ) . n ;
set x = the Element of ( g . n ) /\ ( g . n ) ;
p => ( q => r ) in TAUT ( A ) ;
set \pi = LSeg ( G * ( i1 , j1 ) , G * ( i1 , j1 ) ) ;
set \pi = LSeg ( G * ( i1 , j1 ) , G * ( i1 , j1 ) ) ;
- 1 + 1 <= sqrt ( 2 |^ ( n + 1 ) ) + 1 ;
( reproj ( 1 , z ) . x ) . ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( -
assume b1 . ( r + 1 ) = { c } & b1 . ( r + 1 ) = { c } ;
ex P st a on P & b on P & c on P & P [ b , c ] ;
reconsider g1 = g , g2 = h as strict Subspace of X ;
consider v1 being Element of T such that Q = ( \mathopen { \uparrow } n ) ` and for x being Element of T st x in ( \mathopen { \uparrow } n ) ` holds x in ( \mathopen { \uparrow } n ) ` ;
n in { i where i is Nat : i < n & n <= len f } ;
( F /. ( i + 1 ) ) `1 >= ( ( F /. ( m + 1 ) ) `1 ;
assume K = { p : p `1 >= 1 & p `2 >= 1 & p `2 >= 1 & p `2 >= 1 & p <> 0. K } ;
ConsecutiveDelta ( A , succ ( O , A ) ) = ( ( d , A ) . ( O , A ) ) . ( O , A ) ) . ( O , A ) ;
set I1 = if>0 ( a , k1 , I , J ) ;
for i be Nat st 1 < i & i < len z holds z /. i <> z /. ( i + 1 )
X c= ( the carrier of L1 ) \/ ( the carrier of L2 ) ;
consider x9 being Element of GF ( p ) such that x9 = a and x9 is Element of GF ( p ) ;
reconsider d1 = { e where e is Element of D : e in X & e in X & e in X } as Subset of D ;
ex O being set st O in S & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O c= O & O
consider n being Nat such that for m being Nat st n <= m holds S . m in U ( m ) ;
f * g is_differentiable_on Z ;
defpred P [ Nat ] means A + ( $1 + 1 ) = A ;
the carrier of ( ( TOP-REAL 2 ) | K1 ) = the carrier of ( ( TOP-REAL 2 ) | K1 ) | K1 ;
reconsider p1 = x , p2 = y as Point of TOP-REAL 2 ;
consider g4 such that g2 = y and x <= y and y <= z and z <= z and x <= y ;
for n being Element of NAT , r being Element of REAL n st X [ n , r ] holds X [ n + 1 ]
len ( x2 ^ y2 ) = len ( x2 ^ y2 ) + len y2 .= len y2 + len y2 + len y2 + len y2 + len y2 ;
for x being element st x in X holds x in the carrier of ( n + 1 ) \ { x }
LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = {} ;
func <* X ( ) *> -> set equals { <* X ( ) } \/ { X ( ) } ;
len ( <* C *> ^ ( <* D *> ^ ( <* E *> ^ ( <* E *> ^ ( <* F *> ) ) ) ) ) <= len <* E *> + len <* F *> ;
attr K is with_addition means : Def3 : for i st i <> n holds ( ( i + 1 ) |^ ( n + 1 ) ) |^ ( i + 1 ) = i * ( ( i |^ n ) |^ ( i + 1 ) ) ;
consider o being OperSymbol of S such that t = [ o , the carrier of S ] and t = [ o , the carrier of S ] ;
for x st x in X holds ex y being Element of X st y c= X & y in X & y in X & x in X ;
IC Comput ( P , s , k ) in dom ( P +* I ) ;
attr q < s & r < q & s < q & q in ]. p , q .[ implies ]. p , q .[ \ ]. p , q .[ c= ]. p , q .[ ;
consider c being Element of Class ( f , c ) such that Y = ( F . ( c , d ) ) . c and for i being Element of NAT holds ( F . i ) . c = F . ( c , d ) ;
the ResultSort of S2 = id the carrier of S2 & the carrier' of S2 = the carrier' of S2 ;
set y9 = [ <* y , z *> , f2 ] ;
assume x in dom ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) * ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( (
{ |[ f , g ]| : [ f , g ] in Indices GoB f } c= L~ f ;
( q `1 ) ^2 >= ( ( ( Cage ( C , n ) * ( i , j ) `1 ) ^2 ;
set Y = { a "/\" b where a is Element of L : a in X } ;
i -' len f <= len f + 1 - len f ;
for n holds x in N & h . n = x0 + r
set i0 = ( > ( ( a , I , J ) . a , I ) . a , p ) ;
( p . k ) . 0 = 1 or ( p . 0 ) . 0 = 1 ;
u + Sum ( ( U \ { u } ) \/ { v } ) in ( U \ { u } ) \/ { v } ;
consider x9 being set such that x in x9 and x9 in B and x9 in B and x9 in C and x9 in C and x9 in D and x9 in D and x9 in C and x9 in D ;
( p ^ q ) . ( m + 1 ) = ( ( q | ( k + 1 ) ) . m + ( q | ( k + 1 ) ) . m ;
g + h = g1 + g2 & g + h = g1 + g2 implies g is h
L1 is distributive & L2 is distributive implies L1 + L2 is distributive
hence x in rng f & y in rng f implies f . x = f . y
assume that 1 < p and p < 1 and p = |[ a , b ]| and p <> 0. TOP-REAL 2 and p <> 0. TOP-REAL 2 and p <> 0. TOP-REAL 2 ;
F! ( f , t ) = rpoly ( 1 , t ) *' ( f , t ) .= Exec ( 1 , t ) *' ( f , t ) ;
let X being set , A being Subset of X , B being Subset of X st A = {} & B is open & A is open & A is open holds A is open
( ( ( N-min L~ Cage ( C , n ) ) .. Cage ( C , n ) ) `1 <= ( ( E-max L~ Cage ( C , n ) ) `1 ;
let c being Element of the bound A , a , b being Element of the bound A ;
s1 . intpos ( 0 + 1 ) = ( Exec ( i , s1 ) . intpos ( 0 + 1 ) ) . intpos ( 0 + 1 ) .= s . intpos ( 0 + 1 ) ;
let a , b , c , d be Real ;
for x , y being Element of X holds ( x \ y ) \ ( x \ y ) = ( x \ y ) \ ( x \ y )
mode InternalRel of i , j , m , n , m , n , m , k be Nat ;
set x2 = ( Re ( y ) ) | ( ( Im y ) | ( ( x | ( ( x | ( ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( y | ( ) ) ) ) ) ) ) ) ) ) ;
[ y , x ] in dom g & [ y , x ] in g . ( y , x ) ;
]. ( D , k ) / ( n + 1 ) , r .[ c= A ;
0 <= delta ( 2 ) & |. seq . n .| < r / 2 ;
( - ( q `1 ) / ( |. q .| - cn ) ) ^2 <= ( - ( q .| / ( |. q .| - cn ) ) ^2 ;
set A = sqrt 2 ;
for x , y being set st x in R & y in R & x in R & y in R & x in R holds x in R
deffunc { F ( Nat ) = b . ( $1 + 1 ) * ( M . ( $1 + 1 ) ) ;
for s being element holds s in |= ( f ) iff s in |= ( f ) \/ |= ( f )
let S being non empty non empty TopSpace ;
max ( ( ( |. z .| - |. z .| ) / ( |. z .| - cn ) ) + ( ( |. z .| - cn ) / ( ( |. z .| - cn ) ) ) ) / ( ( |. z .| - cn ) ) ) ) >= 0 ;
consider n1 being Nat such that for k being Nat holds ( for n be Nat holds ( seq . n ) . k ) `1 < r ;
Lin ( A /\ B ) is Subspace of V & Lin ( B ) is Subspace of V
set nX1 = { n } '&' ( ( M . ( n + 1 ) ) . ( x ) where n is Element of NAT ) : n <= len ( ( M . n ) . x ) } ;
f " ( [#] X ) in SX & f " ( [#] X ) in D & f " ( [#] X ) in D ;
rng ( a ^\ c ) c= { a } \/ { b } ;
consider y being Vertex of G such that y = y & dom ( y .--> x ) = dom ( y .--> x ) ;
dom ( ( ( ( 1 / 2 ) (#) ( ( ( ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( - 2 ) / 2 ) * ( ( ( 1 - 2 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) /\ ( ( ( 1 / 2 ) * ( ( ( ( ( ( - 2 ) / 2 ) * ( ( (
AffineMap ( i , j , r ) is Element of REAL n ;
v ^ ( ( n |-> ( ( m + 1 ) ) |-> ( a , b ) ) ) . ( v , ( n + 1 ) ) . ( v , ( m + 1 ) ) . ( v , ( v + 1 ) ) ) . ( v , ( m + 1 ) ) ) . ( v , ( m + 1 ) ) . ( v , ( v + 1 ) ) .
ex a , k1 , k2 being Nat st i = a & k1 = b & k2 = b & k2 = b & k2 = b & k2 = c & k2 = c & k2 = c & k2 = d & k2 = c & k2 = d & k2 = c & k2 = d & k2 = d & k2 = c & k2 = d & k2 = c & k2 = d ;
t . NAT = ( NAT .--> ( i1 + 1 ) ) . NAT .= ( i1 + 1 ) . NAT .= i1 + 1 .= i1 + 1 ;
assume that F is bbfamily and rng p = { n } and dom p = { n } and dom p = { n } ;
not b , c // b , c & b , c // b , c
( L1 ! ) . O O O O O O O O O O ) c= ( L1 L1 ) . O & ( L1 L1 ) . O O O O O O O O O O O ) . O ;
consider F being ManySortedSet of E such that for d being Element of E holds F . d = F ( d ) ;
consider a , b such that a * ( is_differentiable_on Z ) and 0 < b and a < b and b < 1 ;
defpred P [ FinSequence of D ] means |. Sum ( $1 ) .| <= Sum ( $1 ) - ( Sum ( $1 ) ) ;
u = cos ^ ( x , y ) .= cos . ( x , y ) .= cos . ( x , y ) ;
dist ( seq . n ) + dist ( seq . n ) <= dist ( seq . n ) + dist ( seq . n ) ;
P [ p , |. p .| ] ;
consider X being Subset of [: A , B :] such that X c= Y and X is open and Y is open and X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open or X is open ;
|. b .| * |. b .| >= |. b .| * |. b .| ;
1 < ( E-max L~ Cage ( C , n ) ) .. Cage ( C , n ) ;
l in { l1 where l1 is Real : l <= len h & l <= len h } ;
( Partial_Sums ( G ) . ( n + 1 ) <= ( Partial_Sums ( G ) ) . ( n + 1 ) ;
f . y = x * ( 0. L ) .= x * ( 0. L ) .= x ;
NIC ( i , l ) = { i1 , i2 } ;
LSeg ( p1 , p2 ) /\ LSeg ( p2 , p2 ) = { p2 } ;
( ( ( ( the support of ( I ) +* ( i , I ) ) +* ( i , I ) ) +* ( i , I ) ) ) . ( ( i + 1 ) ) . ( ( i + 1 ) ) . ( ( i + 1 ) + 1 ) in { i } ;
Following ( s , n ) | ( the carrier of S1 ) = Following ( s , n ) | ( the carrier of S2 ) ;
W is_not to ( q `1 ) & W is open & W is open & ( ex W being Subset of TOP-REAL 2 st W is_not empty /\ ( ( ) & ( for p being Point of TOP-REAL 2 st p in W holds W is open & ( for p being Point of TOP-REAL 2 st p in W holds W . p <= ( ( ( ( ( ) `1 ) / |. p .| - sn ) ^2 ) ) * ( ( 1 - sn ) ^2 )
f /. i2 <> f /. ( i1 + 1 ) ;
M |= ( f . ( x , y ) ) . ( ( x , y ) . ( x , y ) ) . ( ( x , y ) . ( x , y ) ) . ( x , y ) ) ;
len ( ( P ^ <* a *> ) ^ <* a *> ) in dom ( P ^ <* a *> ) ;
A |^ ( m + 1 ) c= A |^ ( m + 1 ) & A |^ ( m + 1 ) c= A |^ ( m + 1 ) ;
{ { q : |. q .| < 1 & |. q .| < 1 } c= { q : |. q .| >= 1 } ;
consider n1 being element such that n1 in dom p1 and p1 = p1 . n1 and p1 = p1 . n1 and p2 = p1 . n1 and p1 = p1 . n1 ;
consider X being set such that X in Q and for x being set st x in Q holds X [ x , X . x ] ;
CurInstr ( P3 , Comput ( P3 , s3 , l ) ) <> halt SCM+FSA ;
for v being VECTOR of l1 , w being Element of l , a , b , c being Element of l , d be Real st a = b & b = c holds ( a + b ) * ( ( b + c ) = ( a + b ) * ( b + c )
for \varphi st \varphi in X holds \varphi . \varphi in X & \varphi . \varphi in X
rng ( ( Sgm dom ( ( dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ) ) * ( ( ( ( ( ) ) * ( ( ( ( ( ) ) / ( ( ( ( ) ) * ( ( ( ( ) ) / ( ( ( ) ) * ( (
ex c being FinSequence of D st len c = k & for k being Nat st k in dom c holds ( for n being Nat st n in dom c holds ( ( for n be Nat st n in dom c holds ( ( for n be Nat st n in dom c holds ( ( for n be Nat st n in dom c holds ( ( for n be Nat st n in dom c holds c . n = a ) ) implies ( ( for n be Nat
the_arity_of ( a , b ) = <* <* <* b , c *> , <* c , d *> *> , <* d , e *> *> , <* d *> *> ;
consider f1 , f2 being Function of X , Y such that f1 = |. f1 - f2 .| and f1 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous and f2 is continuous & f2 is continuous & f2 is continuous & f2 is continuous & f2 is continuous
a1 = b1 & b1 = b2 & b1 = b1 implies b1 = b2 & b1 = b1 & b1 = b2
D2 . indx ( D2 , D1 , j1 ) = D2 . ( indx ( D2 , D1 , j1 ) + 1 ) ;
f . ( |. r .| ) = ||. ( r ) . ( |. r .| ) .= ||. f .|| . ( |. r .| ) .= ||. f .|| ;
consider n being Nat such that for m being Nat st n <= m holds seq . m = seq . ( m + n ) ;
consider d being Real such that for a , b , c being Real st a in X & b in X & c in Y holds a <= b ;
||. L /. ( K + n ) - L /. ( K + n ) .|| <= ||. L /. ( K + n ) - L /. ( K + n ) .|| + L /. ( K + n ) .|| + L /. ( K + n ) - L /. ( K + n ) .|| ;
attr F is commutative associative means : Def3 : for b being Element of X holds F . ( b , c ) = f . b ;
p = 1- ( p `2 ) + ( p `2 ) .= 1 + ( p `2 ) .= ( - p `2 ) * ( 1 + ( p `2 ) ) .= ( - p `2 ) * ( 1 + 1 ) ;
consider z1 such that b , z1 // o , z1 and o , z1 // o , z1 and o , z1 // o , z1 ;
consider i such that Arg ( ( ( ( r ) . ( q `1 ) ) ) = s + r * ( r * ( ( q `1 / 2 ) `1 ) ) ) ;
consider g such that g is one-to-one and dom g = dom f and rng g = { x } and rng g = { x } ;
assume A = P2 \/ { p } & B <> { p } & A <> { p } & A <> { p } & A <> { p } & B <> { p } & A <> { p } & A <> { p } & A <> { p } & B <> { p } & A <> { p } & A <> { p } & B <> { p } & A <> { p } & A <> { p } &
attr F is associative means : only : F .: ( F .: ( rng f ) ) = F .: ( f .: ( rng f ) ) ;
ex x being Element of NAT st m = x & x in z & y in z & z = x & z in z & x in z & y in z ;
consider k2 being Nat such that k2 in dom ( P . ( n + 1 ) ) and l = ( P . ( n + 1 ) ) . k2 ;
seq = r * ( seq . n ) + ( r * ( seq . n ) ) .= r * ( seq . n ) + r * ( seq . n ) ;
F1 . ( [ a , b ] , [ a , b ] , [ b , c ] ) = [ f , g ] . [ a , b ] ;
{ p } "\/" { p } = { p } "\/" { q } ;
consider z being element such that z in dom ( ( F . 0 ) | ( dom F ) ) and ( F . 0 ) | ( dom F ) . z = y ;
for x , y being element st x in dom f & y in dom f & f . x = f . y holds f . x = f . y
cell ( G , i , j ) = { |[ r , s ]| : G * ( i , j ) `1 <= r & r <= s & s <= G * ( i + 1 , j ) `2 } ;
consider e being element such that e in dom ( T | ( E \ { x } ) ) and ( T | ( E \ { x } ) ) . e = v ;
( F `1 ) . ( b1 , b2 ) = ( ( Mx2Tran ( J , b1 , b2 ) ) . ( b1 , b2 ) ) . ( b1 , b2 ) ;
- 1 = ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * ( ( - 1 ) * (
hence for x being set st x in dom f /\ dom g holds ( f | ( dom g ) ) . x <= f . x
len ( ( f1 ^ f2 ) . ( j + 1 ) ) = len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len ( f1 ^ f2 ) .= len f1 + len f2 ;
All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) '&' All ( x , A ) )
LSeg ( E . ( k + 1 ) , F . ( k + 1 ) ) c= Cl ( ( L~ Cage ( C , n ) * ( i + 1 ) ) ` ;
x \ ( a \ b ) = x \ ( a \ b ) .= ( x \ b ) \ ( a \ b ) ;
k - ( Sum ( I ) ) = ( ( I . ( n + 1 ) ) - ( I . ( n + 1 ) ) ) * ( I . ( n + 1 ) ) .= ( I . ( n + 1 ) ) * ( I . ( n + 1 ) ) .= ( I . ( n + 1 ) ) * ( I . ( n + 1 ) ) ;
for s being State of A , n being Nat holds Following ( s , n ) . ( ( ) ) is stable
for x st x in Z holds ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( (
( support ( ( ( ( n ) * ( ( ( ( m , n ) , 1 ) , ( ( n , 1 ) , ( ( m , 1 ) , ( ( n , 1 ) , ( ( n , 1 ) , ( ( m , 1 ) , ( ( n , 1 ) , ( ( n , 1 ) , ( ( m , 1 ) , ( ( n , 1 ) , ( ( n , 1 ) ) , ( ( n , 1 ) ) , (
reconsider t = u as Function of the carrier of A , the carrier of B ;
- ( a * sqrt ( b * ( sqrt ( b + c ) ) ) <= - ( sqrt ( b + c ) * sqrt ( b + c ) ) ;
\varphi . a = g . a & \varphi . a = f . a & \varphi . a = g . a ;
assume that i in dom ( F ^ <* p *> ) and j in dom ( F ^ <* p *> ) ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 } \/ { x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } ;
the Sorts of U1 /\ ( the Sorts of U1 ) c= the Sorts of U2 /\ ( the Sorts of U2 ) ;
( - 2 * a ) * ( ( 2 * a ) * ( ( - 2 * a ) * ( ( - 2 * a ) * ( - 2 * b ) ) ) ) > 0 ;
consider W1 such that for z being element st z in W1 holds ( for z being element st z in W1 holds z in W1 . z iff ex n being Nat st n <= n & n <= len z & n <= len z & ( for z being Nat st z in W1 . z holds P [ z , n ] ) implies P [ z , n ] ;
assume ( the ResultSort of S ) . o = <* a *> . o & ( the ResultSort of S ) . o = <* a *> . o ;
Z = dom ( ( ( ( ( - 1 ) (#) ( ( #Z 2 ) * ( ( ( n + 1 ) * ( ( ( ( ( n + 1 ) * ( ( ( 1 / 2 ) * ( ( ( ( n + 1 ) * ( ( ( n + 1 ) * ( ( ( n + 1 ) * ( ( ( ( n + 1 ) / ( 2 * ( ( ( ( n + 1 ) / ( 2 * ( 2
integral ( f , S ) is convergent & lim ( f , S ) = integral ( f , S ) ;
{ [ a , b ] : [ a , b ] in the InternalRel of G & [ a , b ] in the InternalRel of G & [ a , b ] in the InternalRel of G & [ a , b ] in the InternalRel of G & [ a , b ] in the InternalRel of G & [ a , b ] in the InternalRel of G ;
len ( M2 * M2 ) = n & width ( M2 * M2 ) = n ;
attr X1 \/ X2 is open means : Def3 : ( X \/ Y ) /\ Y is open & X is open implies X is open & Y is open & X is open & Y is open & X is open & Y is open & X is open & Y is open & X is open & Y is open & Y is open & X is open & Y is open & X is open & Y is open & X is open & Y is open & X is open & Y is open
let L being lower-bounded RelStr , X , Y be Subset of L ;
reconsider f3 = ( F . ( b , c ) ) . ( b , c ) as Function of [: M , M :] , M , M :] . ( b , c , d ) ;
consider w being FinSequence of I such that the carrier of M = { <* s *> where s is Element of I : s is state of M & w is state of M & s is state of M & s is state of M } ;
g . ( a |^ 0 ) = g . ( a |^ 0 ) .= g . ( a |^ 0 ) ;
assume for i being Nat st i in dom f holds f . i = rpoly ( 1 , z ) . ( i + 1 ) ;
ex L being Subset of X st L = L & for x being Element of X st x in L holds x in L . x
( the carrier of C1 ) /\ ( the carrier of C1 ) c= the carrier of C1 /\ the carrier of C1 ;
reconsider o9 = o as Element of TS ( V , V ) ;
1 * ( x1 + x2 ) + ( 0 * x2 ) = x1 + x2 * x2 + x3 * x3 .= x1 + x2 * x3 + x3 * x4 * x4 + x4 * x4 .= x1 + x2 * x3 + x4 * x4 + x4 * x4 * x4 + x4 * x4 + x4 * x4 + x4 * x4 + x4 * x4 .= 1 ;
E ` ` ` = ( ( E ` ) ` ) ` .= ( E ` ) ` .= ( E ` ) ` .= ( E ` ) ` .= ( E ` ) ` .= ( E ` ) ` ;
reconsider u1 = the carrier of U , q2 = the carrier of U , q1 = the carrier of U , q2 = the carrier of U , q2 = the carrier of U , q2 = the carrier of U ;
( x "/\" z ) "\/" ( x "/\" z ) <= ( x "/\" z ) "\/" ( x "/\" z ) ;
|. f . ( s1 . ( n + 1 ) - f . ( n + 1 ) .| < r / 2 ;
LSeg ( ( <* ( ( Gauge ( C , n ) * ( i , j ) , i ) , G * ( i + 1 , j ) ) `1 , G * ( i + 1 , j ) ) ) is vertical ;
( f | Z ) /. ( x + h ) = L /. ( x + h ) + R /. ( x + h ) .= L /. ( x + h ) + R /. ( x + h ) ;
g . c * ( g . c ) <= h . c * ( f . c ) ;
( f + g ) | divset ( D , i ) = f | divset ( D , i ) ;
cluster ColVec2Mx f -> non empty Line ( A , j ) & len f = n ;
len ( - ( ( - ( n + 1 ) ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) ) = len ( ( ( - 1 ) * ( ( n + 1 ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) ) ) & ( ( n + 1 ) * ( ( n + 1 ) ) * ( ( n + 1 ) ) ) * ( ( n + 1 ) * ( ( n + 1 ) ) ) * ( ( ( n
let n , i being Nat ;
pdiff1 ( f1 , 2 ) is_differentiable_in x0 & pdiff1 ( f2 , 2 ) is_differentiable_in x0 & SVF1 ( 2 , 2 , 2 ) . x0 = L . ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * (
attr a <> 0 & b <> 0 & a <> 0 implies Arg a = - ( b * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( a * ( - b ) ) ) ) ) ) ) = Arg ( a * ( a * ( a * ( a * ( a * ( a * b ) ) ) ) ) & ( a * ( a * ( a *
for c being set st c in [. a , b .[ holds not c in { a , b }
assume that v1 is linearly zero and V is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and W is open and
z * ( x1 + y1 ) + ( z * y1 ) * ( x1 + y1 ) in N & z * ( y1 + y2 ) + ( z * y2 ) * ( x1 + y2 ) in N ;
rng ( ( P +* ( ( i , 1 ) .--> ( i , 1 ) ) ) = Seg ( len ( ( ( ( i , 1 ) |-> ( ( i , 1 ) ) ) ) ) ;
consider s2 being convergent convergent convergent convergent convergent convergent and b = lim s2 and for n holds ( for n holds ( for n holds ( n >= k ) . n ) implies ( ( for n holds ( ( for n holds n >= k ) . n ) implies ( ( for n holds ( ( for n holds n >= k ) . n ) implies ( ( ( for n holds n >= k ) . n ) implies ( ( for n holds k <= k ) implies ( ( ( for n be Nat holds k <= n ) . n ) implies
h2 " . n = h2 . ( n + 1 ) & h . n = h2 . ( n + 1 ) ;
( Partial_Sums ( seq ) ) . ( n + 1 ) = ( Partial_Sums ( seq ) ) . ( n + 1 ) .= ( Partial_Sums ( seq ) ) . ( n + 1 ) ;
( Comput ( P1 , s1 , i ) ) . b = ( Comput ( P2 , s2 , i ) ) . b .= ( Comput ( P2 , s2 , i ) ) . b ;
- v = - ( - v ) & - ( - v ) = - ( - ( - v ) * ( - v ) * ( - v ) ;
ex_sup_of { k } , D , D , k } = [: { k } , { k } , { k } :] ;
A |^ ( k + l ) = ( A |^ ( n + l ) ) |^ ( n + l ) .= ( A |^ ( n + l ) ) |^ ( n + l ) ;
let R being add-associative right_zeroed right_complementable associative associative distributive non empty double loop R , a , b , c , d being Element of R , f being Element of R , g being Element of R st f = g & g = h & f is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & f is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one & g is one-to-one &
( f . p ) `1 = ( f . p ) `1 + ( f . p ) `1 ;
let a , b , c , d , e , f , g , h , h being Element of NAT , g , h being Element of NAT , f , g , h being Element of NAT st h = v & g = v & h = v & h = v & g is prime & h is prime & f is prime & g is prime & h is prime & h is prime & h is prime & h is prime & h is prime holds h is prime
consider A5 being countable countable countable Subset of [: A , B :] such that r is [: A , B :] and for n being Nat holds r <= n & n <= len A & n <= len B holds r is [: A , B :]
for X being non empty addLoopStr , M being Subset of X , N being Subset of X st N is open & M c= N holds ( M /\ N is open iff M /\ N is open
{ [ x1 , x2 ] , [ y1 , y2 ] } c= { [ x1 , x2 ] , [ y1 , y2 ] }
h . O = [ A * ( ( f . O ) `1 , B ] ) + B * ( f . O ) ] .= B * ( f . O ) + D * ( f . O ) `2 .= B * ( f . O ) + D ;
( Gauge ( C , n ) * ( k , j ) , i ) `2 in LSeg ( G * ( k , j ) , G * ( k , j ) ) ;
cluster m , n -> prime ;
( f * F ) . ( x , y ) = f . ( x , y ) & ( f * F ) . ( y , z ) = f . ( x , y ) ;
let L being Lattice , a , b , c , d being Element of L , a , b , c being Element of L st a <= b & b <= c & d <= c holds a "/\" b <= c
consider b being element such that b in dom ( H . ( x , y ) ) and z = ( H . ( x , y ) ) . ( ( x , y ) . ( x , y ) ) ;
assume x in dom ( F * ( g ) ) & ( F * ( g ) ) . x = ( F * ( g ) ) . x ;
assume that e Joins W . 1 , W . ( len W + 1 ) , G and W . ( len W + 1 ) in G . ( len W + 1 ) ;
( ( ( ( ( ( ( ( f ) * ( h ) ) ) `| Z ) ) . n ) `| Z ) . x = ( ( ( ( f * ( h ) `| Z ) . x ) `| Z ) . n ) / ( ( ( ( f * ( h + ( h + ( n + 1 ) ) * ( ( n + 1 ) ) ) ^2 ) ) ) . x ) ;
j + 1 = i + 1 .= i + 1 .= i + 1 .= i + 1 ;
( S /* ( S ^\ k ) ) . ( f . k ) = S /* ( ( S ^\ k ) . ( f . k ) ) .= S . ( f . k ) ;
consider H such that H is one-to-one & rng ( ( L2 ) * ( H ) ) = the carrier of ( TOP-REAL 2 ) | K1 & ( ( L2 ) | K1 ) | K1 ) | K1 & ( ( L2 ) | K1 ) | K1 = ( ( L2 ) | K1 ) | K1 ) | K1 ;
attr R is Rev means : Def3 : for p , q being Element of R st p in R & q <> p holds p is special & q <> p & p <> q & p <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & p <> q & p <> q & p <> q & p <> q & q <> q & p <> q & p <> q & q <> q & p <> q & p <> q
dom ( ( ( ( X --> f ) | ( ( X --> ( X --> f ) ) | ( X --> ( X --> f ) ) ) ) ) = ( ( X --> f ) | ( X --> ( X --> f ) ) ) | ( X --> f ) ) .= ( X --> f ) | ( X --> ( X --> f ) ) .= ( X --> f ) | ( X --> f ) ) ;
upper_bound ( ( proj2 .: ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) ) ) <= ( ( cn ) | K1 ) | K1 ) ^2 + ( ( cn ) | K1 ) ^2 + ( ( cn ) | K1 ) ^2 ;
for r be Real st 0 < r ex m be Nat st for n be Nat st n <= m holds |. S . n - g .| < r
i * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( of of of ) ) * ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ( ( of of of of ) ) * ( ( ( ( ( ( of of of n , 1 ) * ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ) )
consider f being Function such that dom f = 2 -tuples_on X and for x being element st x in X holds f . x = F ( x ) ;
consider g1 , g2 being element such that g1 in [#] ( Y | ( X /\ Y ) ) and g1 in Y and g2 in Y ;
func d \! > 0 -> Nat equals n -|^ ( n + 1 ) ;
f . [ 0 , t ] = f . ( 0 + 1 ) .= f . ( 0 + 1 ) .= a ;
t = h . B or t = h . B or t = h . C or t = h . D or t = h . D or t = h . E ;
consider m1 be Nat such that for n be Nat st n >= m1 holds dist ( ( seq . n ) - ( seq . n ) ) < r ;
( ( ( q `1 ) / |. q .| - cn ) / ( ( 1 + cn ) ^2 ) <= ( ( ( q `1 / |. q .| - cn ) ^2 ) / ( 1 + cn ) ^2 ) ) / ( ( 1 + cn ) ^2 ) ;
( h . ( i + 1 ) ) . ( i + 1 ) = h . ( i + 1 ) .= h . ( i + 1 ) ;
consider o being Element of the carrier' of S such that a = [ o , the carrier' of S ] and o = [ o , the carrier' of S ] ;
let L being RelStr , a , b , c being Element of L , d being Element of L st a <= b & b <= c & d <= c holds a <= b
||. h1 .|| . n = ||. ( h . n ) . n - ( h . n ) . n .|| .= ||. ( h . n ) - ( h . n ) .|| .= ||. ( h . n ) - ( ( h . n ) . n ) .|| .= ||. ( h . n ) - ( h . n ) .|| ;
( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( 1 / 2 ) / 2 ) * ( ( ( ( ( 1 / 2 ) / 2 ) * ( ( ( ( 1 + 2 ) / 2 ) * ( ( ( ( 1 + 2 ) / 2 ) * ( ( ( ( ( ( 1 + 2 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( (
attr r = F .: ( p , q ) ;
sqrt ( r + ( r - ( r - ( r ) ) ) ^2 ) + ( r - ( r - ( r ) ) ^2 ) <= r ;
for i being Nat , M being Matrix of n , K , n , K , m , k being Nat st ( for i being Nat st i in Seg n holds ( ( ( Line ( M , i ) ) * ( ( Line ( M , i ) ) * ( ( Line ( M , i ) ) * ( ( Line ( M , i ) ) * ( ( Line ( M , i ) ) * ( ( Line ( M , i ) ) ) ) ) ) ) ) . ( ( i , j ) ) ) . ( i , j ) ) = ( ( Line ( M , i ) ) * ( ( i , j ) ) ) . ( ( i , j
then a <> 0. R & a " * ( a * b ) = 1 * ( a " * b ) ;
p . ( j -' 1 ) * ( q . ( j + 1 ) ) = Sum ( p . ( j + 1 ) ) * ( q . ( j + 1 ) ) ;
deffunc F ( Nat ) = L . ( ( " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) * ( ( h " ) ) ) ) ) ) ) ;
assume the carrier of ( ( TOP-REAL 2 ) | K1 ) = f .: K1 & ( the carrier of ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) | K1 ) | K1 = f .: K1 ;
Args ( o , Free ( S , X ) = ( ( the Sorts of Free ( S , X ) ) * the ResultSort of S ) . o ;
H = n + 1 .= n + 1 .= n + 1 .= n + 1 ;
( O ) `1 = 0 & ( O ) `2 = 0 & ( O ) `2 = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & O = 0 & Q is 0 & Q is I & Q is O & Q is O & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is
F1 .: ( dom ( F | ( ( n + 1 ) ) /\ ( F | ( n + 1 ) ) ) = { f } /\ ( F | ( n + 1 ) ) .= { f } ;
attr b <> 0 & b <> 0 implies b - a = b - a
dom ( ( f +* g ) +* ( g , D ) ) = dom ( ( f +* g ) +* ( g , D ) ) .= dom ( ( f +* g ) +* ( g , D ) ) \/ ( ( f +* g ) +* ( g , D ) ) .= ( ( f +* g ) +* ( g , D ) ) +* ( g , D ) ) ;
for i being set st i in dom g ex a , b being Element of L st g /. i = u * a & g /. i = u * b
g " * P = g " * P .= g " * P .= P * P ;
consider i , s1 such that f . i = s1 & ( f . i ) `1 = s1 & s1 . ( i + 1 ) `1 <> s2 & s2 . ( i + 1 ) <> s2 . ( i + 1 ) `1 & s2 . ( i + 1 ) <> s2 . ( i + 1 ) `1 ;
( h | ]. a , b .[ ) | ]. a , b .[ = ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ .= ( g | ]. a , b .[ ) | ]. a , b .[ ) | ]. a , b .[ ;
[ s1 , s2 ] in [: { 1 } , { 2 } :] & [ s1 , s2 ] in [: { 2 } , { 2 } :] ;
then H is negative & H is negative & H is negative implies H is negative
attr f1 is total means : only : f1 is total & f2 is total & f2 is total & f1 is total & f2 is total & f1 is total & f2 is total & f1 is total & f2 is total & f1 is total & f2 is total & f2 is total & f2 is total & f2 is total & f2 is total & f1 is total & f2 is total & f2 is total & f2 is total & f2 is total & f2 is total & f2 is total & f1 is total & f2 is total & f2 is total & f2 is total & f2 is total & f1 is total & f2 is total & f2 is
z1 in W2 or z2 in W1 & z1 in W2 & z2 in W2 & z2 in W2 & ( ex F being Subset of X st F = ( F ) /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F /\ ( F
p = 1 * p .= a * p .= a * p .= p * p .= p * ( 1 - p ) ;
for r be sequence of X , s being sequence of X , n being Nat st n <= m holds ( for m be Nat st n <= m holds r . m <= r ) implies ( for n be Nat holds r <= s . n ) implies for n be Nat holds r <= n )
Index ( E-max C , E-max C ) meets L~ Cage ( C , n ) or ( E-max C ) `1 meets L~ Cage ( C , n ) or ( E-max C ) `1 meets L~ Cage ( C , n ) ) `1 ;
||. f . ( g . ( k + 1 ) - f . ( k + 1 ) .|| <= ||. g . ( k + 1 ) - f /. ( k + 1 ) .|| * ||. g . ( k + 1 ) - f /. ( k + 1 ) .|| ;
assume h = ( B .--> ( B .--> C ) ) +* ( D .--> E ) ) +* ( F .--> E ) ;
|. ( ( ( H . n ) . k ) . k ) .| <= e * ( ( H . n ) . k ) ;
( ( ( the Sorts of Free ( S , X ) ) . e ) . ( ( the Sorts of Free ( S , X ) ) . e ) = [ ( the Sorts of Free ( S , X ) . e ] ;
{ x1 , x2 , x3 , x4 , x5 , x5 , x5 , x5 , x5 , x5 , x5 , x5 } = { x1 , x2 , x4 , x5 , x5 , x5 , x5 , x5 } ;
assume A = [. 0 , PI / 2 * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI * PI / 2 * PI * PI * PI * PI * PI / 2 * PI * PI * PI ;
p `1 is Permutation of dom ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m ) /. /. j ) ) ) ) ) ) /. ( i + 1 ) ) ) ) ) /. ( i + 1 ) ) ) ) & ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m m m ) ) /. j ) ) /. j ) ) /. ( i , j ) ) /. j ) )
for x , y being Element of A st x in A & y in B holds |. ( f . x ) - f . y .| <= 1
( ( q `1 ) ^2 - ( q `2 ) ^2 ) = |. q .| * ( ( q `1 ) ^2 - ( q `2 ) ^2 ) ;
let f being PartFunc of C , REAL , a , b be Real st dom f = the carrier of C & for n being Nat holds f . n = a * f . n + b * f . n
assume for x being Element of Y st x in EqClass ( z , CompF ( PA , G ) ) holds ( u , G ) . x = TRUE ) . x
consider F1 such that dom ( F | ( n + 1 ) ) = n1 and for k being Nat st k in dom F holds F . k = F . ( k + 1 ) ;
ex u , v st u <> v & u in v & v in w & w , v |= u , v ) implies u , v |= w , v
let G being Group , A being Subset of G , B being Subset of G st A = N holds ( N ` ) ` = N ` `
for s be Real st s in dom F holds F . s = ( ( ( ( f + g ) / ( n + 1 ) ) * ( ( f + g ) / ( n + 1 ) ) ) * ( ( f + g ) / ( n + 1 ) ) )
width ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m m , n , m ) , 1 ) , 1 ) , 1 ) , ( ( ( ( ( ( ( m , len ( ( ( ( ( ( ( ( m , len ( ( ( ( ( ( m , len ( ( ( ( ( m , len ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = len ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( m ,
f | ]. - 1 , 1 .[ = f | ]. - 1 , 1 .[ & f | ]. - 1 , 1 .[ = f | ]. - 1 , 1 .[ ;
assume that X is closed and a in X and b in X and a in X & b in X & b in X & c in X & a in X & b in X & c in X & b in X & c in X & d in X & d in X & d in X & b in X & d in X ;
Z = dom ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( ( ( ( ( ) ) ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( 1 / 2 ) ) * ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
func bound_QC-variables ( V ) -> Subset of V equals { l where l is Element of V : l in A & l in A & l in A & l in A } ;
let L being non empty TopSpace , N be net of L , p being Element of L , x being Element of L st x in N & p is Element of L & x is Element of L & x is Element of L & x is Element of L holds x is Element of L
for s being Element of NAT , v being Element of V holds ( ( ( ( ( the carrier of V ) | ( the carrier of V ) ) | ( the carrier of V ) ) | ( the carrier of V ) ) . ( ( the carrier of V ) | ( the carrier of V ) ) . ( ( the carrier of V ) | ( the carrier of V ) ) . ( ( the carrier of V ) ) = ( ( the carrier of V ) | ( the carrier of V ) ) . ( ( the carrier of V ) ) ;
then z /. 1 = ( N-min L~ z ) `1 & ( E-max L~ z ) `1 <= ( E-max L~ z ) `1 or ( E-max L~ z ) `1 <= ( E-max L~ z ) `1 ;
len ( p ^ <* 0 *> ) = len p + len <* 0 *> .= len <* 0 *> + 1 ;
assume that Z c= dom ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) * ( ( ( ( ( ( ( ( ( ) / 2 ) * ( ( ( ( ) ) ) * ( ( ( ( ( ( ) ) * ( ( ( ) ) ) * ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) and for x st x in Z holds ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
let R being add-associative right_zeroed right_complementable associative associative distributive non empty double loop R , a , b , c , d being Element of R , f being Function of the carrier of R , the carrier of R , the carrier of R , the carrier of R , d be Element of R ;
consider f being Function of B , C such that for x being Element of A , y being Element of B , z being Element of A , f being Function of B , C st f . x = F ( x , y ) & f . ( x , y ) = F ( x , y ) ;
dom ( x2 + y2 ) = Seg len ( x2 + y2 ) .= dom ( x2 + y2 ) .= dom ( x2 + y2 ) ;
for S being category , B being non empty category , A being ManySortedSet of I , B being ManySortedSet of I , B being ManySortedSet of I st A is \mathord { A } & B is \mathord { A } holds A is \mathord { A }
ex a st a = { b } & a in { b } & b in { a } & { b } c= { b } ;
a in Free ( H ) \/ { x } \/ { y } ;
let C1 , C2 be C2 non empty set , f , g be stable Function of C1 , C2 , C2 , h being Function of C1 , C2 st C1 = C2 & g = h & h = f & h = g & h = h & h = h & f = g holds f = g
( ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) | K1 = K1 & ( ( ( TOP-REAL 2 ) | K1 ) | K1 ) | K1 ) | K1 ) | K1 = K1 ;
assume that u = <* x0 , y0 , z0 , z0 , 1 , 2 , 3 *> and f is_differentiable_in z0 and u = <* z , y0 , z0 , 1 , 2 *> ;
then ( t . {} ) `1 = ( t . {} ) `1 & t . {} = t . {} ) `1 ;
Valid ( p , J ) . ( v , J ) = Valid . ( v , J ) . ( v , J ) .= J . ( v , J ) ;
assume for x , y being Element of S st x <= y & y in f . x holds a >= f . y
func Class ( R , A ) -> Subset-Family of R means : Def3 : for a , b being Element of R st a in b & b in A & a in A & b in A holds it = a ;
defpred P [ Nat ] means ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) . . . ( n + 1 ) ) ) ) ) ) ) ) ) . ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( \ \ ( n + 1 ) ) . ( n + 1 ) ) ) ) ) ) ) ) ) . ( ( n + 1 ) ) ) ) ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) `1 ) ^2 ) `1 ) `1 ) `1 ) `1 <= ( ( ( ( ( n + 1 ) ) ^2 ) ) `1 ) `1 ) `1
assume that dim ( W1 , W2 ) = 0 and dim ( W1 , W2 ) = 0 ;
mam in ( m ! ) . ( t ! ) .= ( ( m ! ) . ( t ! ) ) . ( t ! ) .= ( m ! ) . ( t ! ) .= ( m ! ) . ( t ! ) ;
d = { x } ^ ( <* y *> ^ <* x *> ) .= <* x *> ^ <* y *> .= <* x *> ^ <* y *> ^ <* y *> .= <* x *> ^ <* y *> ;
consider g such that x = g and dom g = dom ( f | X ) and for x st x in X holds g . x = f . x ;
x + 0. V = x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + ( len x ) .= x + x ;
k1 -' 1 in dom ( f /^ ( len f -' 1 ) ) ;
assume that P1 is walk of product I and I c= P and I c= P and J c= P and I c= P and J c= P and P c= P and I c= P and P c= P and I c= P and P c= P and P c= Q and P c= Q and P c= Q and P c= Q and P c= Q and P c= Q and P c= Q and P c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c= Q and Q c=
reconsider a1 = a , b1 = b , b2 = c , b3 = d , b3 = b , b3 = c , b3 = d , b1 = c , b3 = d , b3 = d , b3 = c , b1 = d , b3 = c , b3 = d , b3 = d , b3 = c , b3 = d , b3 = c , b1 = d , b3 = c , b3 = d , b3 = c , b3 = d + 6 + 6 + 6 + 6 + 6 * 6 , b1 = d + 6 * w1 , b3 = d + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 * w1 + 6 *
reconsider GFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF
LSeg ( f , i + 1 ) = LSeg ( f , i + 1 ) ;
\int ( P . ( m + 1 ) , ( P . ( n + 1 ) ) ) <= \int ( P . ( m + 1 ) ) + ( P . ( n + 1 ) ) ;
assume dom ( ( f1 + f2 ) `| Z ) = dom ( f1 + f2 ) & for x st x in Z holds f1 . x = f1 . x + f2 . x ;
consider v such that v = y and dist ( u , v ) < r and dist ( v , v ) < r ;
let G being Group , H being Element of G , a , b , c , d being Element of H st a = b & b = c & c = d holds a * b = b
consider B being Function of Seg ( len S + 1 ) , the carrier of V such that for x being Element of V st x in Seg ( len S + 1 ) holds P [ x , B . x , B . x ] ;
reconsider K = { p where p is Point of TOP-REAL 2 : p `1 <= 1 & p `2 <= 1 & p `2 <= 1 & p `2 <= 1 } as Subset of TOP-REAL 2 ;
sqrt ( ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) <= ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) ^2 + ( ( ( ( ( ( TOP-REAL 2 ) | K1 ) ^2 ) ^2 ) ^2 ) ) ;
for x being Element of X , y being Element of X st |. ( F . n ) . x - y .| <= P . ( x - y ) holds |. ( F . n ) . x - y .| <= P . ( x - y )
len ( <* 2 *> ^ ( <* 2 *> ^ ( <* 2 *> ^ ( <* 2 *> ) ) ) = len ( ( <* 2 *> ^ ( <* 2 *> ^ ( <* 2 *> ^ ( <* 2 *> ) ) ) ) .= len <* 2 *> ^ ( <* 2 *> ^ ( <* 2 *> ) ) .= len <* 2 *> + ( 2 * ( <* 2 *> ) ) .= len <* 2 *> + ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 * ( 2 *
v / ( ( ( x , y ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( , , ( , m ) , m ) , m ) , m ) ) , ( ( ( ( ( ( ( ( m , m ) , m ) ) , ( ( ) ) ) ) , ( ( ( ( ( ( ( ( m , m ) ) , m ) ) ) , ( ( ) ) ) , ( ( ( ) ) ) , ( ( ( ) ) ) , ( ( ( ( ( ( ( ) ) ) , ( ( ( ) ) ) ) , ( ( ( ) ) ) , ( (
consider r being Element of M such that M , v |= ( { ( { x } \leftarrow ( { x } , { y } ) } ) . ( { y } ) ) . ( { x } ) = r & ( M , v ) . ( { y } ) = r ) . ( ( { x } , { y } ) . ( { x } ) ) ;
func { w where w is Element of Union G : for i being Element of NAT holds ( ( the Element of G ) . ( i + 1 ) ) `1 = ( ( the Element of G ) . ( i + 1 ) ) `1 ;
s2 . b = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 2 ) / 2 ) ) ) * ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 2 ) / 2 ) ) * ( ( ( ( ( - 2 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
for n be Nat holds 0 <= ( Partial_Sums ( seq ) ) . n
set F = S \! \mathop { 0 } ;
( Partial_Sums ( s ) ) . ( n + 1 ) >= ( Partial_Sums ( s ) ) . ( n + 1 ) + ( Partial_Sums ( s ) ) . n + 1 ) ;
consider L , R such that for x st x in N holds ( f | Z ) . x = L . ( - R . x ) + R . x ;
the carrier of ( TOP-REAL 2 ) | K1 = ( TOP-REAL 2 ) | K1 .= K1 ;
a * b + ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) + ( b * c ) * ( b * c ) * ( b * c ) *
v / ( ( ( x1 , x2 ) / ( m , k ) ) ) = v / ( m , k ) ;
<% Q %> ^ <* x %> = ( <% ( Q ^ <* x *> ) ^ <* y *> ) ^ <* x *> ) ^ <* y *> .= ( <% x %> ^ <* y *> ) ^ <* x *> ^ <* y *> ) ^ <* y *> ^ <* x *> ^ <* y *> .= <* x *> ^ <* y *> ^ <* y *> ^ <* y *> ;
Sum ( F ) = r |^ ( n + 1 ) .= r * ( ( F |^ ( n + 1 ) ) ) .= r * ( ( F |^ ( n + 1 ) ) ) .= r * ( ( F |^ ( n + 1 ) ) ) * ( ( F |^ ( n + 1 ) ) ) .= r * ( ( F |^ ( n + 1 ) ) * ( ( F |^ ( n + 1 ) ) ) ;
( ( ( ( GoB f ) * ( 1 , 1 ) , 1 ) `1 ) `1 = ( GoB f ) * ( 1 , 1 ) `1 ) `1 ;
defpred X [ Element of NAT ] means ( Partial_Sums ( s ) ) . $1 = ( a * $1 ) . $1 + b * $1 + b * $1 + b * $1 ;
the_arity_of g . ( ( the connectives of S ) . ( o ) ) = ( the connectives of S ) . ( o ) .= ( the connectives of S ) . ( o ) ;
( X \times Y ) \/ ( X \times Y ) c= X \/ Y
for a , b being Element of S , s being Element of T , t being Element of T st s = F . ( s , t ) holds t = F ( s , t )
E |= All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) => All ( x , H ) ) => All ( x , H ) \Rightarrow All ( x , H ) \Rightarrow All ( x , H ) \Rightarrow All ( x , H ) ) = All ( x , H ) => All ( x , H ) ;
ex R2 being 1-sorted , R being Relation of n , ( the carrier of R ) , a , b being Element of R st ( the InternalRel of R ) . ( a , b ) = a & ( the InternalRel of R ) . ( a , b ) = b ) & ( ( the InternalRel of R ) . ( a , b ) = a ) & ( ( the InternalRel of R ) . ( a , b ) = b ) implies ( ( the InternalRel of R ) . ( a , b ) = a ) & ( ( the InternalRel of R ) . ( a , b ) . ( a , b ) . ( a , b ) . ( a , b ) . ( a , b ) = b ) . ( a ,
[. a , b .] + ( 1 - a ) is Element of the qua .[ & ( ( ( 1 - a ) / ( b - a ) ) * ( ( 1 - a ) / ( b - a ) ) ) * ( ( 1 - a ) / ( b - a ) ) ) is Element of the carrier of X ) ;
Comput ( P , s , 2 + 1 ) . ( 2 + 1 ) = Exec ( i , Comput ( P , s , 2 ) . 2 ) . ( 2 + 1 ) .= Exec ( i , s ) . ( 2 + 1 ) ;
card ( ( h . k ) . ( i + 1 ) ) = ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) * ( ( ( ( ( ( ( (
( ( f /. c ) * ( g /. c ) ) = f /. c * ( g /. c ) .= f /. c * ( g /. c ) .= f /. c * ( g /. c ) .= f /. c * ( g /. c ) .= f /. c * ( g /. c ) ;
len ( ( ( len C ) -' 1 ) -' 1 ) - 1 ) = len ( ( ( ( len C ) -' 1 ) -' 1 ) - 1 ) - 1 ) + 1 ) - 1 .= len ( ( ( ( len C ) -' 1 ) - 1 ) - 1 ) - 1 ) + 1 ) + 1 .= len ( ( ( ( len C ) -' 1 ) - 1 ) - 1 ) + 1 ) ;
dom ( r (#) ( f | X ) ) = dom ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) /\ ( ( r (#) ( f | X ) ) .= dom ( r (#) ( f | X ) ) .= dom ( r (#) ( ( r (#) ( f | X ) | X ) ) .= dom ( r (#) ( ( r (#) ( f | X ) | X ) ) ;
defpred P [ Nat ] means 2 * ( n + 1 ) + 1 - $1 * ( n + 1 ) = Fib ( n + 1 ) - ( n + 1 ) * ( n + 1 ) * ( n + 1 ) ;
consider f being Function of { n + 1 } , { n } such that f = f and for k being Nat st k in { n } holds f . k = F ( k ) ;
consider c being Function of S , BOOLEAN such that { c } = { \raise , b } \/ { c , d } and c in { d , b } ;
consider y being Element of [: Y , Y :] such that a = "\/" ( { y } , Y ) and y in Y and y in Y ;
assume A c= dom f & f = ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( ( ( ( ( 1 ) ) (#) ( ( - 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( (
( ( GoB f ) * ( i , j ) `2 = ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 .= ( GoB f ) * ( i , j ) `2 ;
dom ( ( ( Seq ( q ) ) | ( ( len q ) + 1 ) ) ) = { ( ( ( ( ( q ) | ( len q ) ) . ( len q ) + 1 ) ) . ( len q ) } ;
consider G1 , G2 being Element of V such that G1 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G1 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G2 & G2 <= G1 & G1 <= G1 & G2 = G2 & G1 = G2 & G1 = G2 = G1 & G1 = G2 & G1 = G2 & G1 = G1 & G2 = G1 & G2 = G1 & G1 = G2 & G2 = G1 & G1 = G2 & G2 = G1 & G2 = G1 & G1 = G1 & G2 = G1 & G1 = G2 & G1 = G2 & G1
func - f -> PartFunc of C , REAL means : Def3 : for c being Element of C st c in dom it holds it . c = - f . c + f . c
consider \varphi such that \varphi is increasing & \varphi . a = a & for b st b in dom \varphi holds \varphi . b = b & \varphi . b = a & \varphi . b = b & \varphi . b = b & \varphi . b = b & L . b = a & L . b = b & L . b = a & L . b = b & L . b = b & L . b = b & L . b = b & L . b = b & L . b = a & L . b = b & L . b = b & L . b & L . b & L . b = b & L . b = b & L . b & L . b = a & L . b = b & L . b & L . b = b & L
consider i1 , j1 being Nat such that [ i1 , j1 ] in Indices GoB f and f /. ( i1 + 1 ) = ( GoB f ) * ( i1 , j1 ) `1 ;
consider i , n such that n <> 0 & n <= len p and p . i = ( i + n ) * ( n + 1 ) ;
assume that 0 in Z and for x st x in Z holds ( ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( 1 ) ) ) ) ) ) ) )
cell ( G , i1 , j1 -' 1 , j1 -' 1 ) \ { |[ 0 , 1 ]| } c= cell ( G , i1 , j1 -' 1 ) \ { |[ 0 , 1 ]| } ;
ex Q being open Subset of X st s = Q & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & P is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & Q is open & ( ex P being Subset of X st P is open & ( ( ex Q being Subset of X st Q is open & ( ( ( ) implies ( ( ( ex Q being Subset of X ) implies ( ( ( ex Q being Subset of X ) implies Q is open implies Q is open & ( ( ( X ) /\ Q is open implies ( ( ( ( X ) /\ Q is open implies ( ( ( X ) /\ Q is open implies
gcd ( A , gcd ( A , B ) ) = 1 ;
R1 = ( the InternalRel of ( 2 , m ) . ( ( 2 , n ) + 1 ) ) . ( ( 2 , n ) + 1 ) .= ( ( the InternalRel of ( 2 , m ) ) . ( ( 2 , n ) + 1 ) ) . ( ( 2 , n ) + 1 ) ) . ( ( 2 , n ) + 1 ) .= ( ( ( 2 , n ) + 1 ) + 1 ) . ( ( 2 * n ) ) . ( ( 2 * n ) ) . ( 2 * ( 2 * n ) ) . ( 2 * ( 2 * n ) ) . ( 2 * n ) ) . ( 2 * ( 2 * ( 2 * n ) ) . ( 2 * ( 2 * n ) ) . ( 2 * ( 2 * n ) . ( 2 * ( 2
CurInstr ( P3 , Comput ( P3 , s , m ) ) = CurInstr ( P3 , Comput ( P3 , s , m ) ) .= ( halt SCMPDS ) . ( m + 1 ) .= halt SCMPDS ;
P1 /\ P2 = ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( 1 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
func not still ( f , p ) -> Subset of [: A , A :] means : Def3 : for i , j st i in dom f & j in dom f holds it . i = f . ( j , i ) ;
let a , b , c , d be Real ;
defpred P [ Nat ] means ( for i , j st 1 <= i & i <= n & j <= len g holds ( G * ( i , j ) , ( G * ( i , j ) ) `1 = G * ( i , j ) `1 ;
cluster C1 , C2 , C1 , C2 , C2 , C1 , C2 *> -> associative & C1 is associative & C2 is associative & C2 is associative & C1 is associative & C2 is associative & C2 is associative & C1 is associative & C2 is associative implies C1 is associative & C2 is associative & C2 is associative & C1 is transitive & C2 is associative & C1 is transitive & C2 is transitive & C1 is transitive & C2 is transitive & C1 is transitive & C2 is associative & C1 is associative & C1 is transitive & C2 is transitive & C1 is transitive & C2 is transitive & C1 is transitive & C1 is transitive & C2 is transitive & C1 is transitive & C1 is transitive & C2 is transitive & C1 is transitive & C2 is transitive & C1 is transitive & C2 is transitive & C2 is transitive & C1 is transitive & C1 is transitive & C2 is
( ||. f .|| | X ) . ( n + 1 ) = ||. ( f | X ) . ( n + 1 ) - f /. ( n + 1 ) .|| .= ||. f /. ( n + 1 ) - f /. ( n + 1 ) .|| .= ||. f /. ( n + 1 ) - f /. ( n + 1 ) .|| ;
|. q .| ^2 + ( ( q `1 ) ^2 + ( q `2 ) ^2 + ( q `2 ) ^2 ) = ( ( q `1 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) + ( q `2 ) ^2 + ( q `2 ) ^2 ) + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 ) ^2 + ( ( q `2 ) ^2 + ( q `2 ) ^2 ) + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 ) ^2 + ( q `2 ) ^2 + ( ( q `2 ) ^2
for F being Subset-Family of T st F is open & for A being Subset of T st A in F holds A /\ F = F /\ ( A /\ F )
assume that len F >= 1 and len F = k + 1 and len F = k + 1 and len F = n and len F = n + 1 and len F = n and F is one-to-one and F is one-to-one and F is one-to-one & F is one-to-one & G is one-to-one & G is one-to-one & F is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & F is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is one-to-one & G is
( i |^ ( n + 1 ) ) |^ ( n + 1 ) = i |^ ( n + 1 ) .= i |^ ( n + 1 ) ;
consider q being FinSequence such that r = q and for n being Element of NAT st n <= len q holds ( ( G * ( len G , 1 ) ) `1 = v & ( G * ( len G , 1 ) `1 <= v ) `1 ;
defpred P [ Element of NAT ] means ( for n st n <= $1 holds ( ( for m st n <= $1 holds ( ( for n st n <= $1 holds ( ( ( for n st n >= 1 holds ( ( ( for n st n >= 1 holds ( ( ( for n holds ( ( for n st n <= m holds ( ( ( for n st n >= 1 holds ( ( ( for n holds n <= m holds ( ( ( for n st n >= 1 ) ) * ( ( ( n ) holds ( ( ( n ) holds ( ( for n ) holds ( ( ( n ) ) * ( ( for n ) ) * ( ( ( for n ) holds ( ( ( n ) ) * ( ( ( ( ( n ) ) * ( ( n ) ) * ( ( ( ( n ) ) * ( ( n ) ) * ( ( ( ( n ) ) * (
let A being Matrix of n , m , k , D , i , j , Nat , Nat , j , k be Nat st i in Seg n & j in Seg n & k in Seg n & k in Seg n & k in Seg n & i = ( n , k ) * ( i , j ) holds ( ( n , j ) * ( i , j ) = ( n , j ) * ( i , j )
consider s being FinSequence of the carrier of R such that Sum s = a * s and for i being Element of NAT st i in dom s holds s . i = a * s . i ;
func x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | | ( x | ( | ( ( | ( | | ( x | ( | ( | ( | ( | ( | ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) -> Element of REAL ) ) ) | ( ( ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x | ( x |
consider g1 being FinSequence of ( len g1 ) -tuples_on REAL such that g1 = g1 and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is continuous and g1 is compact and g1 <> 0 & g1 <> 0 <= g1 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g1 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g1 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g2 <> 0 & g2 <>
then n1 >= len ( p1 , p2 , n1 ) & n2 >= len ( p1 , p2 , n1 , n2 ) = len ( p1 , p2 , n1 , n2 , n1 , n1 ) ;
( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( ( q `1 ) * ( q `1 ) ) ) ) ) ) ) ) <= ( q `1 ) * ( ( q `1 ) * ( q `1 ) ) ) & ( q `1 ) * ( ( q `1 ) * ( q `1 ) ) ) * ( ( ( q `1 ) ) * ( ( q `1 ) ) * ( ( q `1 ) ) * ( ( q `1 ) ) * ( ( q `1 ) ) ) ) * ( ( ( q `1 ) * ( ( q `1 ) ) * ( ( ( ( q `1 ) ) * ( ( q `1 ) ) ) * ( ( q `1 ) ) ) * ( ( q `1 ) ) ) * ( ( ( q `1
( F . ( len F ) ) . ( len F + 1 ) = ( F . ( len F ) ) . ( len F + 1 ) .= F . ( len F + 1 ) .= F . ( len F + 1 ) ;
consider k1 being Nat such that k1 + 1 = ( <* a *> ^ k1 ) /. ( k1 + 1 ) and a = ( <* a *> ^ k1 ) /. ( k1 + 1 ) ;
consider B8 being Subset of [: { A , B , C , D } , { D , C , E , F , G , G , F , G , G , F , G , G , F , G , G , F , G , G , F , G , G , F , G , G } such that { F , G , F , G } = { F , G } ;
v2 . ( F2 , F2 ) = ( F2 , F2 ) . ( F2 , F2 ) .= ( F2 , F2 ) . ( F2 , F2 ) .= ( F2 , F2 ) . ( F2 , F2 ) .= ( F2 , F2 ) . ( F2 , F2 ) .= ( F2 , F2 ) . ( F2 , F2 ) .= ( F2 , F2 ) . ( F2 , F2 ) ;
dom ( ( Initialize ( s ) ) | ( ( carrier of S ) \/ { IC S } ) ) = dom ( ( the Sorts of A ) | ( ( the Sorts of A ) \/ { S } ) ;
ex d be Real st d > 0 & for n be Nat st n >= m holds |. ( h . n ) - ( h . n ) .| < d
LSeg ( G * ( len G , 1 ) `1 , G * ( len G , 1 ) `1 ) `1 c= ( G * ( len G , 1 ) `1 ;
LSeg ( h , i ) = LSeg ( h , i ) .= LSeg ( h , i ) ;
A = { q : |. q .| = 1 & |. q .| <= 1 & |. q .| <= 1 & q <> 0. TOP-REAL 2 } ;
( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( ( - x ) | ( ( - x ) | ( ( ( - x ) | ( ( - x ) | ( ( - x ) | ( ( ( x ) | ( ( - x ) | ( ( -
0 * sqrt ( ( 1 - ( ( ( ( ( ( p `1 / |. p .| - cn ) / ( ( 1 + cn ) / ( ( 1 + cn ) / ( ( 1 + cn ) / ( ( 1 + cn ) ) ) ) ) ) ) ) ) = ( ( ( ( - ( ( ( ( ( ( ( ( ( ( ( ( ( ( p ) / |. p .| - cn ) / ( ( 1 + cn ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( p ) / ( ( ( ( ( ( ) ) / ( ( ( ( ( ) ) / ( ( ( ( ( ( ( ( ( ( ( ( ( ( p ) / ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ) ) ) ) = ( ( ( ( ( ( ( ( ( (
( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( - 1 ) (#) ( ( - 1 ) (#) ( ( ( - 1 ) (#) ( ( ( ( - 1 ) (#) ( ( ( - 1 ) / ( ( ( - 1 ) / ( ( ( - 1 ) / ( ( ( ( - 1 ) / ( ( ( ( ( 1 ) ) * ( ( ( - 1 ) / ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = ( ( ( - 1 ) * ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / ( ( - 1 ) * ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) * * ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / ( ( ( (
redefine func f . ( x + h ) -> PartFunc of REAL , REAL n equals f . ( x + h ) .= f . ( x + h ) ;
assume that 1 <= k and k <= len f and k <= len f and f /. k = G * ( i , j ) ;
assume that not y in Free H and x in { y } and y in { y } ;
defpred P [ Element of NAT ] means ( for n being Element of NAT holds ( for m being Element of NAT st n >= $1 holds ( n <= m ) implies ( ( 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 + 1 ) ) * ( ( ( n + 1 ) ) * ( ( ( n + 1 ) ) * ( ( ( ( n + 1 ) ) * ( ( n + 1 ) ) * ( ( ( n + 1 ) ) * ( ( n + 1 ) ) * ( ( ( n + 1 ) ! ) ) * ( ( ( n + 1 )
func \sigma ( C , D ) -> non empty Subset of X equals ( the carrier of X ) \/ { {} } ;
[#] ( ( ( ( dist ( ( ( ( ( ( ) ) ) | Q ) ) | Q ) ) | Q ) ) ) .: Q ) = ( ( dist ( ( ( dist ( ( ( dist ( ( ( Q ) ) | Q ) ) ) ) | Q ) ) .: Q ) .: Q ) .: Q ) ;
rng ( F | ( ( S | ( ( S | ( ( S ) | ( ( S | ( ( S | ( ( S ) | ( ( S | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S | ( ( S | ( ( S | ( ( S ) | ( ( S ) | ( ( S ) | ( ( S ) | ( ( ( ) ) | ( ( S ) | ( ( ) ) | ( ( S ) | ( ( ) ) | ( ( S ) | ( ( S ) | ( ( ) | ( ( ) ) | ( ( S ) | ( ( S ) | ( ( ( ) | ( ( S ) | ( ( S ) | ( ( ( ) ) | ( ( S ) | ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) = { p ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) =
( f " ) . ( ( rng f ) . i ) = f . ( ( i + 1 ) ) .= f . ( i + 1 ) .= f . ( i + 1 ) .= f . ( i + 1 ) .= f . ( i + 1 ) .= f . ( i + 1 ) ;
consider P1 , P2 being Subset of TOP-REAL 2 such that P1 /\ P2 = { p1 : p1 in P & p2 in P & P [ P1 , p2 ] } and P [ P1 , p2 ] ;
f . ( p2 + 1 ) = [ sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) , ( p2 `2 ) ^2 ) ] .= [ sqrt ( ( p2 `1 ) ^2 + ( p2 `2 ) ^2 ) , sqrt ( ( p2 `2 ) ^2 ) ] ;
( \langle a , b *> " ) . ( x + 1 ) = ( ( \langle a , b *> . ( x + 1 ) ) * ( ( \langle a , b *> . ( x + 1 ) ) ) . ( x + 1 ) .= ( ( \langle a , b *> . ( x + 1 ) ) * ( x + 1 ) .= ( ( \langle a , b *> . ( x + 1 ) ) * ( x + 1 ) ) * ( x + 1 ) ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) .= ( x + 1 ) * ( x + 1 ) .= ( ( x + 1 ) * ( x + 1 ) * ( x + 1 ) * ( x + 1 ) ) * ( x + 1 ) .= ( x + 1 ) * ( x + 1 ) * ( x + 1 ) .= ( x + 1 ) * ( x +
let T being non empty TopSpace , A , B being Subset of T , B being Subset of T , C being Subset of T st A <> B & C c= A & A is open & B is open & C is open & C is open & B is open & C is open & A is open & C is open & B is open & C is open & C is open & A is open & C is open & B is open & A c= B & A c= B & B is open & B c= B & A is open & B c= B & B is open & B is open & B c= B & A c= B & B c= B & A c= B & B c= B & B c= B & A is open & B is open & B c= B & B c= B & A c= B & B c= B & A is open & B is open & A is open & B is open & B c= B & A is open & B c= B & A is open & B c= B & A is open & B c= B & B is open & B c= B & A is open & B c= B & B is
for i , j being strict Subgroup of G st i + 1 in dom F & j in dom F & F . i = F . j holds F . j = F . j
for x st x in Z holds ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( 1 + 2 ) ) * ( ( ( ( ( ( ( ) ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) ) ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
synonym f is right & f is right & f is right & f is right implies f is right & g is right & g is convergent & f is convergent & f is convergent & g in dom f & g in dom f & g in dom f & r in dom f & r < g & r < x0 ) implies ex r st r < r & for n st n <= n holds |. f . n - r < x0 )
then X1 misses X2 & X1 misses X2 & X1 union X2 = X1 union X2 & X1 union X2 = X2 union X1 & X1 union X2 = X1 union X2 ;
ex N be Neighbourhood of x0 st N c= dom ( ( 1 / 2 ) (#) ( ( f `| Z ) `| Z ) ) . ( n ) - L . ( n + 1 ) )
sqrt ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) / 2 ) ) ) ) ) ) ) ) ^2 ) ) ) ^2 ) ) + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) / 2 ) ) ) / 2 ) ) ) ) ) ) ) ) ) ^2 ) ) ) ) ^2 ) ) + ( ( ( ( ( ) ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ) ) / 2 ) ) ^2 ) ) ) ^2 ) ) ^2 ) ) ) + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ^2 ) ) ) ) ^2 ) ) ^2 ) + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) / 2 ) ) ) ^2 ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ) ) ^2 ) ) ^2 ) ) )
( ( ( 1 - ( ( ( ( ( ( ( ( ( 1 / 2 ) * ( ( ( ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ) ) / 2 ) ) * ( ( ( ( ( ( ( - 2 ) / 2 ) * ( ( ( ( ( ( 1 / 2 ) * ( ( ( ( ( ( ) ) ) * ( ( ( ( ( ( ) ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) = ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
cluster ( ( - 1 ) (#) ( ( ( ( ( ( ( ( ( - 1 ) (#) ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( - 1 ) / 2 ) * ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
consider X1 being open Subset of X such that t = ( Y | X1 ) /\ ( X | X1 ) and X1 is open and Y is open and Y is open ;
card ( S . ( n + 1 ) ) = card { ( { d } |^ ( n + 1 ) where d is Element of GF ( p ) : d in { d } & d in { p } ;
sqrt ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) ) ) ) ) ) ) ^2 ) = ( ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) * ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) ) / 2 ) ) ) ) ) ) ) ) ) ) ) ^2 ) ) ) ) ) .= ( ( ( ( ( - 1 ) / 2 ) ) ) ^2 ) ) ) .= ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) * ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( - 1 ) / 2 ) ) ) ) ) ) ) ) ) ) ) ) ) )